Proceedings Book of ICRAPAM (2018)

Editor Ekrem SAVAS

Associate Editors Mahpeyker OZTURK, Rahmet SAVAS, Veli CAPALI

ISBN Number: 978-605-68969

1 5Th INTERNATIONAL CONFERENCE ON RECENT ADVANCES IN PURE AND APPLIED MATHEMATICS (ICRAPAM 2018) JULY 23-27 2018 at Karadeniz Technical University Prof. Dr. Osman Turan Convention Center in Trabzon, TURKEY ICRAPAM CONFERENCE PROCEEDING

1 5Th INTERNATIONAL CONFERENCE ON RECENT ADVANCES IN PURE AND APPLIED MATHEMATICS (ICRAPAM 2018) JULY 23-27 2018 at Karadeniz Technical University Prof. Dr. Osman Turan Convention Center in Trabzon, TURKEY ICRAPAM CONFERENCE PROCEEDING

PREFACE

International Conference on Recent Advances in Pure and Applied Mathematics (ICRAPAM 2018) was held in Trabzon, Turkey, at the Faculty of Arts and Science, from July 23 to 27, 2018. It was the fifth edition of such conferences. The chairman of the Organizing Committee of ICRAPAM 2018 was Professor Ekrem Savas, and the Scientic Committee consisted of mathematicians from 16 countries. 250 participants from 30 countries attended the conference and 150 papers have been presented, including 6 plenary lectures and 25 presentations in Poster Session. The conference was devoted to almost all fields of mathematics and variety of its applications. The organizers gratefully acknowledge a partial financial support by Turkish Academy of Science (TUBA) and Karadeniz Technical University. This issue of the procceding contains 38 papers presented at the conference and selected by the usual editorial procedure of scientific committe. We would like to express our gratitude to the authors of articles published in this issue and to the referees for their kind assistance and help in evaluation of contributions. I would like to thank to the following my colleagues and students who helped us at every stage of International Conference on Recent Advances in Pure and Applied Mathematics (ICRAPAM 2018).

Editor: Ekrem SAVAS

Usak University, Usak – Turkey

2 5Th INTERNATIONAL CONFERENCE ON RECENT ADVANCES IN PURE AND APPLIED MATHEMATICS (ICRAPAM 2018) JULY 23-27 2018 at Karadeniz Technical University Prof. Dr. Osman Turan Convention Center in Trabzon, TURKEY ICRAPAM CONFERENCE PROCEEDING

AIM OF THE CONFERENCE

International Conference on Recent Advances in Pure and Applied Mathematics (ICRAPAM 2018) is aimed to bring researchers and professionals to discuss recent developments in both theoretical and applied mathematics and to create a professional knowledge exchange platform between mathematicians. The conference is supported by The Turkish Academiy of Sciences (Türkiye Bilimler Akademisi - TÜBA / Conference's Co- Organizer) and Karadeniz Technical University.

SCOPUS

Prospective authors are invited to submit their one-page abstracts on the related, but not limited, following topics of interest:

Numerical Analysis, Ordinary and Partial Differential Equations, Scientific computing, Boundary Value Problems, Approximation Theory, Sequence Spaces and Summability, Real Analysis, Functional Analysis, Fixed Point Theory, Optimization, Geometry, Computational Geometry, Differential Geometry, Applied Algebra, Combinatorics, Complex Analysis, Flow Dynamics, Control, Mathematical modelling in scientific disciplines, Computing Theory, Numerical and Semi-Numerical Algorithms, Game Theory, Operations Research, Optimization Techniques, Fuzzy sequence spaces, Symbolic Computation, Fractals and Bifurcations, Analysis and design tools, Cryptography, Number Theory and Mathematics Education, Finance Mathematics, Fractional Dynamics, Fuzzy systems and fuzzy control, Dynamical systems and chaos, Biomathematics & modeling. Soft Computing, Cryptology & Security Analysis, Image Processing, etc.

PROCEEDING BOOK

The full texts contained in this proceeding book contain all oral presentations in ICRAPAM 2018 Conference.

3 5Th INTERNATIONAL CONFERENCE ON RECENT ADVANCES IN PURE AND APPLIED MATHEMATICS (ICRAPAM 2018) JULY 23-27 2018 at Karadeniz Technical University Prof. Dr. Osman Turan Convention Center in Trabzon, TURKEY ICRAPAM CONFERENCE PROCEEDING

COMMITTEE

Honorary Committee

Prof. Dr. B. E. Rhoades Prof. Dr. W. Sintunavarat (USA) (Thailand)

Prof. Dr. H. M. Srivastava Prof. Dr. V. Kalantarov (Canada) (Turkey)

Prof. Dr. Ljubisa Kocinac Prof. Dr. Cihan Orhan (Serbia) (Turkey)

Prof. Dr. Sadek Bouroubi Prof. Dr. Metin Basarir (Algeria) (Turkey)

Prof. Dr. Ali M. Akhmedov Prof. Dr. Taras Banakh (Azerbaijan) (Polska)

Prof. Dr. Werner Varnhorn Prof. Dr. Mohammed Al-Gwaiz (Germany ) (Kingdom of Saudi Arabia)

Prof. Dr. Emine Mısırlı Prof. Dr. Gradimir V. Milovanovic (Turkey) (Serbia)

Prof. Dr. Huseyin Cakalli Prof. Dr. Said Melliani (Turkey) (Morrocco)

Prof. Dr. G. Das Prof. Dr. Reza Langari (India) (USA)

Prof. Dr. M. Perestyuk Prof. Dr. Rovshan Aliyev (Ukraine) (Azerbaycan)

Prof. Dr. O. Boichuk Prof. Dr. Ahmet Cevat ACAR (Ukraine) (Turkey)

Prof. Dr. I. Shevchuk Prof. Dr. Ahmet Nuri YURDUSEV (Ukraine ) (Turkey)

Prof. Dr. Anatoliy M. Samoilenko Prof. Dr. Mustafa SOLAK (Ukraine) (Turkey)

Prof. Dr. V. Guliyev Prof. Dr. Fikrettin ŞAHİN (Turkey) (Turkey)

Prof. Dr. M. Abbas Prof. Dr. İzzet ÖZTÜRK (S.Africa) (Turkey)

Prof. Dr. M. Mursaleen Prof. Dr. Ercümend Arvas (India) (Turkey)

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Scientific Committee

Prof. Dr. Huseyin Cakalli Prof. Dr. Husamettin Coskun (Turkey) (Turkey)

Prof. Dr. Jeff Connor Prof. Dr. Maria Zeltser (USA) (Estonia)

Prof. Dr. Lubomira Softova Prof. Dr. Kamalmani Baral (Italy) (Nepal)

Prof. Dr. Reza Langari Prof. Dr. Ants Aasma (USA) (Estonia)

Prof. Dr. Mikail Et Prof. Dr. Ismail N. Cangul (Turkey) (Turkey)

Prof. Dr. S. A. Mohiuddine Prof. Dr. Murat Tosun (S. Arabia) (Turkey)

Prof. Dr. Narendra Kumar Govil Prof. Dr. Yilmaz Simsek (USA) (Turkey)

Prof. Dr. T. A. Chishti Prof. Dr. Harry Miller (India) (Bosnia)

Prof. Dr. Ayhan Serbetci Prof. Dr. Ali Fares (Turkey) (France)

Prof. Dr. Bilal Altay Prof. Dr. Ibrahim Canak (Turkey) (Turkey)

Prof. Dr. Ismail Ekincioglu Prof. Dr. Naim Braha (Turkey) (Kosova)

Prof. Dr. A. Sinan Cevik Prof. Dr. Mustapha Cheggag (Turkey) (Algeria)

Prof. Dr. Leiki Loone Prof. Dr. Fahrettin Abdullayev (Estonia) (Kırgizistan)

Prof. Dr. Akbar B. Aliyev Prof. Dr. Praveen Agarwal (Azerbaijan) (India)

Prof. Dr. Vali M. Gurbanov Prof. Dr. P. D. Srivastava (Azerbaijan) (India)

Prof. Dr. Faqir M. Bhatti Prof. Dr. Mehmet Akbaş (Pakistan) (Turkey)

Prof. Dr. Said Melliani Prof. Dr. Erhan Coşkun (Morocco) (Turkey)

Prof. Dr. Abdalah Rababah Prof. Dr. Funda Karaçal (Jordan) (Turkey)

Prof. Dr. Radouane YAFIA Prof. Dr. Vasile Berinde (Morocco) (Romania)

Prof. Dr. Sudarsan Nanda Prof. Dr. Haskız Coşkun (India) (Turkey)

Prof. Dr. Seyit Temir Prof. Dr. Erdal Karapınar

5 5Th INTERNATIONAL CONFERENCE ON RECENT ADVANCES IN PURE AND APPLIED MATHEMATICS (ICRAPAM 2018) JULY 23-27 2018 at Karadeniz Technical University Prof. Dr. Osman Turan Convention Center in Trabzon, TURKEY ICRAPAM CONFERENCE PROCEEDING

(Turkey) (Turkey)

Prof. Dr. Halit Orhan Prof. Dr. Sultan Yamak (Turkey) (Turkey)

Prof. Dr. Vatan Karakaya Prof. Dr. Selçuk Han Aydın (Turkey) (Turkey)

Prof. Dr. Amir Khosravi Associate Prof. Ayhan Aydın (Iran) (Turkey)

Prof. Dr. Seifedine Kadry Assoc. Prof. Dr. Yasemin Sağıroğlu (Kuwait) (Turkey)

Prof. Dr. Ali M. Akhmedov Assoc. Prof. Dr. Tülay Kesemen (Azerbaijan) (Turkey)

Prof. Dr. Ziyatkan Aliyev Assist. Prof. Dr. Hafize Gök (Azerbaijan) (Turkey)

Prof. Dr. Poom Kumam Assist. Prof. Dr. Sukran Konca (Thailand) (Turkey)

Prof. Dr. Agacik Zafer Dr. Mayssa Alqurashi (Kuwait) (S. Arabia)

Prof. Dr. Tunay Bilgin Dr. Fardous Taoufic (Turkey) (S. Arabia)

Prof. Dr. Gangaram S. Ladde Dr. Dr.K.V.L.N.Acharyulu (USA) (India)

Prof. Dr. Claudio Cuevas Dr. Ammar Edress Mohamed (Brazil) (IRAK)

Prof. Dr. Reza Saadati Prof. Dr. Abdelmejid Bayad (Iran) (France)

Prof. Dr. Salih Aytar Prof. Dr. Amiran Gogatishvili (Turkey) (Czech)

Prof. Dr. Charles Swartz Prof. Dr. Oktay Duman (USA) (Turkey)

Prof. Dr. Yagub A. Sharifov Prof. Dr. Abdelmejid Bayad (Azerbaijan) (France)

Prof. Dr. Niyazi A. Ilyasov Assoc.Prof. Dr. Azhar Hussain (Azerbaijan) (Pakistan)

Prof. Dr. Aref Jeribi (Tunisia)

6 5Th INTERNATIONAL CONFERENCE ON RECENT ADVANCES IN PURE AND APPLIED MATHEMATICS (ICRAPAM 2018) JULY 23-27 2018 at Karadeniz Technical University Prof. Dr. Osman Turan Convention Center in Trabzon, TURKEY ICRAPAM CONFERENCE PROCEEDING

Organizing Committee

Prof. Dr. Ekrem Savas Prof. Dr. Debasis Giri (Turkey) (India)

Prof. Dr. Hamdullah Şevli Prof. Dr. Naim Braha (Turkey) (Kosovo)

Prof. Dr. Necip Şimşek Assoc. Prof. Dr. Rahmet Savas (Turkey) (Turkey)

Prof. Dr. Ömer Pekşen Assoc. Prof. Dr. Erhan Deniz (Turkey) (Turkey)

Prof. Dr. Richard Patterson Assoc. Prof. Dr. Mahpeyker Ozturk (USA) (Turkey)

Prof. Dr. Mehmet Gurdal Assoc. Prof. Dr. Bahadır Özgür Güler (Turkey) (Turkey)

Prof. Dr. Martin Bohner Assoc. Prof. Dr. İdris Ören (USA) (Turkey)

Prof. Dr. Ram Mohapatra Assits. Prof. Dr. Ali Hikmet Değer (USA) (Turkey)

Prof. Dr. Fairouz Tchier Assist. Prof. Dr. Emel Aşıcı (S. Arabia) (Turkey)

Prof. Dr. Mehmet Dik Assist. Prof. Dr. Gökhan Çuvalcıoğlu (USA) (Turkey)

Prof. Dr. Lubomira Softova Assist. Prof. Dr. Arzu Akgun (Italy) (Turkey)

Prof. Dr. Agron Tato Assist. Prof. Dr. Veli Çapalı (Albania) (Turkey)

Dr. Lakhdar Ragoub (S. Arabia)

Local Organizing Committee

Abdurrahman Büyükkaya Sefa Anıl Sezer (Turkey) (Turkey)

Melek Eriş Büyükkaya Tuğba Yıldırım (Turkey) (Turkey)

Uğur Gözütok Ekber Girgin (Turkey) (Turkey)

Ümmügülsün Akbaba Neslihan Kaplan (Turkey) (Turkey)

Fatih Yılmaz Safa Güney (Turkey) (Turkey)

Rabia Savaş Madiha Rashid (Turkey) (Pakistan)

Bouchra BEN AMMA (Morocco)

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INVITED SPEAKERS

Prof. Dr. Abdalah Rababah Jordan University of Science and Technology, Department of Mathematics and Statistics / Jordan

Best Uniform Approximation with 2n+1 Equioscillations

Prof. Dr. Vasile Berinde North University Center at Baia Mare, Technical University of Cluj-Napoca, Department of Mathematics and Computer Sciences / Romania

The Distance Between Two Sets and its Amazing Applications in Science and Technology

Prof. Dr. Amiran Gogatishvili Institute of Mathematics, Czech Academy of Sciences

Weighted norm inequalities for positive operators restricted on the cone of ρ-quasiconcave functions

Prof. Dr. C. M. Khalique North-West University, International Institute for Symmetry Analysis and Mathematical Modelling, Department of Mathematical Sciences / South Africa

Symmetry Methods and Conservation Laws for Differential Equations

8 5Th INTERNATIONAL CONFERENCE ON RECENT ADVANCES IN PURE AND APPLIED MATHEMATICS (ICRAPAM 2018) JULY 23-27 2018 at Karadeniz Technical University Prof. Dr. Osman Turan Convention Center in Trabzon, TURKEY ICRAPAM CONFERENCE PROCEEDING

Prof. Dr. Hüseyin Çakalli Maltepe University / Turkey

Totally boundedness and uniform continuity via quasi Cauchy sequences

Prof. Dr. Erdal Karapınar Atılım University, Department of Mathematics / Turkey

Indispensable remarks on some recent results in metric fixed point theory

Prof. Dr. Robin Harte Trinity College, School of Mathematics / Dublin 2, Ireland

Cofactor Matrix Theory

9 5Th INTERNATIONAL CONFERENCE ON RECENT ADVANCES IN PURE AND APPLIED MATHEMATICS (ICRAPAM 2018) JULY 23-27 2018 at Karadeniz Technical University Prof. Dr. Osman Turan Convention Center in Trabzon, TURKEY ICRAPAM CONFERENCE PROCEEDING

SPECIAL SESSIONS

Special Session Name: Second Modern and Classical Optimization Techniques in Multi Disciplinary Studies

The aim of this symposium is to increase the interaction between scholars, scientists and researchers in both theoretical and practical aspects of optimization as well as encouraging multidisciplinary studies. Using modern and classical techniques in the modelling of real life problems are also aimed. The Symposium covers the following topics, but not limited to: Applications in optimization, Interdisciplinary studies, Convex optimization, Non-Convex optimization, Mathematical modelling, Fuzzy Logic modelling, Data analysis, Artificial Neural Network modelling, other modern and classical modelling and optimization techniques.

Special Session Speaker: Prof. Dr. Ahmet ŞAHİNER, Suleyman Demirel University

Online Sessions Havva KIRGIZ, Selcuk University (k-Generalized Pell Numbers and Its Properties)

Ahmet UĞIR, Selcuk University (Bounds for Atom-bond Connectivity Index)

Amina KALSOOM, International Islamic University Islamabad (Approximating fixed point of Certain Nonlinear mappings in CAT(0) Spaces)

Awais ASIF, International Islamic University Islamabad (Some Fixed Point Theorems in Generalized b-metric Spaces)

Muhammad Imran ASJAD, University of Management and Technology Lahore (Influence of Thermo Diffusion, Heat Absorption and Chemical Reaction Effect on Non- Newtonian Fluid Flow)

10 5Th INTERNATIONAL CONFERENCE ON RECENT ADVANCES IN PURE AND APPLIED MATHEMATICS (ICRAPAM 2018) JULY 23-27 2018 at Karadeniz Technical University Prof. Dr. Osman Turan Convention Center in Trabzon, TURKEY ICRAPAM CONFERENCE PROCEEDING

CONTENT

Preface 2

Committee 4

Invited Speakers 8

Special Sessions 10

Content 11 Application of the Chebyshev polynomials to coefficient estimates of analytic functions Nizami Mustafa, Veysel Nezir, Erdi Akbulut 13 The Arrowhead-Pell Sequences in Finite Groups Yeşim Aküzüm, Taha Doğan, Ömür Deveci 21 Estimates of Faber Polynomial Coefficients for an Unified Class of Bi-Univalent Functions Defined Through the Jackson (p,q)-Derivative Operator 24 Şahsene Altınkaya, Sibel Yalçın On the Coefficient Bounds of Gamma and Beta Starlike Functions of Order Alpha Nizami Mustafa, Veysel Nezir, Mustafa Ateş 39 Infinite Dimensional Subspaces of a Degenerate Lorentz-Marcinkiewicz Space with The Fixed Point Property 42 Veysel Nezir, Nizami Mustafa, Merve Delibaş The Arrowhead-Pell-Random-Type Numbers Modulo m Taha Doğan, Özgür Erdağ, Ömür Deveci 49 On f - Lacunary Statistical Boundedness of Order α Mikail Et, Hüseyin Sönmez 51 Aczél Type Inequalities for Hilbert Space Operators Ulas Yamancı, Mehmet Gürdal 55 Coefficient Bound Estimates for Beta-Starlike Functions of Order Alpha Nizami Mustafa, Veysel Nezir 61 On the fixed point property for a degenerate Lorentz-Marcinkiewicz Space Veysel Nezir, Nizami Mustafa 64 On the Coefficient Bounds of Certain Subclasses of Analytic Functions of Complex Order Nizami Mustafa, Tarkan Öztürk 72 Korovkin Theory For Extraordinary Test Functions via Power Series Method Tuğba Yurdakadim, Emre Taş 77 Spaces of Strongly λ - Invariant Summable Sequences Ekrem Savaş 81 Fixed Point Theorems for a-F-Suzuki Contraction Mapping with Rational Expressions in Branciari b-Metric Spaces 84 Neslihan Kaplan Kuru, Mahpeyker Öztürk

Fixed Point Theorems for Mappings Satisfying B-Contraction Abdurrahman Büyükkaya, Mahpeyker Öztürk Analysis of Neutron Capture Cross Section Veli Çapalı 91

11 5Th INTERNATIONAL CONFERENCE ON RECENT ADVANCES IN PURE AND APPLIED MATHEMATICS (ICRAPAM 2018) JULY 23-27 2018 at Karadeniz Technical University Prof. Dr. Osman Turan Convention Center in Trabzon, TURKEY ICRAPAM CONFERENCE PROCEEDING

Some Fixed Point Theorems in Generalized b-metric Spaces Awais Asif, Ekrem Savaş, Waqas Ahmad, Muhammad Arshad 93 Influence of Thermo Diffusion, Heat Absorption and Chemical Reaction Effect on Non-Newtonian Fluid Flow 99 Hina Khursheed, Muhammad Imran Asjad, Rabia Naz

On The Fekete–Szegö Problem For a New Class of m-Fold Symmetric Bi-Univalent Functions Satisfying Subordination Condition Given by Symmetric Q-Derivative Operator 122 Arzu Akgül

The full texts contained in this proceeding book contain all oral presentations in ICRAPAM 2018 Conference.

12 5Th INTERNATIONAL CONFERENCE ON RECENT ADVANCES IN PURE AND APPLIED MATHEMATICS (ICRAPAM 2018) JULY 23-27 2018 at Karadeniz Technical University Prof. Dr. Osman Turan Convention Center in Trabzon, TURKEY ICRAPAM CONFERENCE PROCEEDING

Application of the Chebyshev Polynomials to Coefficient Estimates of Analytic Functions

Nizami Mustafa1, Veysel Nezir1, Erdi Akbulut∗1 1Kafkas University, Faculty of Science and Letters, Department of Mathematics, Kars, Turkey

Keywords Abstract: In this paper, making use of the Chebyshev polynomials, we introduce and Analytic function, investigate new subclasses of the analytic functions in the open unit disk in the complex Coefficient bound, plane. Here, we obtain upper bound estimates for the initial second coefficients of the Chebyshev polynomials, functions belonging to these classes.

1. Introduction and preliminaries related to specific orthogonal polynomials of Chebyshev family, contain essentially results of Chebyshev polyno- Let A denote the class of all complex valued functions f (z) mials of first and second kinds Tn (x) and Un (x), and their normalized by f (0) = 0 = f 0(0) − 1 and given by numerous uses in different applications (see [1, 5]). The well-known kinds of the Chebyshev polynomials are ∞ 2 3 n the first and second kinds. In this paper, we will use sec- f (z) = z + a2z + a3z + ··· = z + ∑ anz , an ∈ C (1) n=2 ond kind Chebyshev polynomials to investigation of the coefficient estimates of the analytic functions. which are analytic in the open unit disk U = It is well-known that, in the case of real variable x on {z ∈ C : |z| < 1} in the complex plane . (−1,1), the second kinds of the Chebyshev polynomials Furthermore, let S be the class of all functions in A which are defined by are univalent in U. Some of the important and well- investigated subclasses of S are classes S∗ and C given sin[(n + 1)arccosx] sin[(n + 1)arccosx] ∗ Un (x) = = √ . below (see also [2, 3, 7] such that S is the class of starlike sin(arccosx) 1 − x2 functions and C is the class of convex functions We consider the function G(t,z) = 1/1 − 2tz + z2
, t ∈  z f 0 (z)  S∗ = f ∈ S : ℜ > 0,z ∈ U (1/2,1), z ∈ U. f (z) It is well-known that if t = cosα, α ∈ (0,π/3), then and G(t,z) = 1 + 2cosαz + 3cos2 α − sin2 α
z2   z f 00 (z)  +8cos3 α − 4cosα
z3 + ···, z ∈ U. C = f ∈ S : ℜ 1 + > 0,z ∈ U . f 0 (z) That is, An analytic function f is subordinate to an analytic func- 2 3 tion φ and written f (z) ≺ φ (z), provided that there is an an- G(t,z) = 1 +U1 (t)z +U2 (t)z +U3 (t)z + ···(2) alytic function (that is, Schwarz function) ω defined on U , t ∈ (1/2,1),z ∈ U, with ω (0) = 0 and |ω (z)| < 1 satisfying f (z) = φ (ω (z)). Ma and Minda [4] unified various subclasses of star- where Un (t), n ∈ N are the second kind Chebyshev poly- nomials. like and convex functions for which either of the quan- 0 tity z f 0 (z)/ f (z) or 1 + z f 00 (z)/ f 0 (z) is subordinate to It is clear that G(t,0)= 0 and Gz (t,0) > 0 a more general function. For this purpose, they consid- From the definition of the second kind Chebyshev poly- ered an analytic function φ with positive real part in U, nomials, we easily obtain that U0 (t) = 1, U1 (t) = 2t, 2 φ (0) = 1, φ 0 (0) > 0 and φ maps U onto a region starlike U2 (t) = 4t − 1. Also, it is well-known that with respect to 1 and symmetric with respect to the real axis. U (t) = 2tU (t) −U (t) (3) The class of Ma-Minda starlike and Ma-Minda convex n+1 n n−1 functions consists of functions f ∈ A satisfying the subordi- 0 00 0 for all n ∈ N − {1}. nation z f (z)/ f (z) ≺ φ (z) and 1 + z f (z)/ f (z) ≺ φ (z), Inspired by the aforementioned works, making use of the respectively. Chebyshev polynomials, we define a subclass of univalent Chebyshev polynomials, which are used by us in this paper, functions as follows. play a considerable act in numerical analysis and math- ematical physics. It is well-known that the Chebyshev Definition 1.1. A function f ∈ S given by (1) is said to be polynomials are four kinds. The most of research articles in the class M (G;β,t), β ≥ 0,t ∈ (1/2,1), where G is an

∗ Corresponding author: [email protected] 13 / analytic function given by (2), if the following condition 2. Upper bound estimates for the coefficients is satisfied In this section, we prove the following theorem on up- z f 0 (z)1−β  z f 00 (z)β per bound estimates for the coefficients of the functions 1 + ≺ G(t,z), z ∈ U. f (z) f 0 (z) belonging to the class M (G;β,t). Theorem 2.1. Let the function f (z) given by (1) be in the Remark 1.2. Taking β = 0, we have the function class ∗ class M (G;β,t), β ∈ [0,1],t ∈ (1/2,1). Then, M (G;0,t) ≡ S (G;t), t ∈ (1/2,1); that is, 2 2 2 2 2t 2(β +11β+4)t +2(1+β) t−(1+β) |a2| ≤ 1+β and |a3| ≤ 2 . z f 0 (z) 2(1+β) (1+2β) f ∈ S∗ (G;t) ⇔ ≺ G(t,z), z ∈ U. f (z) Proof. Let f ∈ M (G;β,t), β ∈ [0,1],t ∈ (1/2,1). Then, according to Definition 1.1, there is an analytic function ω : Remark 1.3. Taking β = 1, we have the function class U → U with ω (0) = 0, |ω (z)| < 1 satisfying the following M (G;1,t) ≡ C (G;t), t ∈ (1/2,1); that is, condition

z f 00 (z) z f 0 (z)1−β  z f 00 (z)β f ∈ C (G;t) ⇔ 1 + ≺ G(t,z), z ∈ U. 1 + = G(t,ω (z)), z ∈ U. (4) f 0 (z) f (z) f 0 (z)

Remark 1.4. As you can see that the class M (G;β,t), de- It follows that fined by Definition 1.1, is a generalization of the Ma-Minda 1 + (1 + β)a2z starlike and Ma-Minda convex functions such that in the  2  special case for β = 0 and β = 1 the classes M (G;0,t) and β − 7β − 2 2 2 + 2(1 + 2β)a3 + a2 z + ··· M (G;1,t) are Ma-Minda starlike and Ma-Minda convex 2 functions, respectively, when there function φ (z) is G(t,z) = G(t,ω (z)). (5) for fixed value t ∈ (1/2,1). Let the function p ∈ P be define as follows In this paper, making use of the Chebyshev polynomials, ∞ we introduce and investigate new subclasses M (G;β,t), 1 + ω(z) 2 n ∗ p(z) := = 1 + p1z + p2z + ··· = 1 + ∑ pnz . β ≥ 0,t ∈ (1/2,1), S (G;t), t ∈ (1/2,1) and C (G;t), t ∈ 1 − ω(z) n=1 (1/2,1) of the analytic functions in the open unit disk in the complex plane. Here, we will obtain upper bound From this, we have estimates for the initial second coefficients of the functions p(z) − 1 belonging to these classes. ω(z) : = p(z) + 1 To prove our main results, we shall require the following   2   p1 2 well known lemma. 1 p1z + p2 − 2 z =   3  . (6) 2 p1 3 Lemma 1.5. [2] Let P be the class of all analytic functions + p3 − p1 p2 + 4 z + ··· p(z) of the form Taking z ≡ ω(z) in (2), we get ∞ 2 n p(z) = 1 + p1z + p2z + ··· = 1 + pnz U (t) ∑ G(t,ω(z)) = 1 + 1 p z n=1 2 1 U (t)  p2  U (t)  satisfying ℜ(p(z)) > 0, z ∈ U and p(0) = 1. Then, |pn| ≤ 1 1 2 2 2 + p2 − + p1 z + ···. 2, for every n = 1,2,3,... . These inequalities are sharp 2 2 4 for each n. (7) Moreover, Thus, by substituting the expression G(t, (z)) in (5), we 2p = p2 + 4 − p2
x, ω 2 1 1 can easily write

3 2
2
2 h β 2−7β−2 2i 2 4p3 = p1 + 2 4 − p1 p1x − 4 − p1 p1x 1 + (1 + β)a2z + 2(1 + 2β)a3 + 2 a2 z + ··· = h  2  i 2
 2 U1(t) U1(t) p1 U2(t) 2 2 +2 4 − p1 1 − |x| z 1 + 2 p1z + 2 p2 − 2 + 4 p1 z + ··· . (8) for some x, z with |x| ≤ 1, |z| ≤ 1. Comparing the coefficients of the like power of z in both sides of (8), we have Lemma 1.6. [6] Let p ∈ P. Then |pn| ≤ 2 for all n = 1,2,3,... and U1(t) (1 + β)a = p , (9) 2 2 1

λ 2 2 p2 − p1 ≤ max{2,2|λ − 1|} β − 7β − 2 2 2 2(1 + 2β)a3 + a2  2 2, if λ ∈ [0,2],  2  = U1(t) p1 U2(t) 2 2|λ − 1|, elsewhere. = p2 − + p . (10) 2 2 4 1 14 /

It follows that the upper bound estimate for |a4| can be found. So us- U (t) a = 1 p , (11) ing this work we can find upper bound estimate for the 2 2(1 + β) 1 2 second Hankel determinant a2a4 − a3 for the functions belonging in the class M (G;β,t). 2 2
2 2(1 + β) U2 (t) + 2 + 7β − β U1 (t) 2 a3 = p1 16(1 + β)2 (1 + 2β) References  2  U1(t) p1 [1] Doha, E. H. 1994. The first and second kind Cheby- + p2 − . (12) 4(1 + 2β) 2 shev coefficients of the moments of the general order derivative of an infinitely differentiable function. In- From (11), since |p1| ≤ 2, inequality for |a2| is clear. Then, 2 2 tern. J. Comput. Math., 51, 21-35. since the coefficients of p1 and p2 − p1/2 are positive each β ∈ [0,1] and t ∈ (1/2,1), using Lemma 1.6 to equality [2] Duren, P. L. 1983. Univalent Functions. Grundlehren (12), we obtain der Mathematischen Wissenschaften. Springer-Verlag, New York, 259p. 2β 2 + 11β + 4
t2 + 2(1 + β)2 t − (1 + β)2 |a3| ≤ . [3] Goodman, A. W. 1983. Univalent Functions. Volume 2(1 + β)2 (1 + 2β) I, Polygonal. Mariner Comp., Washington. With this, the proof of Theorem 2.1 is completed. [4] Ma, W. C., Minda, D. 1994. A unified treatment of some special classes of functions. pp. 157-169. Li, Z., Setting = 0 and = 1 in Theorem 2.1, we can readily β β Ren, F., Yang, L., Zhang, S., ed. 1994. Proceedings of deduce the following results, respectively. the Conference on Complex Analysis, Int. Press. Corollary 2.2. Let the function f (z) given by (1) be in the ∗ [5] Lewin, M. 1967. On a coefficient problem for bi- function class M (G;0,t) ≡ S (G;t), t ∈ (1/2,1). Then, univalent functions. Proc. Amer. Math. Soc., 18, 63- 8t2+2t−1 |a2| ≤ 2t and |a3| ≤ 2 . 68. Corollary 2.3. Let the function f (z) given by (1) be in the [6] Pommerenke, C. H. 1975. Univalent Functions. Van- function class ℵ(G;1,t) ≡ C (G;t), t ∈ (1/2,1). Then, denhoeck and Rupercht, Göttingen. 2 |a | ≤ t and |a | ≤ 8t +2t−1 . 2 3 6 [7] Srivastava, H. M., Owa, S. 1992. Current Topics in An- 2 alytic Function Theory. World Scientific, Singapore. Remark 2.4. Using this work, one can examine a3 − µa2 the Fekete - Szegö problem for the coefficients of the func- tion class M (G;β,t). Moreover, using the same technique,

15 5Th INTERNATIONAL CONFERENCE ON RECENT ADVANCES IN PURE AND APPLIED MATHEMATICS (ICRAPAM 2018) JULY 23-27 2018 at Karadeniz Technical University Prof. Dr. Osman Turan Convention Center in Trabzon, TURKEY ICRAPAM CONFERENCE PROCEEDING

The Arrowhead-Pell Sequences in Finite Groups

Ye¸simAküzüm∗1, Taha Dogan˘ 2, Ömür Deveci3 1,2,3Kafkas University,Science and Letters Faculty, Department of Mathematics, Turkey

Keywords Abstract: In [1], Akuzum and Deveci defined the arrowhead-Pell sequences and obtain The arrowhead-Pell miscellaneous properties of these sequences. In this work, we redefine the arrowhead-Pell Sequence,Period,Group. sequences by means of the elements of groups which is called arrowhead-Pell-orbit. Then we examine the arrowhead-Pell-orbit of the finite groups in detail. Finally, we obtain the periods of the arrowhead-Pell orbit of the semidihedral group SD2m as applications of the results obtained.

1. Introduction

k+1 Akuzum and Deveci [1] defined the arrowhead-Pell se- It is important to note that detMk = (−1) . quence as shown: The linear recurrence sequences in groups were firstly ak+1 (n + k + 1) = ak+1 (n + k)−2ak+1 (n + k − 1)−ak+1 (n + k − 2)−···−ak+1 (n) studied by Wall [10] who calculated the periods of the Fibonacci sequences in cyclic groups. As a natural gen- for n ≥ 1 and k ≥ 2, with initial conditions a (1) = ··· = k+1 eralization of the problem, Wilcox [11] investigated the a (k) = 0, a (k + 1) = 1. k+1 k+1 Fibonacci lengths to abelian groups. Recently, many au- In [1], the arrowhead-Pell matrix had been given as: thors studied the linear recurrence sequences in groups   1 −2 −1 −1 ··· −1 −1 and fields; see for example, [3–9]. In this work, we study  1 0 0 ··· 0 0 0  the arrowhead-Pell sequence in groups and then we de-    0 1 0 0 ··· 0 0  fine the arrowhead-Pell-orbit of 2-generator groups. Also,   M =  0 0 1 0 0 ··· 0  we obtain the periods of the arrowhead-Pell orbit of the k  . . .   ......  semidihedral group SD2m as applications of the results  ......    obtained.  0 0 ··· 0 1 0 0  0 0 0 ··· 0 1 0 (k+1)×(k+1) 2. The Arrowhead-Pell Sequences in Finite Groups Also, by an inductive argument they obtained that (i). For k = 2, Let G be a finite p-generator group and let X be the subset of G × G × ··· × G such that (x1,x2,...,xp) ∈ X if and only  aα+3 aα+4 − aα+3 −aα+2  | {z } 3 3 3 3 p times (M )α = aα+2 aα+3 − aα+2 −aα+1 2  3 3 3 3  G x ,x ,...,x (x ,x ,...,x ) α+1 α+2 α+1 α if is generated by 1 2 p. 1 2 p is said to a3 a3 − a3 −a3 be a generating p-tuple for G. (ii). For k ≥ 3, Definition 2.1. Let G be a 2-generator group and let (x,y)  aα+k+1 aα+k+2 − aα+k+1 −aα+k  k+1 k+1 k+1 k+1 be a generating pair of G. Then we define the arrowhead- aα+k aα+k+1 − aα+k −aα+k−1  k+1 k+1 k+1 k+1  Pell orbit AR (G : x,y) as follows:  +k−1 +k +k−1 +k−2  k+1 α  aα aα − aα M∗ −aα  (Mk) =  k+1 k+1 k+1 k k+1 ,  . . .   −1  −1  −2    . . .  x (n + k + 1) = x (n) x (n + 1) ...x (n + k − 2)
−1 x (n + k − 1) x (n + k)  . . .  k+1 k+1 k+1 k1 k+1 k+1 α+1 α+2 α+1 α ak+1 ak+1 − ak+1 −ak+1 ∗ for n ≥ 1, with initial constants xk+1 (1) = x,xk+1 (2) = where Mk is a (k + 1) × (k − 2) matrix as follows: y,xk+1 (3) = e,...,xk+1 (k + 1) = e.  −aα+2 − aα+3 − ··· − aα+k −aα+3 − aα+4 − ··· − aα+k k+1 k+1 k+1 k+1 k+1 k+1  α+1 α+2 α+k−1 α+2 α+3 α+k−1  −a − a − ··· − a −a − a − ··· − a  k+1 k+1 k+1 k+1 k+1 k+1 Theorem 2.2. A the arrowhead-Pell orbit ARk+1 (G : x,y)  α α+1 α+k−2 α+1 α+2 α+k−2 ∗  −a − a − ··· − a −a − a − ··· − a M =  k+1 k+1 k+1 k+1 k+1 k+1 k  of a finite group G is simply periodic.  . .  . .   . . −aα+2−k − aα+3−k − ··· − aα −aα+3−k − aα+4−k − ··· − aα k+1 k+1 k+1 k+1 k+1 k+1 Proof. Let p be the order of the group G, then it is clear k+1 ··· −aα+k−1 − aα+k  p k + G. k+1 k+1 that there are distinct 1-tuples of elements of α+k−2 α+k−1  ··· −a − a  k k+1 k+1  Then, it is easy to see that at least one of the + 1-tuples α+k−3 α+k−2  ··· −a − a  k+1 k+1 . appears twice in the arrowhead-Pell orbit. Because of  .  .  the repeating, the arrowhead-Pell orbit of the group G is  .  ··· −aα−1 − aα periodic. Since the orbit ARk+ (G : x,y) is periodic, there k+1 k+1 1

∗ Corresponding author: [email protected] 16 / exist natural numbers g and h with g ≡ h(modk + 1), such Using the relations of the SD2m , this sequence becomes: that x3 (1) = x,x3 (2) = y,x3 (3) = e,..., xk+1 (g + 1) = xk+1 (h + 1),xk+1 (g + 2) = xk+1 (h + 2),...,xk+1 (g + k + 1) = xk+1 (h + k + 1). 8i+1 −4i By the definition of the arrowhead-Pell orbit x3 (14i + 1) = x ,x3 (14i + 2) = yx ,x3 (14i + 3) = e,.... AR (G : x,y), we can easily derive m− k+1 So we need the smallest i ∈ N such that 4i = 2 1k (k ∈ N).    −1  −1  −2   −1 m−3 xk+1 (n) = xk+1 (n + 1) ... xk+1 (n + k − 2) xk+1 (n + k − 1) xk+1 (n + k) xk+1 (n + k + 1) . If we choose i = 2 , we obtain

m−2
m−2
m−2
Thus, xk+1 (g) = xk+1 (h), and it then follows that x3 2 .7 + 1 = x,x3 2 .7 + 2 = y,x3 2 .7 + 3 = e.

xk+1 (g − h + 1) = xk+1 (1),xk+1 (g − h + 2) = xk+1 (2),...,xk+1 (g − h + k + 1) = xk+1 (k + 1). Since the elements succeeding So the proof is complete. m−2
m−2
m−2
x3 2 .7 + 1 ,x3 2 .7 + 2 ,x3 2 .7 + 3 de- We denote the length of the period of the orbit pend on x and y for their values, the cycle begins ARk+1 (G : x,y) by LARk+1 (G : x,y). again with the 2m−2.7
nd element. So we get It is well-known that the semidihedral group SD2m , (m ≥ 4) m−2 LAR3 (SD2m : x,y) = 2 · 7. is defined by the presentation Example 2.2. Since AR4 (SD 5 : x,y) is D 2m−1 2 −1+2m−2 E 2 SD2m = x,y| x = y = e, yxy = x . x4 (1) = x,x4 (2) = y,x4 (3) = e,x4 (4) = e,..., 13 −4 8 12 In [2], Akuzum et al. denoted the period of the sequence x4 (13) = x ,x4 (14) = yx ,x4 (15) = x ,x4 (16) = x ,..., {ak+1 (n)} when read modulo m by Lk+1 (m). 9 8 8 x4 (25) = x ,x4 (26) = yx ,x4 (27) = e,x4 (28) = x ,..., We now address the length of the period of the arrowhead- 5 4 8 4 x4 (37) = x ,x4 (38) = yx ,x4 (39) = x ,x4 (40) = x ,..., Pell orbit in the semidihedral group SD2m . x4 (49) = x,x4 (50) = y,x4 (51) = e,x4 (52) = e,..., Theorem 2.3. The length of the period of the arrowhead- m−2 Pell orbit in the semidihedral group SD2m is 2 · the length of the period of the arrowhead-Pell orbit Lk+1 (2). LAR4 (SD25 : x,y) is 48.

Proof. We consider the length of the period of the Acknowledgment arrowhead-Pell orbit in the semidihedral group by the aid of the period Lk+1 (2). The orbit ARk+1 (SD2m : x,y) is This Project was supported by the Commission for the Scientific Research Projects of Kafkas University. The x (1) = x,x (2) = y,x (3) = e,...,x (k + 1) = e. k+1 k+1 k+1 k+1 Project number. 2017-FM-21. Thus, we also have

λ14i+1 λ24i xk+1 (2Lk+1 (2)i + 1) = x ,xk+1 (2Lk+1 (2)i + 2) = yx ,

λ34i λk+14i xk+1 (2Lk+1 (2)i + 3) = x ,...,xk+1 (2Lk+1 (2)i + k + 1) = x , References where λ ,λ ,...,λ are positive integers such that 1 2 k+1 [1] Akuzum, Y., Deveci, O. The Arrowhead-Pell Se- gcd(λ1,λ2,...,λk+1) = 1. So we need the smallest i ∈ N m−1 m−3 quences. Ars Combinatoria, in press. such that 4i = 2 k (k ∈ N). If we choose i = 2 , we obtain [2] Akuzum, Y., Deveci, O. 2018. The Arrowhead-Pell m−2
m−2
xk+1 2 .Lk+1 (2) + 1 = x,xk+1 2 .Lk+1 (2) + 2 = y, Sequences Modulo m. AIP Conference Proceedings, m−2
m−2
xk+1 2 .Lk+1 (2) + 3 = e,...,xk+1 2 .Lk+1 (2) + k + 1 = e. 1(1991), 020032. Since the elements succeeding [3] Aydın, H., Aydın, R. 1998. General Fibonacci Se- m−2
m−2
xk+1 2 .Lk+1 (2) + 1 , xk+1 2 .Lk+1 (2) + 2 , quences in Finite Groups. Fibonacci Quarterly, 36(3), m−2
m−2
xk+1 2 .Lk+1 (2) + 3 ,...,xk+1 2 .Lk+1 (2) + k + 1 216-221. depend on x and y for their values, the cycle begins again [4] Campbell, C. M., Doostie, H., Robertson, E. F. 1990. with the 2m−2.L (2)
nd element. Thus it is verified k+1 Fibonacci Length of Generating Pairs in Groups in that m−2 Applications of Fibonacci Numbers. Vol. 3 Eds. G. E. LAR (SD m : x,y) = 2 · L (2). k+1 2 k+1 Bergum et al. Kluwer Academic Publishers, 27-35. [5] Deveci, O. 2015. The Pell-Padovan Sequences and Example 2.1. For k = 2, we consider the length of the The Jacobsthal-Padovan Sequences in Finite Groups. period of the arrowhead-Pell orbit in the semidihedral Utilitas Mathematica, 98, 257-270. group SD2m . It is easy to see that L3 (2) = 7. We have the sequence [6] Deveci, O., Karaduman, E. 2015. The Pell Sequences in Finite Groups. Utilitas Mathematica, 96, 263-276. x3 (1) = x,x3 (2) = y,x3 (3) = e, −1 −1 −3 [7] Doostie, H., Campbell, C. M. 2000. Fibonacci Length x3 (4) = x ,x3 (5) = yx ,x3 (6) = yx , −4 2m+1 −6 of Automorphism Groups Involving Tribonacci Num- x3 (7) = yx ,x3 (8) = x ,x3 (9) = yx , bers. Vietnam Journal of Mathematics, 28, 57-65. m−2 m−2 x (10) = x−8,x (11) = x5.2 −5,x (12) = yx−5.2 +5, 3 3 3 [8] Karaduman, E., Aydın, H. 2003. On Fibonacci Se- 5.2m−2+19 −2 9 x3 (13) = yx ,x3 (14) = yx ,x3 (15) = x , quences in Nilpotent Groups. Mathematica Balkanica, −4 −9 x3 (16) = yx ,x3 (17) = e,x3 (18) = x ,.... 17(3-4), 207-214. 17 /

[9] Knox, S. W. 1992. Fibonacci Sequences in Finite American Mathematical Monthly, 67(6), 525-532. Groups. Fibonacci Quarterly, 30, 116-120. [11] Wilcox, H. J. 1986. Fibonacci Sequences of Period n [10] Wall, D. D. 1960. Fibonacci Series Modulo m. The in Groups. Fibonacci Quarterly, 24(4), 356-361.

18 5Th INTERNATIONAL CONFERENCE ON RECENT ADVANCES IN PURE AND APPLIED MATHEMATICS (ICRAPAM 2018) JULY 23-27 2018 at Karadeniz Technical University Prof. Dr. Osman Turan Convention Center in Trabzon, TURKEY ICRAPAM CONFERENCE PROCEEDING

Estimates of Faber polynomial coefficients for an unified class of Bi-univalent functions defined through the Jackson (p,q)-derivative operator

¸SahseneALTINKAYA∗1, Sibel YALÇIN2

1Bursa Uludag University, Department of Mathematics, Turkey 2Bursa Uludag University, Department of Mathematics, Turkey

Keywords Abstract: In this conference paper, a new subclass of bi-univalent functions involving Bi-univalent function, the Jackson (p,q)-derivative is defined. Afterwards, by using the Faber polynomial Faber polynomial expansion, expansions, we determine upper bounds for |an| (n > 3) coefficients of functions belonging (p,q)-derivative operator to the class.

1. Introduction Definition 1.3. For f and g analytic in D, we say that f is subordinate to g, written f ≺ g, if there exists a Schwarz Let C be the complex plane and D = {z ∈ C : |z| < 1} be function ∞ open unit disc in C. A function f is analytic at a point n ϖ(z) = cnz z0 ∈ D if it is differentiable in some neighbourhood of z0 ∑ n=1 and it is analytic in a domain D if it is analytic at all points in domain D. with |ϖ(z)| < 1 in D, such that f (z) = g(ϖ (z)). For the Schwarz function ϖ (z) we note that |cn| < 1 (see [7]). Definition 1.1. An analytic function f is called univalent Many researchers (see [3], [5], [6], [11], [12], [13]) in a domain D if it does not take the same value twice, so have recently introduced and investigated several inter- that for z1,z2 ∈ D, esting subclasses of bi-univalent function class σ and they have found non-sharp estimates on the first two Taylor- f (z1) 6= f (z2) for z1 6= z2. Maclaurin coefficients |a2| and |a3|. However, there are Denote by A the class of functions analytic in D, satisfying only a few works determining the general coefficient the condition bounds |an| for the analytic bi-univalent functions in the literature (see [4], [9]). The coefficient estimates problem 0 f (0) = f (0) − 1 = 0. for each of Then each function f in has the Taylor expansion A |an| (n ∈ N\{1,2};N = {1,2,3,...}) ∞ n is still an open problem. f (z) = z + ∑ anz . (1) n=2 In the field of Geometric Function Theory, various sub- classes of analytic function class A have been studied We denote by S the subclass of A consisting of functions from different viewpoints. The q-calculus as well as the the form (1) which are also univalent in . D fractional q-calculus provide important tools that have been It is well known that every function f ∈ S has an inverse used in order to investigate various subclasses of . His- f −1, satisfying A torically speaking, a firm footing of the usage of the q- −1 f ( f (z)) = z (z ∈ D) calculus in the context of Geometric Function Theory was actually provided and the basic (or q-) hypergeometric and functions were first used in Geometric Function Theory in  1 a book chapter by Srivastava (see, for details, [14]). In fact, f f −1 (w)
= w |w| < r ( f ),r ( f ) ≥ , 0 0 4 the theory of univalent functions can be described by using the theory of the q-calculus. Moreover, in recent years, where such q-calculus operators as the fractional q-integral and −1 2 2
3 fractional q-derivative operators were used to construct g(w) = f (w) = w − a2w + 2a2 − a3 w (2) several subclasses of analytic functions. 3
4 For the convenience, we provide some basic definitions − 5a − 5a2a3 + a4 w + ··· . 2 and concept details of q-calculus which are used in this pa- Definition 1.2. A function f ∈ A is said to be bi-univalent per. We suppose throughout the paper that 0 < q < p ≤ 1. − in D if both f and f 1 are univalent in D. Let σ be the We shall follow the notation and terminology in [8]. class of bi-univalent functions defined in the unit disc D.

∗ Corresponding author: [email protected] 19 /

p We recall the definitions of fractional q-calculus operators In general, for any p ∈ N and n ≥ 2, an expansion of Kn−1 of complex valued function f . is as, [1], p(p − 1) p! p! K p = pa + E2 + E3 + ... + En−1, Definition 1.4. (Chakrabarti and Jagannathan [10]) The n−1 n 2 n−1 (p − 3)!3! n−1 (p − n + 1)!(n − 1)! n−1 Jackson (p,q)-derivative of the function f is defined as (6) p p where En−1 = En−1 (a2,a3,...) and  f (pz)− f (qz)  ; z 6= 0 ∞ µ1 µn−1  (p−q)z m m!(a2) ...(an) (Dp,q f )(z) = . (3) En−1 (a2,...,an) = ∑ , for m ≤ n µ1!...µn−1!  f 0(0); z = 0 n=2 while a1 = 1, and the sum is taken over all nonnegative From (3), we deduce that integers µ1,..., µn satisfying

∞ µ1 + µ2 + ... + µn−1 = m, n−1 (Dp,q f )(z) = 1 + [n] anz , ∑ p,q µ1 + 2µ2 + ... + (n − 1) µn−1 = n − 1. n=2 n−1 n−1 Evidently, E (a2,...,an) = a , while a1 = 1, and the where the symbol [n] denotes the so-called (p,q)- n−1 2 p,q sum is taken over all nonnegative integers µ1,..., µn satis- bracket or twin-basic number fying pn − qn [n]p,q = . p − q µ1 + µ2 + ... + µn = m,

n n−1 µ1 + 2µ2 + ... + nµn = n. It happens clearly that Dp,qz = [n]p,q z . From (2) and n n (3), we also deduce that It is clear that En (a1,a2,...,an) = a1. The first and the last polynomials are: g(pw) − g(qw) (Dp,qg)(w) = 1 n n (p − q)w En = an, En = a1. Next, let Φ be an analytic function with positive real part 2
2 = 1 − [2] a w + [3] 2a − a w 0 p,q 2 p,q 2 3 in D with Φ(0) = 1 and Φ (0) > 0. Also, let Φ(D) be starlike with respect to 1 and symmetric with respect to the 3
3 real axis. Thus, Φ has the Taylor series expansion −[4]p,q 5a2 − 5a2a3 + a4 w + ··· Φ(z) = 1 + Φ z + Φ z2 + Φ z3 + ··· (Φ > 0). (7) where the function g is given by (2). 1 2 3 1 On the other hand, by using the Faber polynomial expan- Suppose that u(z) and v(w) are analytic in the unit disc D sion of functions f ∈ A of the form (1), the coefficients with u(0) = v(0) = 0, |u(z)| < 1, |v(w)| < 1, and suppose of its inverse map g = f −1 may be expressed as follows that ∞ n (see [1]): u(z) = u1z + ∑n=2 unz (|z| < 1), (8) ∞ ∞ n −1 1 −n n v(w) = v1w + ∑n=2 vnw (|w| < 1) g(w) = f (w) = w + ∑ Kn−1 (a2,a3,...)w , n=2 n It is well known that where 2 2 |u1| ≤ 1, |u2| ≤ 1−|u1| , |v1| ≤ 1, |v2| ≤ 1−|v1| . (9) (−n)! (−n)! K−n = an−1 + an−3a n−1 (−2n + 1)!(n − 1)! 2 [2(−n + 1)]!(n − 3)! 2 3 Next, the equations (7) and (8) lead to (−n)! + an−4a (−2n + 3)!(n − 4)! 2 4 2 2 (−n)! Φ(u(z)) = 1 + Φ1u(z) + Φ2(u(z)) z + ··· + an−5 a + (−n + 2)a2 (4) [2(−n + 2)]!(n − 5)! 2 5 3

(−n)! n−6 + a [a6 + (−2n + 5)a3a4] 2
2 (−2n + 5)!(n − 6)! 2 = 1 + Φ1u1z + Φ1u2 + Φ2u1 z + ···(10) n− j +∑a2 Vj, j≥7 ∞ n k n = 1 + ∑ ∑ΦkEn (u1,u2,...,un)z , n=1k=1 such that Vj with 7 ≤ j ≤ n is a homogeneous polynomial in the variables a2,a3,...,an [2]. In particular, the first and three terms of K−n are given below: n−1 2 2 Φ(v(w)) = 1 + Φ1v(w) + Φ2(v(w)) z + ···

1 −2 K = −a2, 2
2 2 1 = 1 + Φ1v1w + Φ1v2 + Φ2v1 w + ···(11) 1 −3 2 K2 = 2a2 − a3, (5) ∞ n 3 k n 1 = 1 + ∑ ∑ΦkEn (v1,v2,...,vn)w . K−4 = −5a3 − 5a a + a
. n=1k=1 4 3 2 2 3 4 20 /

2. Main Result 3. Initial coefficient estimates

Definition 2.1. A function f ∈ σ is said to be in the class In this section we obtain initial coefficient estimates for p,q p,q functions belonging to the class Rσ,γ (Φ). Rσ,γ (Φ)(γ ∈ C\{0}, 0 < q < p ≤ 1, z,w ∈ D) Theorem 3.1. Let f ∈ Rp,q (Φ). Then if the following subordination relationships are satisfied: σ,γ  √   1   |γ|Φ1 Φ1  |a2| ≤ min K(p,q), q 1 + ((Dp,q f )(z) − 1) ≺ Φ(z) (12)  (Φ2 − Φ )(p2 + q2) + (Φ2 − 2Φ )pq + Φ (p2 + 2pq + q2)  γ 1γ 2 1γ 2 1 and and  1  1 + ((Dp qg)(w) − 1) ≺ Φ(w) (13) |a3| ≤ min{L(p,q),M(p,q)}, γ , where where the function g is given by (2).  q Φ1|γ| Our first main result is given by Theorem 2.2 below.  2 2 ; |Φ2| ≤ Φ1  p +pq+q K(p,q) = , Let f ∈ p,q ( ). If a = , ≤ m ≤ n− , Theorem 2.2. Rσ,γ Φ m 0 2 1  q |Φ ||γ|  2 | | > then  p2+pq+q2 ; Φ2 Φ1

Φ1 |γ| |a | ≤ , n ≥ 3.  Φ1|γ| n 2 2 ; |Φ2| ≤ Φ1 [n]p,q  p +pq+q L(p,q) = , Proof. Let f be given by (1). We have  |Φ2||γ| | | > p2+pq+q2 ; Φ2 Φ1 ∞ n−1 and (Dp,q f )(z) − 1 = ∑ [n]p,q anz (14) n=2 M(p,q) = and, for its inverse map g = f −1, it is seen that  Φ |γ| 2 2  1 ; Φ ≤ p +2pq+q ∞ ∞  p2 + pq + q2 1 (p2+pq+q2)|γ| (D g)(w)− = K−n (a ,a ,...)wn−1 = [n] b wn−1.  p,q 1 ∑ n−1 2 3 ∑ p,q n  n=2 n=2  |γ|Φ [|(Φ2γ−Φ )(p2+q2)+(Φ2γ−2Φ )pq|+Φ2|γ|(p2+pq+q2)] (15) 1 1 2 1 2 1 ; [|(Φ2γ−Φ )(p2+q2)+(Φ2γ−2Φ )pq|+Φ (p2+2pq+q2)](p2+pq+q2) From (12) and (13) yields  1 2 1 2 1    p2+2pq+q2 1  Φ1 > . 1 + ((Dp q f )(z) − 1) = Φ(u(z)) (16) (p2+pq+q2)|γ| γ , Proof. Replacing n by 2 and 3 in (18) and (19), respectively, and 1 we find that 1 + ((Dp,qg)(w) − 1) = Φ(v(w)). (17) 1 γ [2] a = Φ u , (20) γ p,q 2 1 1 Comparing the corresponding coefficients of (16) and (17) yields 1 [3] a = Φ u + Φ u2, (21) 1 γ p,q 3 1 2 2 1 [n] an = Φ u (18) γ p,q 1 n−1, 1 and − [2]p,q a2 = Φ1v1, (22) 1 γ [n] bn = Φ v . (19) γ p,q 1 n−1 1 2 2 [3]p,q (2a2 − a3) = Φ1v2 + Φ2v1. (23) Note that for am = 0,2 ≤ m ≤ n − 1, we obtain bn = −an γ and so From (20) and (22), we obtain 1 [n] an = Φ1un−1, γ p,q u1 = −v1. (24) 1 − [n] an = Φ1vn−1. By adding (23) to (21), further computations using (24) γ p,q lead to Now taking the absolute values of either of the above two 2 equations and from (9), we obtain [3] a2 = Φ (u + v ) + 2Φ u2. (25) γ p,q 2 1 2 2 2 1 Φ1 |un−1||γ| Φ1 |vn−1||γ| Φ1 |γ| |an| = = ≤ . Making use of (20) in the above equality (25), we get [n]p,q [n]p,q [n]p,q h 2 2 i 2 2 3 2γΦ1 [3]p,q − 2Φ2 [2]p,q a2 = γ Φ1 (u2 + v2). (26) 21 /

Combining (26) and (9), we obtain Hence, we write

 | | 2 2 2 2 3 Φ1 γ 2 γΦ [3] − Φ [2] |a | ≤ |γ| Φ (|u | + |v |)  [3] ; |Φ2| ≤ Φ1 1 p,q 2 p,q 2 1 2 2  p,q |a3| ≤ .    |Φ2||γ| 2 3 2  ; |Φ2| > Φ1 ≤ 2|γ| Φ1 1 − |u1| (27)  [3]p,q On the other hand, by using (9) and (29), we have 2 3 2 3 2 = 2|γ| Φ1 − 2|γ| Φ1 |u1| . 2 2 [3] |a | ≤ [3] |a |2 + Φ (|u | + |v |) It follows from (20) that |γ| p,q 3 |γ| p,q 2 1 2 2 √   |γ|Φ1 Φ1 2 2 2 |a | ≤ . (28) ≤ [3]p,q |a2| + 2Φ1 1 − |u1| . 2 r |γ| 2 2 2 Φ1γ [3]p,q − Φ2 [2]p,q + Φ1 [2]p,q Then, with the help of (20), we have

Moreover, by (9) and (25) h 2 i 2 2 2 Φ1 |γ|[3]p,q |a3| ≤ Φ1 |γ|[3]p,q − [2]p,q |a2| + Φ1 |γ| . 2 2 2 [3] |a2| ≤ Φ1 (|u2| + |v2|) + 2|Φ2||u1| Now, from (28), we obtain |γ| p,q    h 2 i  2 2 Φ1 Φ1 |γ|[3]p,q − [2]p,q ≤ 2Φ1 1 − |u1| + 2|Φ2||u1| Φ1 |γ|   |a3| ≤ 1 + . 2 2 [3]p,q 2  Φ1γ [3]p,q − Φ2 [2]p,q + Φ1 [2]p,q  2 = 2Φ1 + 2|u1| (|Φ2| − Φ1) References  Φ ; |Φ | ≤ Φ 1  1 2 1 [1] H. Airault, H. Bouali, Differential calculus on the [3] |a |2 ≤ . | | p,q 2 Faber polynomials, Bulletin des Sciences Mathema- γ  | | | | > Φ2 ; Φ2 Φ1 tiques, (2006) 179-222. Clearly, we can see that [2] H. Airault, J. Ren, An algebra of differential opera- tors and generating functions on the set of univalent  r Φ |γ| functions, Bulletin des Sciences Mathematiques, 126  1 ; |Φ | ≤ Φ  [3] 2 1  p,q (2002) 343-367. |a2| ≤ .  r [3] ¸S.Altınkaya, S. Yalçın, Coefficient bounds for a sub-  |Φ2||γ| ; | | > class of bi-univalent functions, TWMS Journal of  [3] Φ2 Φ1 p,q Pure and Applied Mathematics, 6 (2) (2015) 180- 185. Next, in order to find the bound on |a3|, by subtracting (23) from (21), we have [4] ¸S.Altınkaya, S. Yalçın, Faber polynomial coefficient bounds for a subclass of bi-univalent functions, C.R. 2 2 [3] a = [3] a2 + Φ (u − v ). (29) Acad. Sci. Paris, Ser.I 353 (2015) 1075-1080. γ p,q 3 γ p,q 2 1 2 2 [5] Brannan, D. A. and Clunie, J. G. Aspects of contempo- rary complex analysis, Proceedings of the NATO Ad- Clearly, from (25), we have that vanced Study Institute Held at University of Durham, New York: Academic Press, 1979.  2 γ Φ1 (u2 + v2) + 2Φ2u1 γΦ1 (u2 − v2) a3 = + [6] Brannan, D. A. and Taha, T. S. On some classes of 2[3] 2[3] p,q p,q bi-univalent functions, Studia Universitatis Babe¸s- γΦ u + γΦ u2 Bolyai Mathematica, 31 (2), 70-77, 1986. = 1 2 2 1 [3]p,q [7] P.L. Duren, Univalent Functions, Grundlehren der Mathematischen Wissenschaften, Springer, New and consequently York, USA, 259 (1983).

2 [8] G. Gasper, M. Rahman, Basic Hypergeometric Series, |γ|Φ1 |u2| + |γ||Φ2||u1| |a3| ≤ Cambridge Univ. Press, Cambridge, MA, 1990. [3]p,q [9] S.G. Hamidi, J.M. Jahangiri, Faber polynomial coef-  2 2 |γ|Φ1 1 − |u1| + |γ||Φ2||u1| ficient estimates for analytic bi-close-to-convex func- ≤ tions, C. R. Acad. Sci. Paris, Ser.I 352 (2014) 17–20. [3]p,q [10] R. Chakrabarti, R. Jagannathan, A (p,q)-oscillator 2 |γ|Φ1 + |γ||u1| (|Φ2| − Φ1) realization of two-parameter quantum algebras, J. = . [3]p,q Phys. A: Math. Gen. 24 (1991) L711-L718. 22 /

[11] Lewin, M. On a coefficient problem for bi-univalent Applied Mathematics Letters, 23 (2010) 1188-1192. functions 18, , Proc. Amer. Math. Soc., 63-68, 1967. [14] H. M. Srivastava, Univalent functions, fractional cal- [12] Netanyahu, E. The minimal distance of the image culus, and associated generalized hypergeometric boundary from the origin and the second coefficient functions, in Univalent Functions; Fractional Cal- of a univalent function in |z| < 1, Archive for Rational culus; and Their Applications (H. M. Srivastava and Mechanics and Analysis, 32, 100-112, 1969. S. Owa, Editors), Halsted Press (Ellis Horwood Lim- [13] H.M. Srivastava, A.K. Mishra, P. Gochhayat, Cer- ited, Chichester), pp. 329-354, John Wiley and Sons, tain subclasses of analytic and bi-univalent functions, New York, Chichester, Brisbane and Toronto, 1989.

23 5Th INTERNATIONAL CONFERENCE ON RECENT ADVANCES IN PURE AND APPLIED MATHEMATICS (ICRAPAM 2018) JULY 23-27 2018 at Karadeniz Technical University Prof. Dr. Osman Turan Convention Center in Trabzon, TURKEY ICRAPAM CONFERENCE PROCEEDING

On the coefficient bounds of gamma and beta starlike functions of order alpha

Nizami Mustafa1, Veysel Nezir1, Mustafa Ate¸s∗1 1Kafkas University, Faculty of Science and Letters, Department of Mathematics, Kars, Turkey

Keywords Abstract: In this paper, we introduce and investigate new subclass of the analytic Analytic function, functions in the open unit disk in the complex plane. Here, we obtain upper bound Coefficient bound estimate, estimates for the first second coefficients for the functions belonging to this class. Gamma and beta–starlike functions of order alpha,

1. Introduction and preliminaries classes were studied by Moustafa [9] and Porwal and Dixit [14]. Also, the class S∗(α,β) recently was studied by In this section, we give the necessary information which Porwal [13]. shall need in our investigation. Inspired by the studies mentioned above, we introduce Let A be the class of analytic functions f (z) in the open a generalization of the function classes S∗ (α,β) and U = {z ∈ |z| < } f ( ) = unit disk C : 1 , normalized by 0 C (α,β) defined as follows. 0 = f 0(0) − 1 of the form Definition 1.1. A function f ∈ A given by (1) is said to be ∞ 2 3 n n in the class ℵγ (α) = S∗C (α,β;γ), α,β ∈ [0,1), γ ∈ [0,1] f (z) = z+a2z +a3z +···+anz +··· = z+ ∑ anz , an ∈ C. β n=2 gamma and beta – starlike function of order alpha if the (1) following condition is satisfied We denote by S the subclass of A consisting of functions  z f 0(z) + γz2 f 00(z)  which are also univalent in U. ℜ > α,z ∈ U. γz[ f 0(z) + βz f 00(z)] + (1 − γ)[βz f 0(z) + (1 − β) f (z)] Some of the important and well-investigated subclasses (2) of the univalent function class S include the classes S∗(α) and C(α), respectively, starlike and convex of or- Remark 1.2. Taking γ = 0 in Definition 1.1 and con- 0 ∗ der α (α ∈ [0,1)) in the open unit disk U. sidering the above note, we have ℵβ (α) = S (α,β), By definition, we have (see for details [6, 8, 15]) α,β ∈ [0,1); that is,

 z f 0(z)   z f 0(z)  S∗(α) = f ∈ A : ℜ > α, z ∈ U , α ∈ [0,1), f ∈ ℵ0 (α) ⇔ ℜ > α,z ∈ U. f (z) β βz f 0(z) + (1 − β) f (z)

  z f 00(z)   Remark 1.3. Taking γ = 1 in Definition 1.1, we have C(α) = f ∈ A : ℜ 1 + > α, z ∈ U , α ∈ [0,1). f 0(z) 1 ℵβ (α) = C (α,β), α,β ∈ [0,1); that is, For some recent investigations of various subclasses of  0 00  the univalent functions class S, see the works by Altinta¸s f (z) + z f (z) f ∈ ℵ1 (α) ⇔ ℜ > α,z ∈ U. et al. [2], Gao et al. [7], and Owa et al. [11]. β f 0(z) + βz f 00(z) Interesting generalization of the function classes ∗ ∗ 0 ∗ 1 S (α) and C(α), are classes S (α,β) and C(α,β) for Remark 1.4. It is clear that ℵ0(α) = S (α), ℵ0(α) = β ≥ 0 , which are defined by C (α). Remark 1.5. Numerous subclasses of the classes given   0   ∗ z f (z) by the Definition 1.1 can be obtained by specializing the S (α,β) = f ∈ A : ℜ 0 > α, z ∈ U , βz f (z) + (1 − β) f (z) various parameters involved. Many of these classes were α,β ∈ [0,1) studied by earlier researches (cf. e.g. [1–5, 13]). and The object of the present paper is to obtained upper bound   f 0(z) + z f 00(z)   C (α,β) = f ∈ A : ℜ > α, z ∈ U , α,β ∈ [0,1), estimates for the first second coefficients for the functions f 0(z) + βz f 00(z) belonging to the class ℵγ (α), α,β ∈ [0,1), γ ∈ [0,1]. respectively. β ∗ ∗ To prove our main results, we need to require the following It is clear that S (α,0) = S (α) and C (α,0) = C (α). lemmas. The class S∗(α,β) were extensively studied by Altin- ta¸sand Owa [4] and certain conditions for hypergeomet- Lemma 1.6. [12] If p ∈ P, then the estimates |pn| ≤ 2,n = ric functions and generalized Bessel functions for these 1,2,3,... are sharp, where P is the family of all functions p,

∗ Corresponding author: [email protected] 24 / analytic in U for which p(0) = 1 and ℜ(p(z)) > 0 (z ∈ U), Corollary 2.2. Let the function f (z) belong to the class and S∗ (α,β), α, β ∈ [0,1). Then, 2 2(1−α) (1−α)[2(1−α)(1+β)+1−β] p(z) = 1 + p1z + p2z + ··· , z ∈ U. (3) |a2| ≤ and |a3| ≤ . 1−β (1−β)2 Lemma 1.7. [12] If the function p ∈ P is given by the Corollary 2.3. Let the function f (z) belong to the class series (3), then C (α,β),α, β ∈ [0,1). Then, 2 2
1−α (1−α)[2(1−α)(1+β)+1−β] 2p2 = p1 + 4 − p1 x,4p3 |a2| ≤ and |a3| ≤ . 1−β 3(1−β)2 3  2  2 2  2 2 = p1 +2 4 − p1 p1x− 4 − p1 p1x +2 4 − p1 1 − |x| z Choose β = 0 in Corollary 2.2 and 2.3, we can readily for some x and z with |x| ≤ 1 and |z| ≤ 1. deduce the following results, respectively. Corollary 2.4. Let the function f (z) belong to the class 2. Upper bound estimates for the coefficients S∗ (α), α ∈ [0,1). Then, In this section, we will obtain upper bound estimates for |a2| ≤ 2(1 − α) and |a3| ≤ (1 − α)(3 − 2α). the first three coefficient for the functions belonging to the Corollary 2.5. Let the function f (z) belong to the class class ℵγ (α), α,β ∈ [0,1), γ ∈ [0,1]. β C (α), α ∈ [0,1). Then, (1−α)(3−2α) Theorem 2.1. Let the function f (z) given by (1) be in the |a2| ≤ 1 − α and |a3| ≤ 3 . class ℵγ (α), α,β ∈ [0,1), γ ∈ [0,1]. Then, β Remark 2.6. As you can see, results obtained in Corollary 2(1−α) (1−α)[2(1−α)(1+β)+1−β] |a2| ≤ and |a3| ≤ . (1−β)(1+γ) (1−β)2(1+2γ) 2.4 and 2.5 verify results obtained in Corollary 2.2 and 2.4 All inequalities obtained here are sharp. in [10] , respectively, for α = 0. Proof.Let f ∈ ℵγ (α), α, β ∈ [0,1), γ ∈ [0,1]. Then, from Remark 2.7. Using this work, one can examine the Fekete β - Szegö problem for the coefficients of the function class (2) we have γ ℵβ (α). Also, using this work one can find upper bound z f 0(z) + γz2 f 00(z) 2 (4) estimate for the a2a4 − a3 . γz[ f 0(z) + βz f 00(z)] + (1 − γ)[βz f 0(z) + (1 − β) f (z)] = α + (1 − α) p(z),z ∈ U, References ∞ n [1] Altınta¸s,O. 1991. On a subclass of certain starlike where function p(z) = 1 + ∑n=1 pnz is in the class P. Comparing the coefficients of the like power of z in (4), functions with negative coefficient. Math. Japon., 36, we obtain 489-495. 1 − α a = p , (5) [2] Altınta¸s,O., Irmak, H., Owa, S., Srivastava, H. M. 2 (1 − β)(1 + γ) 1 2007. Coefficient bounds for some families of star- 2 1 − α (1 − α) (1 + β) 2 like and convex functions of complex order. Applied a3 = p2 + p1. (6) 2(1 − β)(1 + 2γ) 2(1 − β)2 (1 + 2γ) Mathematics Letters, 20, 1218-1222.

In view of Lemma 1.7, from (5) inequality for |a2| is obvi- [3] Altınta¸s,O., Irmak, H., Srivastava , H. M. 1995. Frac- ous. tional calculus and certain starlike functions with neg- 2 ative coefficients. Computers & Mathematics with Ap- Since coefficients of p2 and p1 are positive for each α, β ∈ [0,1) and γ ∈ [0,1] using triangle inequality to equality (6), plications, 30, no.2, 9-16. we obtain [4] Altınta¸s,O., Owa, S. 1988. On subclasses of univalent functions with negative coefficients. Pusan Kyongnam (1 − α)[2(1 − α)(1 + β) + 1 − β] |a3| ≤ . Mathematical Journal, 4, 41-56. (1 − β)2 (1 + 2γ) [5] Altınta¸s,O., Özkan , Ö., Srivastava , H. M. 2004. Thus, the proof of the inequalities in the theorem is com- Neighbourhoods of a Certain Family of Multivalent pleted. Functions with Negative Coefficients. Computers & To see that inequalities obtained in the theorem are sharp, Mathematics with Applications, 47, 1667-1672 we note that equality is attained in the inequalities, when p1 = p2 = 2. [6] Duren, P. L. 1983. Univalent Functions. Grundlehren Moreover, we can easily show that extremal function is the der Mathematischen Wissenschaften. Springer-Verlag, particular solution of the following linear homogeneous New York, 259p. differential equation [7] Gao, C. Y., Yuan, S. M., Srivastava, H. M. 2015. Some functional inequalities and inclusion relationships as- γ [1 − z − β (1 + (1 − 2α)z)]z2y00 sociated with certain families of integral operators. +[1 − z − (1 + (1 − 2α)z)(β + (1 − β)γ)]zy0 Comput. Math. Appl., 49, 1787-1795. −(1 − β)(1 − γ)(1 + (1 − 2α)z)y = 0. [8] Goodman, A. W. 1983. Univalent Functions. Volume Thus, the proof of Theorem 2.1 is completed. I, Polygonal. Mariner Comp., Washington. Setting γ = 0 and γ = 1 in Theorem 2.1, we can readily [9] Moustafa, A. O. 2009. A study on starlike and convex deduce the following results, respectively. properties for hypergeometric functions. Journal of 25 /

Inequalities in Pure and Applied Mathematics, 10, no. denhoeck and Rupercht, Göttingen. 3, article 87,1-16. [13] Porwal, S. 2014. An application of a Poisson distri- [10] Mustafa, N., Ate¸s, M. 2018. On the coefficient bution series on certain analytic functions. J. Complex Bounds of Gamma and Beta Starlike Functions, Jour- Anal., Art. ID 984135, 1-3. nal of Scientific and Engineering Research, 5, no. 6, [14] Porwal, S., Dixit, K. K. 2013. An application of gen- 333-341. eralized Bessel functions on certain analytic functions. [11] Owa, S., Nunokawa, M., Saitoh, H., Srivastava, H. M. Acta Universitatis Matthiae Belii. Series Mathematics, 2002. Close-to-convexity, starlikeness and convexity 51-57. of certain analytic functions. Applied Mathematics [15] Srivastava, H. M., Owa, S. 1992. Current Topics Letters, 15, 63-69. in Analytic Function Theory. World Scientific, Singa- [12] Pommerenke, C. H. 1975. Univalent Functions. Van- pore.

26 5Th INTERNATIONAL CONFERENCE ON RECENT ADVANCES IN PURE AND APPLIED MATHEMATICS (ICRAPAM 2018) JULY 23-27 2018 at Karadeniz Technical University Prof. Dr. Osman Turan Convention Center in Trabzon, TURKEY ICRAPAM CONFERENCE PROCEEDING

Infinite Dimensional Subspaces of A Degenerate Lorentz-Marcinkiewicz Space with The Fixed Point Property

Veysel Nezir1, Nizami Mustafa1, Merve Deliba¸s∗1 1Kafkas University, Faculty of Science and Letters, Department of Mathematics, Kars, Turkey

Keywords Abstract: “We work on some infinite dimensional subspaces of a degenerate `1- Nonexpansive mapping, analog Lorentz-Marcinkiewicz space ` , where the weight sequence = ( ) = δ,1 δ δn n∈N Nonreflexive Banach space, ∞ 1 (2,1,1,1,···) is a decreasing positive sequence in ` \c0, rather than in c0\` (the usual Fixed point property, Lorentz situation). We show that a Goebel and Kuczumow analogy can be obtained by Closed, bounded, convex set, proving that there exists a large classes of non-weak*, closed, bounded and convex subsets Lorentz-Marcinkiewicz spaces of `δ,1 with the fixed point property for nonexpansive mappings.

1. Introduction Goebel and Kuczumow has the fixed point property for asymptotically nonexpansive mappings. Nezir recently constructed an equivalent renorming of `1 In 2009, it was shown by Dominguez Benavides [2] that 1 every reflexive Banach space can be renormed to have the which turns out to produce a degenerate ` -analog Lorentz- fixed point property for nonexpansive mappings. It was Marcinkiewicz space `δ,1, where the weight sequence δ = (δ ) = (2,1,1,1,···) is a decreasing positive se- proved that the converse of this theorem was not true by n n∈N ∞ 1 Lin’s result [9]: non-reflexive Banach space `1, Banach quence in ` \c0, rather than in c0\` (the usual Lorentz space of absolutely summable sequences can be renormed situation). Then, he obtained its isometrically isomor- `0 ` to have the fixed point property for nonexpansive mappings. phic predual δ,∞ and dual δ,∞, corresponding degenerate ∞ Afterward, many recognized fixed point theorists have c0-analog and ` -analog Lorentz-Marcinkiewicz spaces, asked whether or not there is a wider class containing non- respectively. Then, he investigated all types of fixed point expanding mappings that can provide equivalence, but this properties such as weak, weak*, and regular FPP. So we is still a big open question which has not been solved yet. wonder questions mentioned previously for these spaces That is, it remains an open question as to whether or not but as the first step, we want to work on `δ,1 and Goebel & if a Banach space can be renormed to have the fixed point Kuczumow’s idea. Then, we prove that there exist infinite property for a wider class of mappings containing then it dimensional subspaces of `δ,1 with FPP(n.e.). is reflexive. However, by a joint work of the first author 2. Preliminaries with Lennard [8], it was shown that if a Banach space is 1 + a Banach lattice, or has an unconditional basis, or is a Let w ∈ (c0 \ ` ) , w1 = 1 and (wn)n∈N be decreasing; symmetrically normed ideal of operators on an infinite- that is, consider a scalar sequence given by w = (wn)n∈N, dimensional Hilbert space, then it is reflexive if and only if wn > 0,∀n ∈ N such that 1 = w1 ≥ w2 ≥ w3 ≥ ··· ≥ it has an equivalent norm that has the fixed point property wn ≥ wn+1 ≥ ...,∀n ∈ N with wn −→ 0 as n −→ ∞ and ∞ for cascading nonexpansive mappings. This new class of ∑n=1 wn = ∞. This sequence is called a weight sequence. mappings strictly include nonexpansive mappings. But an 1 For example, wn = n ,∀n ∈ N. equivalance principle has not yet been established without Definition 2.1. any necessity of the mentioned conditions. ( n ? ) In connection with the investigation of this question, re- ∑ x j l := x = (x ) ∈ c kxk := sup j=1 < ∞ . searchers are interested to questionize whether or not there w,∞ n n∈N 0 w,∞ n n∈N ∑ j=1 w j exist other non-reflexive Banach spaces than `1 which can ? be renormed to have the fixed point property for nonexpan- Here, x represents the decreasing rearrangement of the sive mappings. But then it is necessary to recall that Lin sequence x, which is the sequence of |x| = (|x j|) j∈N, ar- inspired by the work of Goebel and Kuczumow [5] that can ranged in non-increasing order, followed by infinitely many be said to let him succeed in his research by using some zeros when |x| has only finitely many non-zero terms. of strategies of theirs. By Goebel and Kuczumow’s study, This space is non-separable and an analogue of l∞ space. existence of a large class of non-weak* closed, bounded Definition 2.2. and convex subsets with fixed point property in `1 was ( n ? ) ∑ x j shown and later in 2003, it was shown by Kaczor and Prus l0 x x c j=1 w,∞ := = ( n)n∈N ∈ 0 limsup n = 0 . [6] that under an extra assumption, the sets developed by n−→∞ ∑ j=1 w j

∗ Corresponding author: [email protected] 27 /

This is a separable subspace of lw,∞ and an analogue of c0 3. Main Results space. In his recent work, Nezir proved the first theorem below Definition 2.3. considering the following example. We will extend his results in this paper. ( ) ∞ l x x c x w x ? Example 3.1. Fix b ∈ (0,1). Define the sequence w,1 := = ( n)n∈N ∈ 0 k kw,1 := ∑ j j < ∞ . j=1 ( fn)n∈N in c0 by setting f1 := be1, f2 := be2 and fn := en, for all integers n ≥ 3. Next, define the 1 This is a separable subspace of lw,∞ and an analogue of l1 closed, bounded, convex subset E = Eb of ` by E := 0 ? ∼ ? ∼   space with the following facts: (lw,∞) = lw,1 and (lw,1) = ∞ ∞ l ∼ ∑ tn fn : ∀ tn ≥ 0 & ∑ tn = 1 . w,∞ where the star denotes the dual of a space while = n=1 n=1 denotes isometrically isomorphic. Theorem 3.2. The set E defined as in the example above More information about Lorentz spaces can be seen in has the fixed point property for ~.~-nonexpansive map- 1 [10, 11]. pings where the norm ~·~ on ` is given as below: ~x~ = 1 Now, we will introduce Nezir’s construction. kxk1 + kxk∞, ∀x ∈ ` . For all x = (x ) ∈ `1, we define x := kxk +kxk = n n∈N ~ ~ 1 ∞ Example 3.3. Let b1, b2 ∈ (0,1), 2b1 ≥ b2 and b2 > b1. ∞ ∑ |xn| + sup|xn| . Clearly ~ · ~ is an equivalent norm on Define the sequence ( fn)n∈N in c0 by setting f1 := b1 e1, n=1 n∈N f2 := b2 e2 and fn := en, for all integers n ≥ 3. Next, define 1 1 1 ` with kxk1 ≤ ~x~ ≤ 2kxk1, ∀x ∈ ` . the closed, bounded, convex subset E = Eb of ` by 1 ∗ ∗ ∗ ∗ ∗ Note that ∀x ∈ ` , ~x~ = 2x1 +x2 +x3 +x4 +··· where z ( ∞ ∞ ) is the decreasing rearrangement of |z| = (|zn|)n∈ , ∀z ∈ c0. N E := ∑ tn fn : each tn ≥ 0 and ∑ tn = 1 . Let δ1 := 2,δ2 := 1,δ3 := 1,··· ,δn := 1, ∀n ≥ 4. n=1 n=1 1 We see that (` ,~ · ~) is a (degenerate) Lorentz space `δ,1, where the weight sequence = ( ) is a decreasing Theorem 3.4. The set E defined as in the example above δ δn n∈N ∞ 1 positive sequence in ` \c0, rather than in c0\` (the usual has the fixed point property for k.k-nonexpansive map- 0 pings. Lorentz situation). This suggests that `δ,∞ = (c0,k · k) is 1 an isometric predual of (` ,~ · ~) where for all z ∈ c0, Proof. n We will be using the proof steps of Goebel and ∗ ∑ z j Kuczumow given in detailed as in Everest’s Ph.D. thesis j=1 kzk := sup n . [3], written under supervision of Lennard. Let T : E → E n∈N ∑ δ j j=1 be a nonexpansive mapping. Then, there exists a sequence   The following lemma will be the main ingrediant in our x(n) ∈ E such that Tx(n) − x(n) −→ 0 and so theorems. We provided this lemma in our recently submit- n∈N n (n) (n)   ted study entitled “Fixed point properties for a degenerate Tx − x −→ 0. Without loss of generality, passing 1 n   Lorentz-Marcinkiewicz Space". to a subsequence if necessary, there exists z ∈ `1 such that x(n) converges to z in weak∗ topology. Then, by the proof Lemma 2.4. Our space, a degenerate Lorentz- 1 1 ∗ of Lemma 2.4, we can define a function s : ` −→ [0,∞) Marcienkiewicz space `δ,1 = (` ,~·~) has the weak fixed by point property for nonexpansive mappings with respect to (n) 1 0 s(y) = limsup x − y , ∀y ∈ ` the predual `δ,∞ = (c0,k · k). n   and so   1 ∗   Proof. In order to prove `δ,1 = (` ,~ · ~) has the weak s(y) = s(z) + ~y − z~ , ∀y ∈ `1. fixed point property for nonexpansive mappings, we use the strategy given in [4, 7, Theorem 8.9, Theorem 4.3; Next, define respectively]. There one can see that proof depends on ( ) 1 w∗ ∞ ∞ Goebel and Kuczumow’s lemma [5] for ` with its usual W := E = tn fn : each tn ≥ 0 and tn ≤ 1 1 ∑ ∑ norm. Their lemma says that if {xn} is a sequence in l n=1 n=1 converging to x in weak-star topology, then for any y ∈ l1, Case 1: z ∈ E. Then, we have s(Tz) = s(z) + ~Tz − z~ and r (y) = r (x) + ky − xk1 where r (y) = limsupkxn − yk1 . n s(Tz) = limsup Tz − x(n) Now we note that ` is isometrically isomorphic to sub- n δ,1   space  (n) ≤ limsupTz − T x X := (x∗,x∗,x∗,x∗,x∗,···) : x = (x ) ∈ (`1,k · k ) . n 1 1 2 3 4 n n∈N 1   Then, clearly we obtain Goebel and Kuczumow’s result  (n)  (n) +limsup x − T x  above for the equivalent norm ~ · ~ and noting that B := n  1 ∗   B 1 := x ∈ ` : ~x~ ≤ 1 is a weak -compact set by  (n)  (` ,~·~) ≤ limsup z− x  the Banach-Alaoglu Theorem [7, Proposition 6.13] which n   concludes our proof. = s(z).     28 /

Therefore, ~z − Tz~ ≤ 0 and so Tz = z. Then, Case 2: z ∈ W \ E. ∞ ∞ ~y − z~ = b1|t1 − γ1| + (b1 − b2)|t1 − γ1| Then, z is of the form ∑ γn fn such that ∑ γn < ∞ n=1 n=1 +b |t − γ | + b |t − γ | + |t − γ | 1 and γn ≥ 0, ∀n ∈ N. 2 1 1 2 2 2 ∑ k k k=3 ∞ ∞ Define δ := 1 − ∑ γn and next define = (2b1 − b2)|t1 − γ1| + b2 ∑ |tk − γk| n=1 k=1 ∞ ∞ hλ := (γ1 + λδ) f1 + (γ2 + (1 − λ)δ) f2 + ∑ γn fn. +(1 − b2) ∑ |tk − γk| n=3 k=3 ≥ (2b1 − b2)|t1 − γ1| + b2δ We want hλ to be in E, so we restrict values of λ to be in ∞ − γ1 , γ2 + 1, then δ δ +(1 − b2) ∑ |tk − γk| k=3 ≥ (2b1 − b2)|t1 − γ1| + b2δ + (1 − b2)|δ ~hλ − z~ = ~λδ f1 + (1 − λ)δ f2~ −(t1 − γ1) − (t2 − γ2)|. = ~(λδb1,(1 − λ)δb2,0,0,···)~ b2δ = {| |b ,| − |b } + b | | Subcase 2.1.1: Assume ≥ |t1 − γ1|. δ max λ 1 1 λ 2 1δ λ b1+b2 Then clearly the last inequality from above says that +b2δ|1 − λ|. b ~y − z~ ≥ (2b − b )|t − γ | + δ + (1 − b )(−1 − 1 )|t − γ | 1 γ2 1 2 1 1 2 b 1 1 Thus, if b −b ≤ + 1 then 2 2 1 δ     b1 b2   γ1
≥ 2b − b − (1 − b ) 1 + + 1 δ 2b2δ(1 − λ) − b1δλ if λ ∈ − , 0 , 1 2 2  h δ  b2 b1 + b2  b2  2b2δ(1 − λ) + b1δλ if λ ∈ 0, ,  b1+b2 3b1b2δ  h  ≥ .  b2 2b1λδ + (1 − λ)b2δ if λ ∈ , 1 , b1 + b2 ~hλ − z~ = max b1+b2  h 1  b  2b1λδ + b2(λ − 1)δ if λ ∈ 1, , 2δ  b2−b1 Subcase 2.1.2: Assume < |t1 − γ1|.  h i b1+b2  1 γ2  2b2(λ − 1)δ + b1λδ if λ ∈ , + 1 Then  b2−b1 δ But if 1 > γ2 + 1 then ~y − z~ ≥ (2b − b )|t − γ | + b δ b2−b1 δ 1 2 1 1 2 ∞   γ1
2b2δ(1 − λ) − b1δλ if λ ∈ − , 0 ,  h δ  +(1 − b2) |tk − γk|  b2 ∑  2b2δ(1 − λ) + b1δλ if λ ∈ 0, ,  b1+b2 k=3 ~hλ − z~ = max h  b2  2b1λδ + (1 − λ)b2δ if λ ∈ , 1 , 3b1b2δ  b1+b2 ≥ .   γ2  2b1λδ + b2(λ − 1)δ if λ ∈ 1, δ + 1 b1 + b2 Define Subcase 2.2: b2|t2 − γ2| ≥ b1|t1 − γ1| and b2|t2 − γ2| ≥ Γ := min ~h − z~. γ γ λ |tk − γk|, ∀k ≥ 3. λ∈[− 1 , 2 +1] δ δ Then, Therefore, ~hλ − z~ is minimized when λ ∈ [0,1] with unique minimizer such that its minimum value would be ~y − z~ = b2|t2 − γ2| + (b1 − b2)|t1 − γ1| Γ = 3b1b2δ . ∞ b1+b2 +b |t − γ | + b |t − γ | + |t − γ | ∞ ∞ 2 1 1 2 2 2 ∑ k k k=3 Now fix y ∈ E of the form ∑ tn fn such that ∑ tn = 1 n=1 n=1 b2 with tn ≥ 0, ∀n ∈ N. ≥ b2|t2 − γ2| + (b1 − b2) |t2 − γ2| Then, b1 ∞ ∞ ∞ +b2|t1 − γ1| + b2|t2 − γ2| + |tk − γk| ~y − z~ = ~ ∑ tk fk − ∑ γk fk~ ∑ k=1 k=1 k=3

= ~(t1 − γ1)b1e1 + (t2 − γ2)b2e2 ≥ (2b1 − b2)|t2 − γ2| + b2|t1 − γ1| + b2|t2 − γ2| ∞ +(t3 − γ3)e3 + (t4 − γ4)e4 + ··· + |tk − γk|  2b1|t1 − γ1| + b2|t2 − γ2| + |t3 − γ3|  ∑   k=3  +|t4 − γ4| + |t5 − γ5| + ··· ,    ∞  2b2|t2 − γ2| + b1|t1 − γ1| + |t3 − γ3|    ≥ (2b1 − b2)|t2 − γ2| + b2 |tk − γk|  +|t4 − γ4| + |t5 − γ5| + ··· ,  ∑   k=1  2|t3 − γ3| + b1|t1 − γ1| + b2|t2 − γ2|  ∞ = max +|t4 − γ4| + |t5 − γ5| + ··· ,  2|t4 − γ4| + b1|t1 − γ1| + b2|t2 − γ2|  +(1 − b2) |tk − γk|   ∑  +|t3 − γ3| + |t5 − γ5| + ··· ,  k=3    2|t5 − γ5| + b1|t1 − γ1| + b2|t2 − γ2| + |t3 − γ3|  ∞    +|t4 − γ4| + |t6 − γ6| + ··· ,  ≥ (2b1 − b2)|t2 − γ2| + b2δ + (1 − b2) |tk − γk|   ∑ ············ k=3 ≥ (2b − b )|t − γ | + b δ + (1 − b )|δ Subcase 2.1: b1|t1 − γ1| ≥ b2|t2 − γ2| and b1|t1 − γ1| ≥ 1 2 2 2 2 2 |tk − γk|, ∀k ≥ 3. −(t1 − γ1) − (t2 − γ2)|. 29 /

b2δ Subcase 2.2.1: Assume ≥ |t2 − γ2|. Hence, b1+b2 Then clearly the last inequality from above says that 2b1 − b2 ~y − z~ ≥ |t3 − γ3| + b2δ + (1 − b2)δ ~y − z~ ≥ (2b1 − b2)|t2 − γ2| + δ b1 b −(1 − b2)|t1 − γ1| − (1 − b2)|t2 − γ2|. +(1 − b )(−1 − 2 )|t − γ | 2 b 2 2 1 Hence,   b  ≥ 2b − b − (1 − b ) 1 + 2 |t − γ | 1 2 2 b 2 2 2b1 − b2 1 ~y − z~ ≥ |t3 − γ3| + b2δ + (1 − b2)δ +δ. b1 1 1    −(1 − b ) |t − γ | − (1 − b ) |t − γ | b2 2 3 3 2 3 3 If 2b1 − b2 − (1 − b2) 1 + ≥ 0 then since b1 b2 b1 2 3b1b2δ 2b1 − b2 b2 < we have that ~y − z~ ≥ but if ≥ |t − γ | + b δ + (1 − b )δ 3 b1+b2 3 3 2 2    b1 b2 2b1 − b2 − (1 − b2) 1 + b < 0 then since  1 1  1 −(1 − b ) + |t − γ |. b2δ 2 3 3 ≥ |t2 − γ2|, b1 b2 b1+b2     b2 b2δ b1b2δ ~y − z~ ≥ 2b − b − (1 − b ) 1 + + 1 δ Subcase 2.3.1: Assume ≥ |t3 − γ3|. 1 2 2 b b + b b1+b2 1 1 2 Then clearly the last inequality from above says that 3b b δ ≥ 1 2 . b1 + b2 2b1 − b2 ~y − z~ ≥ |t3 − γ3| + b2δ + (1 − b2)δ b2δ b Subcase 2.2.2: Assume < |t2 − γ2|. 1 b1+b2   Then 1 1 −(1 − b2) + |t3 − γ3| b1 b2 ~y − z~ ≥ (2b1 − b2)|t2 − γ2| + b2δ    2b1 − b2 1 1 ∞ ≥ − (1 − b ) + |t − γ | b 2 b b 3 3 +(1 − b2) ∑ |tk − γk| 1 1 2 k=3 +δ     3b1b2δ 2b1 − b2 1 1 ≥ . ≥ − (1 − b2) + + 1 δ b1 + b2 b1 b1 b2 3b1b2δ Subcase 2.3: |t3 − γ3| ≥ b1|t1 − γ1|, |t3 − γ3| ≥ b2|t2 − γ2|, ≥ . and |t3 − γ3| ≥ |tk − γk|, ∀k ≥ 4. b1 + b2

Then, b1b2δ Subcase 2.3.2: Assume < |t3 − γ3|. b1+b2 ~y − z~ = |t3 − γ3| + (b1 − b2)|t1 − γ1| Then +b |t − γ | + b |t − γ | + |t − γ | 2b − b 2 1 1 2 2 2 3 3 y − z ≥ 1 2 |t − | + b ∞ ~ ~ 3 γ3 2δ b1 + ∑ |tk − γk| ∞ k=4 +(1 − b2) ∑ |tk − γk| 1 k=3 ≥ |t3 − γ3| + (b1 − b2) |t3 − γ3| b1 3b1b2δ ∞ ≥ . b1 + b2 +b2|t1 − γ1| + b2|t2 − γ2| + ∑ |tk − γk| k=3 Subcase 2.4: |t4 − γ4| ≥ b1|t1 − γ1|, |t4 − γ4| ≥ b2|t2 − γ2|, 2b1 − b2 ≥ |t3 − γ3| + b2|t1 − γ1| + b2|t2 − γ2| b1 and |t4 − γ4| ≥ |tk − γk|, ∀k ≥ 5 and for k = 3. ∞ Then, + ∑ |tk − γk| k=3 ~y − z~ = |t4 − γ4| + (b1 − b2)|t1 − γ1| ∞ 2b1 − b2 +b2|t1 − γ1| + b2|t2 − γ2| ≥ |t3 − γ3| + b2 |tk − γk| ∑ ∞ b1 k=1 ∞ +|t3 − γ3| + ∑ |tk − γk| k=4 +(1 − b2) ∑ |tk − γk| k=3 1 ≥ |t4 − γ4| + (b1 − b2) |t4 − γ4| 2b1 − b2 b1 ≥ |t3 − γ3| + b2δ ∞ b1 ∞ +b2|t1 − γ1| + b2|t2 − γ2| + ∑ |tk − γk| k=3 +(1 − b2) ∑ |tk − γk| k=3 2b1 − b2 ≥ |t4 − γ4| + b2|t1 − γ1| + b2|t2 − γ2| 2b1 − b2 b1 ≥ |t3 − γ3| + b2δ ∞ b1 + ∑ |tk − γk|. +(1 − b2)|δ − (t1 − γ1) − (t2 − γ2)|. k=3 30 /

Thus, Note that Λ ⊆ E is compact as it is the continuous image of compact set [0,1] and there exists unique λ ∈ Λ such 2b − b ∞ 0 ~y − z~ ≥ 1 2 |t − γ | + b |t − γ | that ~h − z~ is minimizer of Γ. Now, we can see that for b 4 4 2 ∑ k k λ0 1 k=1 h ∈ Γ, ∞ +(1 − b2) ∑ |tk − γk| (n) k=3 s(Th) = limsup Th − x n 2b1 − b2    ≥ |t − γ | + b δ  (n) 4 4 2 ≤ limsupTh − T x b1 n ∞    (n)  (n) +(1 − b2) ∑ |tk − γk| +limsup x − T x  n k=3   2b − b  (n)  1 2 ≤ limsup h− x  ≥ |t4 − γ4| + b2δ + (1 − b2)|δ n b1   = s(h).   −(t1 − γ1) − (t2 − γ2)|   2b1 − b2 ≥ |t − γ | + b δ + (1 − b )δ Also, s(Th) = s(z) + ~z − Th~ and s(h) = s(z) + ~z − h~. b 4 4 2 2 1 Hence, −(1 − b2)|t1 − γ1| − (1 − b2)|t2 − γ2| 2b − b ≥ 1 2 |t − γ | + b δ + (1 − b )δ ~z − Th~ ≤ ~z − h~ =⇒ ~z − Th~ = ~z − h~ b 4 4 2 2 1 =⇒ Th ∈ Λ. 1 1 −(1 − b ) |t − γ | − (1 − b ) |t − γ | 2 b 4 4 2 b 4 4 1 2 Therefore, T(Λ) ⊆ Λ and since T is continuous, Brouwer’s 2b1 − b2 ≥ |t4 − γ4| + b2δ + (1 − b2)δ Fixed Point Theorem [1] tells us that T has a fixed b1 point such that h = h is the unique minimizer of   λ0 1 1 ~y − z~ : y ∈ E and Th = h. −(1 − b2) + |t4 − γ4|. b1 b2 Hence, E has FPP (n.e.) as desired.

b1b2δ Combining two theorems above, Theorem 3.2 and Theo- Subcase 2.4.1: Assume ≥ |t4 − γ4|. b1+b2 rem 3.4, we give the following corollary which generalizes Then clearly the last inequality from above says that Theorem 3.2. 2b1 − b2 ~y − z~ ≥ |t4 − γ4| + b2δ + (1 − b2)δ Corollary 3.5. Let b , b ∈ (0,1), 2b ≥ b and b ≥ b . b1 1 2 1 2 2 1 Define the sequence ( f ) in c by setting f = b e ,  1 1  n n∈N 0 1 : 1 1 −(1 − b ) + |t − γ | f := b e and f := e , for all integers n ≥ 3. Next, define 2 b b 4 4 2 2 2 n n 1 2 the closed, bounded, convex subset E = E of `1 by    b 2b1 − b2 1 1 ≥ − (1 − b2) + |t4 − γ4| ( ) b1 b1 b2 ∞ ∞ +δ E := ∑ tn fn : each tn ≥ 0 and ∑ tn = 1 .     n=1 n=1 2b1 − b2 1 1 ≥ − (1 − b2) + + 1 δ b1 b1 b2 Then, the set E defined above has the fixed point property 3b b δ ≥ 1 2 . for k.k-nonexpansive mappings. b1 + b2 Now, we work on different examples. b1b2δ Subcase 2.4.2: Assume < |t4 − γ4|. b1+b2 Then 2
Example 3.6. Fix b ∈ 0, 3 . Define the sequence ( fn)n∈N 2b − b in c0 by setting f1 := be1, f2 := be2, f3 := be3 and fn := ~y − z~ ≥ 1 2 |t − γ | + b δ b 4 4 2 en, for all integers n ≥ 4. Next, define the closed, bounded, 1 1 ∞ convex subset E = Eb of ` by +(1 − b2) ∑ |tk − γk| k=3 ( ∞ ∞ ) 3b1b2δ E := ∑ tn fn : each tn ≥ 0 and ∑ tn = 1 . ≥ . n=1 n=1 b1 + b2

Thus, we continue in this way and see that ~y−z~ ≥ 3b1b2δ Theorem 3.7. The set E defined in the example above has b1+b2 from all cases. the fixed point property for k.k-nonexpansive mappings. Therefore, when λ is chosen to be in [0,1], for any y ∈ E and for z ∈ W \ E, ~y − z~ ≥ Γ. Proof. We use similar strategy to the one in the proof of Therefore, when λ is chosen to be in [0,1], for any y ∈ E Theorem 3.4 and we only have different Case 2 as follows: and for z ∈ W \ E, ~y − z~ ≥ Γ. Then, define Case 2: z ∈ W \ E. ∞ ∞ Then, z is of the form ∑ γn fn such that ∑ γn < Λ := {hλ : λ ∈ [0,1]}. n=1 n=1 31 /

4bδ 1 and γn ≥ 0, ∀n ∈ N. ~y − z~ ≥ 3 . δ ∞ Subcase 2.1.2: Assume 3 < |t1 − γ1|. ∞ Define δ := 1 − γn and next define 4bδ ∑ Then ~y−z~ ≥ bδ +b|t1 −γ1|+(1−b) ∑ |tk −γk| ≥ 3 . n=1 k=3 λ λ Subcase 2.2: |t2 − γ2| ≥ |t1 − γ1|, |t2 − γ2| ≥ |t3 − γ3| and h := (γ1 + δ) f1 + (γ2 + δ) f2 λ 2 2 b|t2 − γ2| ≥ |tk − γk|, ∀k ≥ 4. ∞ Then, +(γ3 + (1 − λ)δ) f3 + ∑ γn fn. n=4 ∞ ∞ ~y − z~ = b|t2 − γ2| + b ∑ |tk − γk| + (1 − b) ∑ |tk − γk| k=1 k=4 We want hλ to be in E, so we restrict values of λ to be in h i ∞ − 2γ1 , γ3 + 1 , then δ δ ≥ bδ + b|t2 − γ2| + (1 − b) ∑ |tk − γk| k=4 λ λ ~h − z~ = δ f1 + δ f2 + (1 − λ)δ f3 ≥ b + b|t − | + (1 − b)| λ 2 2 δ 2 γ2 δ    λ λ   −(t1 − γ1) − (t2 − γ2)| =  bδ, bδ,(1 − λ)δb,0,0,···  2 2    ≥ δ + (3b − 2)|t2 − γ2|.   |λ|   = bδ max ,|1 − λ| + bδ|λ| + bδ|1 − λ|  2  δ Subcase 2.2.1: Assume 3 ≥ |t2 − γ2|.  h 2γ   2(1 − λ)bδ − λbδ if λ ∈ − 1 , 0 , Then clearly the last inequality from above says that  δ 4bδ   2
~y − z~ ≥ . = max 2(1 − λ)bδ + λbδ if λ ∈ 0, 3 , 3 bδλ  2
 bδ + if λ ∈ , 1 , Subcase 2.2.2: Assume δ < |t − |.  2 3 3 2 γ2  5bδλ − bδ if λ ∈ 1, γ3 + 1 ∞ 2 δ 4bδ Then ~y−z~ ≥ bδ +b|t2 −γ2|+(1−b) ∑ |tk −γk| ≥ 3 . Define k=3 Subcase 2.3: |t − γ | ≥ |t − γ |, |t − γ | ≥ |t − γ | and Γ := min ~hλ − z~. 3 3 1 1 3 3 2 2 ∈ − γ1 , γ2 + λ [ δ δ 1] b|t3 − γ3| ≥ |tk − γk|, ∀k ≥ 4. Then, Therefore, ~hλ − z~ is minimized when λ ∈ [0,1] with unique minimizer such that its minimum value would be ∞ ∞ 4bδ ~y − z~ = b|t3 − γ3| + b |tk − γk| + (1 − b) |tk − γk| Γ = 3 . ∑ ∑ ∞ ∞ k=1 k=4 Now fix y ∈ E of the form ∑ tn fn such that ∑ tn = 1 ∞ n=1 n=1 ≥ bδ + b|t3 − γ3| + (1 − b) ∑ |tk − γk| with tn ≥ 0, ∀n ∈ N. Then, k=4 ∞ ∞ ≥ bδ + b|t3 − γ3| + (1 − b)|δ − (t1 − γ1) ~y − z~ = ~ ∑ tk fk − ∑ γk fk~ k=1 k=1 −(t2 − γ2) − (t3 − γ3)| = ~(t1 − γ1)be1 + (t2 − γ2)be2 + (t3 − γ3)be3 + (t4 − γ4)e4 ≥ δ + (4b − 3)|t3 − γ3|. +···  2b|t − γ | + b|t − γ | + b|t − γ |  δ  1 1 2 2 3 3  Subcase 2.3.1: Assume ≥ |t3 − γ3|.  +|t − γ | + |t − γ | + ··· ,  3  4 4 5 5  Then clearly the last inequality from above says that  2b|t2 − γ2| + b|t1 − γ1| + b|t3 − γ3|    4bδ  +|t4 − γ4| + |t5 − γ5| + ··· ,  ~y − z~ ≥ .   3  2b|t3 − γ3| + b|t1 − γ1| + b|t2 − γ2|  δ   Subcase 2.3.2: Assume 3 < |t3 − γ3|. = max +|t4 − γ4| + |t5 − γ5| + ··· , ∞   4bδ  2|t4 − γ4| + b|t1 − γ1| + b|t2 − γ2|  Then ~y−z~ ≥ bδ +b|t3 −γ3|+(1−b) ∑ |tk −γk| ≥ .   3  +b|t3 − γ3| + |t5 − γ5| + ··· ,  k=4  |t − | + b|t − | + b|t − | + b|t − |   2 5 γ5 1 γ1 2 γ2 3 γ3  Subcase 2.4: |t4 − γ4| ≥ b|t1 − γ1|, |t4 − γ4| ≥ b|t2 − γ2|,  +|t − γ | + |t − γ | + ··· ,   4 4 6 6  |t4 − γ4| ≥ b|t3 − γ3| and |t4 − γ4| ≥ |tk − γk|, ∀k ≥ 5.  ············  Then, Subcase 2.1: |t − γ | ≥ |t − γ |, |t − γ | ≥ |t − γ | and 1 1 2 2 1 1 3 3 ∞ ∞ b|t1 − γ1| ≥ |tk − γk|, ∀k ≥ 4. ~y − z~ = |t4 − γ4| + b ∑ |tk − γk| + (1 − b) ∑ |tk − γk| Then, k=1 k=4 ∞ ∞ ∞ ≥ bδ + |t − γ | + (1 − b) |t − γ | ~y − z~ = b|t − γ | + b |t − γ | + (1 − b) |t − γ | 4 4 ∑ k k 1 1 ∑ k k ∑ k k k=4 k=1 k=4 ∞ ≥ bδ + |t4 − γ4| + (1 − b)|δ − (t1 − γ1) ≥ bδ + b|t1 − γ1| + (1 − b) ∑ |tk − γk| −(t2 − γ2) − (t3 − γ3)| k=4 3(1 − b) ≥ bδ + b|t − γ | + (1 − b)|δ ≥ bδ + |t − γ | + (1 − b)δ − |t − γ | 1 1 4 4 b 4 4 −(t1 − γ1) − (t2 − γ2)| 4b − 3 ≥ δ + |t4 − γ4|. ≥ δ + (3b − 2)|t1 − γ1| b

δ bδ Subcase 2.1.1: Assume 3 ≥ |t1 − γ1|. Subcase 2.4.1: Assume 3 ≥ |t3 − γ3|. Then clearly the last inequality from above says that Then clearly the last inequality from above says that 32 /

4bδ ~y − z~ ≥ 3 . [5] Goebel, K., Kuczumow, T. 1979. Irregular convex sets bδ with fixed-point property for nonexpansive mappings. Subcase 2.4.2: Assume 3 < |t4 − γ4|. ∞ 4bδ Colloq. Math., 40, no. 2, 259–264. Then ~y − z~ ≥ bδ + |t4 − γ4| + (1 − b) ∑ |tk − γk| ≥ 3 . k=4 [6] Kaczor, W., Prus, S. 2004. Fixed point properties 4bδ 1 Thus, we continue in this way and see that ~y − z~ ≥ 3 of some sets in l . Proceedings of the International from all cases. Conference on Fixed Point Theory and Applications. Therefore, when λ is chosen to be in [0,1], for any y ∈ E [7] Khamsi, M. A., Kirk, W. A. 2011. An introduction and for z ∈ W \ E, ~y − z~ ≥ Γ. Then the rest follows as to metric spaces and fixed point theory. John Wiley & in the proof of Theorem 3.4. Sons. [8] Lennard, C. J., Nezir, V. 2014. Reflexivity is equiva- References lent to the perturbed fixed point property for cascading nonexpansive maps in Banach lattices. Nonlinear Anal- [1] Brouwer, L. E. J. 1911. Über abbildung von man- ysis: Theory, Methods & Applications, 95, 414–420. nigfaltigkeiten. Mathematische Annalen, 71, no. 1, 97–115. [9] Lin, P. K. 2008. There is an equivalent norm on `1 that has the fixed point property. Nonlinear Analysis: [2] Domínguez Benavides, T. 2009. A renorming of some Theory, Methods & Applications, 68, no. 8, 2303- nonseparable Banach spaces with the fixed point prop- 2308. erty. Intern. J. Math. Anal. Appl., 350, no. 2, 525–530. [10] Lindenstrauss, J., Tzafriri, L. 1977. Classical Banach [3] Everest, T. 2013. Fixed points of nonexpansive maps spaces I: sequence spaces, Ergebnisse der Mathematik on closed, bounded, convex sets in l1. Ph.D., Univer- und ihrer Grenzgebiete, 92, Springer-Verlag. sity of Pittsburgh, Pittsburgh, PA, USA. [11] Lorentz, G. G. 1950. Some new functional spaces. [4] Goebel, K., Kirk, W. A. 1990. Topics in metric fixed Ann. Math., 37-55. point theory. Cambridge University Press.

33 5Th INTERNATIONAL CONFERENCE ON RECENT ADVANCES IN PURE AND APPLIED MATHEMATICS (ICRAPAM 2018) JULY 23-27 2018 at Karadeniz Technical University Prof. Dr. Osman Turan Convention Center in Trabzon, TURKEY ICRAPAM CONFERENCE PROCEEDING

The Arrowhead-Pell-Random-Type Numbers Modulo m

Taha Dogan˘ ∗1, Özgür Erdag˘2, Ömür Deveci3 1,2,3Kafkas University,Science and Letters Faculty, Department of Mathematics, Turkey

Keywords Abstract: The arrowhead-Pell-random-type sequence was defined by Erdag et al. (see Sequence, Modulo, Group. [7]). In this work, we study the arrowhead-Pell-random-type numbers modulo m and we obtain the cyclic groups from the generating matrix of the arrowhead-Pell-random-type numbers when reading modulo m. Then we derive the relationships between the orders of the cyclic groups obtained and the periods of the arrowhead-Pell-random-type numbers modulo m.

1. Introduction where Mk,u
α is a (u + k + 1)×(k − 2) matrix as follows:

 u, +u+1 u, +u+2 u, +u+k−1 u, +u+2 u, +u+3 u, +u+k−1 −a α − a α − ··· − a α −a α − a α − ··· − a α k+1 k+1 k+1 k+1 k+1 k+1  u, +u u, +u+1 u, +u+k−2 u, +u+1 u, +u+2 u, +u+k−2  −a α − a α − ··· − a α −a α − a α − ··· − a α  k+1 k+1 k+1 k+1 k+1 k+1  k,uα  In [7], Erdag et al. defined the arrowhead-Pell-random- M =  . .  . . type sequence for n ≥ u as follows:  . .  u, −k+1 u, −k+2 u, −1 u, −k+2 u, −k+3 u, −1 −a α − a α − ··· − a α −a α − a α − ··· − a α k+1 k+1 k+1 k+1 k+1 k+1 ···  au (n + k + 1) = au (n + k − u) − 2au (n + k − u − 1) − au (n + k − u − 2) − ··· − au (n − u) (1) k+1 k+1 k+1 k+1 k+1 ···   α  ∗k,u  M  ··· u u with initial conditions ak+1 (0) = ··· = ak+1 (u + k − 1) = u such that 0 and ak+1 (u + k) = 1 , where 1 ≤ u ≤ k + 1 and k ≥ 2. Also in [7], they gave the generating matrix of the  u,α+u+k−1  −ak+1 arrowhead-Pell-random-type sequence as follows: u,α+u+k−2 α  −a   ∗k,u  k+1  M =  .   .  (u + 1)th  .  ↓ −au,α−1  0 ··· 0 1 −2 −1 ··· −1  k+1 (u+k+1)×1.  1 0 0 0 0 0 ··· 0     0 1 0 0 0 0 ··· 0  α   k,u
α(u+k+1)  0 0 1 0 0 0 ··· 0  It is important to note that det A = (−1) . k,u  0 0 0 1 0 0 ··· 0  A =   .   The study of recurrence sequences in groups began with  ......   ......   . . . .  the earlier work of Wall [10], where the ordinary Fibonacci  0 0 0 ··· 0 1 0 0  0 0 0 0 ··· 0 1 0 sequences in cyclic groups were investigated. Recently, (u+k+1)×(u+k+1) many authors have studied the linear recurrence sequences in groups and fields; see for example, [1–6, 8, 9]. In this Then by an inductive argument they derived that work, firstly we study the arrowhead-Pell-random-type numbers modulo m. Then, we obtain the cyclic groups

(u + 1)th from the multiplicative orders of the generating matrix of ↓  u, +u+k u, +u+k+1 u, +2u+k the arrowhead-Pell-random-type such that the elements of a α a α ··· a α k+1 k+1 k+1  u,α+u+k−1 u,α+u+k u,α+2u+k−1  a a ··· a the generating matrix when reading modulo m. Also, we  k+1 k+1 k+1   . . .  derive the relationships between the orders of the cyclic  . . .  . . .  α  k,u  u,α+u u,α+u+1 u,α+2u groups obtained and the periods of the arrowhead-Pell- A =  a a ··· a  k+1 k+1 k+1  u,α+u−1 u,α+u u,α+2u−1  a a ··· a random-type numbers modulo m.  k+1 k+1 k+1   . . . .  . . . .  . . . .  u, u, +1 u, +u a α a α ··· a α 2. The Arrowhead-Pell-Random-Type Numbers Mod- k+1 k+1 k+1 u, +2u+k+1 u, +u+k a α − a α  ulo m k+1 k+1 u,α+2u+k u,α+u+k−1  a − a  k+1 k+1   .   Reducing the arrowhead-Pell-random-type number .  .  α   u u,α+2u+1 u,α+u  u,k  a (n) by a modulus m, then we get the repeating a − a M , k+1 k+1 k+1  u,α+2u u,α+u−1  a − a  sequence, denoted by k+1 k+1   .  .   .   u,m  u,m u,m u,m u, +u+1 u, a α − a α a (n) = a (1),a (2),...,a (i),... k+1 k+1 k+1 k+1 k+1 k+1

∗ Corresponding author: [email protected] 34 /

u,m u i+1
i+2
where ak+1 (n) ≡ ak+1 (n)(modm). They have the same So we get that h p 6= h p . To complete the proof recurrence relation as in (1). we may use an inductive method on i.  u,m It is well-known that a sequence is periodic if, after a We next denote the period of the sequence ak+1 (n) by u certain point, it consists only of repetitions of a fixed sub- hk+1 (m). sequence. The number of elements in the shortest repeating subsequence is called the period of the sequence. In partic- Theorem 2.3. Let m be a positive integer and suppose δ ei ular, if the first k elements in the sequence form a repeating that m has the prime factorization m = ∏ (pi) , (δ ≥ 1) subsequence, then the sequence is simply periodic and its i=1 where p ’s are distinct primes. Then period is k. i hu m lcmhu p e1 hu p e2 hu p eδ .  u,m k+1 ( ) = k+1 (( 1) ), k+1 (( 2) ),..., k+1 (( δ ) ) Theorem 2.1. The sequence ak+1 (n) is simply periodic u ei for k ≥ 2. Proof. Since hk+1 ((pi) ) is the length of the period of e e n u,(pi) i o n u,(pi) i o Proof. Let us consider the set X = the sequence ak+1 (n) , the sequence ak+1 (n) u ei { (x1,x2,...,xu+k+1)| xi’s are integers such that 0 ≤ xi ≤ m − 1}. repeats only after blocks of length α.hk+1 ((pi) )(α ∈ N). mu+k+1 u + k + u  u,m Since there are distinct 1-tuples of ele- Also, hk+1 (m) is the length of the period ak+1 (n) , ments of Zm, at least one of the u + k + 1-tuples appears n u,(p )ei o u,m i u  which implies that ak+1 (n) repeats after hk+1 (m) twice in the sequence ak+ (n) . So, the subsequence fol- 1 u lowing this u + k + 1-tuples repeats; that is, the sequence terms for all values i. Thus, hk+1 (m) is the form  u,m u ei ak+1 (n) is periodic. Let α.hk+1 ((pi) ) for all values of i, and since any such num-  u,m u,m u,m u,m u,m u,m u,m ber gives a period of a (n) . So we get a (i + 1) ≡ a ( j + 1),a (i + 2) ≡ a ( j + 2),...,a (i + u + k + 1) ≡ a ( j + u + k + 1) k+1 k+1 k+1 k+1 k+1 k+1 k+1 u  u e1 u e2 u eδ  hk+ (m) = lcm hk+ ((p1) ),hk+ ((p2) ),...,hk+ ((pδ ) ) . such that i > j, then i ≡ j (modu + k + 1). From the defi- 1 1 1 1 nition, we can easily obtain

u,m u,m u,m u,m u,m u,m a (i) ≡ a ( j),a (i − 1) ≡ a ( j − 1),...,a (i − j + 1) ≡ a (1), k+1 k+1 k+1 k+1 k+1 k+1 References which implies that the sequence au,m (n) is simply peri- k+1 [1] Aydın, H., Aydın, R. 1998. General Fibonacci Se- odic. quences in Finite Groups. Fibonacci Quarterly, 36(3), Given an integer matrix B = [b ], B(modm) means i j 216-221. that all entries of B are reduced modulo m, that is, B(modm) = (bi j (modm)). Let us consider the set hBim = [2] Campbell, C. M., Doostie, H., Robertson, E. F. 1990. n {B (modm) | n ≥ 0}. If gcd(m,detB) = 1, then hBim is Fibonacci Length of Generating Pairs in Groups in a cyclic group; if gcd(m,detB) 6= 1, then hBim is a semi- Applications of Fibonacci Numbers. Vol. 3 Eds. G. E. group. We next denote the cardinality of the set hBim by Bergum et al. Kluwer Academic Publishers, 27-35. |hBim|. [3] Deveci, O. 2015. The Pell-Padovan Sequences and k,u u+k+1 k,u Since detA = (−1) it is clear that the set A m The Jacobsthal-Padovan Sequences in Finite Groups. is a cyclic group for every positive integer m. Utilitas Mathematica, 98, 257-270. k,u Theorem 2.2. Let p be a prime and let A pm be the [4] Deveci, O., Karaduman, E. 2015. The Pell Sequences cyclic group. If i be the largest positive integer such that in Finite Groups. Utilitas Mathematica, 96, 263-276.

Ak,u 6= Ak,u , then Ak,u = p j−i · Ak,u p pi p j pi [5] Deveci, O., Akuzum, Y., Karaduman, E. 2015. The for every integer j ≥ i. Pell-Padovan p-Sequences and Its Applications. Utili- tas Mathematica, 98, 327-347. Proof. Suppose that n is a positive integer and Ak,u pm [6] Doostie, H., Campbell, C. M. 2000. Fibonacci Length h(pn+1) is denoted by h(pm). If Ak,u
≡ I modpn+1
, of Automorphism Groups Involving Tribonacci Num- h(pn+1) bers. Vietnam Journal of Mathematics, 28, 57-65. then Ak,u
≡ I (modpn) where I is a (u + k + 1) × (u + k + 1) identity matrix . Thus we obtain that h(pn) [7] Erdag, O., Shannon, A. G., Deveci, O. 2018. The divides hpn+1
. On the other hand, if we denote Arrowhead-Pell-Random-Type Sequences. Notes on h(pn)   Number Theory Discrete Mathematics, 24(1), 109- Ak,u
= I + a(n).pn , then by the binomial expan- i j 119. sion, we may write [8] Knox, S. W. 1992. Fibonacci Sequences in Finite n   p p   i k,u
h(p ).p (n) n p (n) n n+1
Groups. Fibonacci Quarterly, 30, 116-120. A = I + ai j .p = ∑ ai j .p ≡ I modp , i=0 i [9] Lü, K., Wang, J. 2006. k-Step Fibonacci sequence which implies that hpn+1
divides h(pn).p. Then we modulo m. Utilitas Mathematica, 71, 169-177. n+1
n n+1
n easily obtain that h p = h(p ) or h p = h(p ).p, [10] Wall, D. D. 1960. Fibonacci Series Modulo m. The (n) and the latter holds if and only if there is an ai j which is American Mathematical Monthly, 67(6), 525-532. not divisible by p. Since i is the largest positive integer such that h(p) = hpi
, we have hpi
6= hpi+1
. Then (i+1) there exists an integer ai j which is not divisible by p. 35 5Th INTERNATIONAL CONFERENCE ON RECENT ADVANCES IN PURE AND APPLIED MATHEMATICS (ICRAPAM 2018) JULY 23-27 2018 at Karadeniz Technical University Prof. Dr. Osman Turan Convention Center in Trabzon, TURKEY ICRAPAM CONFERENCE PROCEEDING

On f − Lacunary Statistical Boundedness of Order α

Mikail Et∗1, Hüseyin Sönmez2 1Firat University, Science Faculty, Department of Mathematics, Turkey 2Firat University, Science Faculty, Department of Mathematics, Turkey

Keywords Abstract: The main purpose of this work is to introduce and examine the concept of Statistical boundedness, f −lacunary statistical boundedness of order α and give the realtions between statistical Lacunary sequence, boundedness and f −lacunary statistical boundedness of order α. Modulus function

1. Introduction statistical convergence of order α was studied by Çolak ([14],[15]). Let w be the set of all sequences of real or complex num- By a lacunary sequence we mean an increasing inte- bers and `∞, c and c0 be respectively the Banach spaces of ger sequence θ = (kr) of non-negative integers such bounded, convergent and null sequences x = (xk) with the that k0 = 0 and hr = (kr − kr−1) → ∞ as r → ∞. usual norm kxk∞ = sup|xk|, where k ∈ N = {1,2,...}, the set of positive integers. The intervals determined by θ will be denoted by kr Ir = (kr−1,kr] and the ratio k will be abbrevi- The idea of statistical convergence was given by Zygmund r−1 ated by qr, and q1 = k1 for convenience. In re- [36] in the first edition of his monograph puplished in cent years, lacunary sequences have been studied in Warsaw in 1935. The consept of statistical convergence ([2],[3],[4],[10],[11],[12],[17],[18],[19],[21],[23]). was introduced by Steinhaus [35] and Fast [20] and later reintroduced by Schoenberg [31]. Over the years and un- The concept of lacunary statistical convergence of order der different names statistical convergence was discussed α was given by ¸Sengüland Et [32], Das and Savas [17] in the theory of Fourier analysis, Ergodic theory, Num- independently as follows: ber theory, Measure theory, Trigonometric series, Turn- Let θ = (kr) be a lacunary sequence and 0 < α ≤ 1 be pike theory and Banach spaces. Later on it was fur- given. The sequence x = (xk) ∈ w is said to be lacunary ther investigated from the sequence space point of view statistically convergent of order α, if there is a real number and linked with summability theory by Bhardwaj et al. L such that ([5],[6],[7],[8],[9]), Caserta et al. [13], Connor [16], Fridy 1 lim |{k ∈ Ir : |xk − L| ≥ ε}| = 0. [22], Fridy and Orhan [23], Mursaleen [26], Salat [34], r→∞ hα Sava¸s([27],[28],[29],[30]) and many others. r The set of all lacunary statistically convergent sequences The idea of statistical convergence depends upon the den- α of order α will be denoted by Sθ . sity of subsets of the set N of natural numbers. The density A modulus f is a function from [0,∞) to [0,∞) such that of a subset E of N is defined by i) f (x) = 0 if and only if x = 0, ii) f (x + y) ≤ f (x) + f (y) for x,y ≥ 0, 1 n δ( ) = lim χ (k), provided that the limit texists. E n→∞ ∑ E iii) f is increasing, n k=1 iv) f is continuous from the right at 0. A sequence x = (xk) is said to be statistically convergent Hence f must be continuous everywhere on [0,∞).A to L if for every ε > 0, modulus may be unbounded or bounded. For example, δ ({k ∈ : |xk − L| ≥ ε}) = 0. p x N f (x) = x (0 < p ≤ 1) is unbounded, but f (x) = x+1 is The concept of statistical boundedness was given by Fridy bounded. and Orhan [24] as follows: Aizpuru et al. [1] defined f −density of a subset E ⊂ N for any unbounded modulus f by The real number sequence x is statistically bounded if there is a number B such that δ ({k : |x | > B}) = 0. f (|{k ≤ n : k ∈ E}|) k d f (E) = lim ,if the limit exists n→∞ f (n) It is well known that every bounded sequence is statistically bounded, but the converse is not true. and defined f −statistical convergence for any unbounded modulus f by The order of statistical convergence of a sequence of num- f bers was given by Gadjiev and Orhan [25] after then d ({k ∈ N : |xk − L| ≥ ε}) = 0

∗ Corresponding author: [email protected] 36 2 MAIN RESULTS i.e. f −lacunary statistically convergent of order α, if there is a real number L such that 1 lim f (|{k ≤ n : |xk − L| ≥ ε}|) = 0, n→∞ f (n) 1 lim α f (|{k ∈ Ir : |xk − L| ≥ ε}|) = 0, r→∞ f (hr) f f
and we write it as S − limxk = L or xk → L S . Every α f −statistically convergent sequence is statistically con- where Ir = (kr−1,kr] and f (hr) denotes the αth power α
α α vergent, but a statistically convergent sequence does not of f (hr), that is f (hr) = ( f (h1) , f (h2) ,..., need to be f −statistically convergent for every unbounded α f ,α f (hr) ,...). This space will be denoted by Sθ . In this modulus f . f ,α  f ,α  case, we write Sθ − limxk = L or xk → L Sθ . Let X be a sequence space. Then X is called Definition 3 Let f be an unbounded modulus, θ = (kr) be i) Solid (or normal), if (αkxk) ∈ X for all sequences (αk) a lacunary sequence and α be a real number such that 0 < of scalars with |αk| ≤ 1 for all k ∈ N, whenever (xk) ∈ X, α ≤ 1. The sequence x = (xk) ∈ w is said to be f −lacunary ii) Monotone if it contains the canonical preimages of all statistically bounded of order α, if there is a M ≥ 0 such its stepspaces, that Symmetric, (x ) ∈ X (x ) ∈ X, ii) if k implies π(k) where π is 1 a permutation of N, lim α f (|{k ∈ Ir : |xk| > M}|) = 0, (1) r→∞ f (h ) iii) Sequence algebra if x.y ∈ X, whenever x,y ∈ X. r

In the present work we introduce the concept of i.e. |xk| ≤ M,a.a.kr ( f α). The set of all f −lacunary sta- f −lacunary statistical boundedness of order α and give the tisticallly bounded sequences of order α will be denoted f ,α r realtions between statistical boundedness and f −lacunary by Sθ (b). For θ = (2 ) and f (x) = x we shall write statistical boundedness of order α α f ,α S (b) instead of Sθ (b) and in the special case α = 1 and θ = (2r) we shall write S f (b) instead of S f ,α (b). For 2. Main Results θ α = 1 and f (x) = x we obtain the set Sθ (b) of all lacu- nary statistically bounded sequences. In the special cases In this section we give the main results of the work. In α = 1, f (x) = x and θ = (2r) we shall write S(b) instead Theorem 5 we give the relations between f −lacunary sta- of S f ,α (b). tistically bounded sequences of order α and f −lacunary θ statistically bounded sequences of order β. Theorem 1 Every f −lacunary statistically convergent se- quence of order α is f −lacunary statistically bounded of Definition 1 [33] Let f be an unbounded modulus, θ = order α, but the converse is not true. (kr) be a lacunary sequence and α be a real number such that 0 < α ≤ 1. We define lacunary f α−density of subset Proof. Let x = (xk) be a f −lacunary statistically conver- E of N by gent sequence of order α and ε > 0 be given. Then there exists L ∈ C such that f ,α 1 δ (E) = lim f (|{kr−1 < k ≤ kr : k ∈ E}|), 1 θ r f (h )α r lim α f (|{k ∈ Ir : |xk − L| ≥ ε}|) = 0. r→∞ f (hr) provided the limit exists. Lacunary f α−density δ f ,α ( ) θ E Since {k ∈ I : |x | > |L| + ε} ⊃ {k ∈ I : |x − L| ≥ ε} we reduces to natural density δ ( ) in the special case α = 1, r k r k E have f (x) = x and θ = (2r). 1 If x = (xk) is a sequence such that xk satisfies property lim f (|{k ∈ Ir : |xk| > |L| + ε}|) r→∞ f (h )α p(k) for all k except a set of lacunary f α−density zero, r then we say that x satisfies p(k) for “ lacunary almost 1 k ≤ lim α f (|{k ∈ Ir : |xk − L| ≥ ε}|). all k according to α and f ” and we abbreviate this by r→∞ f (hr) "a.a.kr ( f α)". So x = (xk) is f −lacunary statistically bounded of order Proposition 1 Let θ = (kr) be a lacunary sequence and α. To show the strictness of the inclusion, choose θ = f ,β f , r α (2 ), f (x) = x and define a sequence x = (xk) by α,β ∈ (0,1] such that α ≤ β, then δθ (E) ≤ δθ (E). Proof. Proof follows from the following inequality  1, k = 2n x = k,n ∈ . k −1, k 6= 2n N 1 f (|{k < k ≤ k : k ∈ }|) β r−1 r E f ,α f ,α f (hr) Then x ∈ Sθ (b), but x ∈/ Sθ . 1 Theorem 2 Every bounded sequence is f −lacunary statis- ≤ α f (|{kr−1 < k ≤ kr : k ∈ E}|). f (hr) tically bounded of order α, but the converse is not true. Definition 2 [33] Let f be an unbounded modulus, θ = Proof. The proof of the first part follows in view of the (kr) be a lacunary sequence and α be a real number such fact that empty set has zero f α−lacunary density for that 0 < α ≤ 1. We say that a sequence x = (xk) is every unbounded modulus f . To show the strictness of

37 REFERENCES

p r the inclusion, choose f (x) = x (0 < p ≤ 1), θ = (2 ), Proof. Let x = (xk) = α = 1 and define x = (x ) by x = (1,0,0,4,0,0,0,0,9,...). α, f k (1,0,0,2,0,0,0,0,3,0,0,0,0,0,0,4,...) ∈ Sθ (b). f ,α f ,α Then δθ ({k ∈ N : |xk − 0| ≥ ε}) = δθ (A), where A = Let y = (yk) be a rearrangement of (xk), which is defined {1,4,9,...}. Thus as follows: √ (yk) = (x1,x2,x4,x3,x9,x5,x16,x6,x25,x7,x36,x8,x49,x10,...) = |{k ≤ n : k ∈ A}| ≤ n,for all n ∈ N (1,0,2,0,3,0,4,0,5,0,6,0,7,0,...). Clerarly for any M > 0, δ f ,α ({k : |y | > M}) 6= 0, in the special case an so we have θ k α = 1, θ = (2r) and f (x) = x. √ p f (|{k ≤ n : k ∈ A}|) ( n) f ,α ≤ → 0, as n → ∞. ii) Let x = (xk) ∈ δθ (b) and y = (yk) be a sequence f (n) np f ,α such that |yk| ≤ |xk| for all k ∈ N. Since x ∈ δθ (b) there f ,α That is, the sequence x = (1,0,0,4,0,0,0,0,9,...) is exists a number M such that δθ ({k : |xk| > M}) = 0. f ,α f −lacunary statistically bounded of order α, but is not Clearly y ∈ δθ (b) as {k : |yk| > M} ⊂ {k : |xk| > M}. bounded. f ,α So δθ (b) is normal. It is well known that every normal f ,α Theorem 3 Every convergent sequence is f −lacunary sta- space is monotone, so δθ (b) is monotone. tistically bounded of order α, but the converse is not true. f ,α iii) Let x,y ∈ δθ (b). Then there exists K,M > 0 such Proof. Proof is similar to that of Theorem 2. f ,α f ,α that δθ ({k : |xk| ≥ K}) = 0 and δθ ({k : |yk| ≥ M}) = Theorem 4 Every f −lacunary statistically bounded se- 0. The proof follows from the following inclusion quence of order α is lacunary statistically bounded of {k : |xkyk| ≥ KM} ⊂ {k : |xk| > K} ∪ {k : |yk| > M}. order α, but the converse is not true. Remark. A subsequence of an f −lacunary statistically Proof. The first part of the proof follows in view of bounded sequence of order α need not be f −lacunary f ,α α the fact that E ⊂ N, δθ (E) = 0 implies δθ (E) = 0. statistically bounded sequence of order α. Really the se- To show the strictness of the inclusion, choose f (x) = quence x = (xk) = (1,0,0,4,0,0,0,0,9,...) is f −lacunary r log(x + 1), θ = (2 ), α = 1 and define x = (xk) by statistically bounded sequence of order α whereas x = (1,0,0,4,0,0,0,0,9,...). Let A = {1,4,9,...}. For any (1,4,9,...) is a subsequence of it which is not f −lacunary M > 0, statistically bounded sequence of order α.

{k ∈ N : |xk| ≥ M} = A − a finite subset of N. References [1] A. Aizpuru, M. C. Listán-García and F. Rambla- Since δ f ,α (A) = 1 6= 0 and δ α (A) = 0, for θ = (2r), α = θ 2 θ Barreno, Density by moduli and statistical conver- 1. So x = (x ) is lacunary statistically bounded of order α, k gence, Quaest. Math. 37(4) (2014) 525–530. but is not f −lacunary statistically bounded sequence of order α. [2] H. Altinok, Y. Altin, M. Et, Lacunary Almost Statisti- cal Convergence of Fuzzy Numbers, Thai Journal of Theorem 5 Let f be an unbounded modulus, θ = (kr) be Mathematics , 2 (2), 265-274, 2004. a lacunary sequence and the parameters α and β are fixed real numbers such that 0 < α ≤ β ≤ 1, then S f ,α (b) ⊆ [3] H. Altinok, R. Colak, Almost lacunary statistical and θ strongly almost lacunary convergence of generalized S f ,β (b) and the inclusion is strict. θ difference sequences of fuzzy numbers Journal of Proof. The inclusion part of proof follows from Propo- Fuzzy Mathematics, 17 (4), 951-968, 2009. sition 1. To show the strictness of the inclusion, choose [4] H. Altinok, M. Et, Y. Altin, Lacunary statistical bound- f (x) = x, θ = (kr) be a lacunary sequence and defined a edness of order β for sequences of fuzzy numbers, sequence x = (xk) by Journal of Intelligent & Fuzzy Systems, 35 (2018) √ √ 2383–2390.   h , k = 1,2,3,..., h  x = r r . k 0, otherwise [5] V. K. Bhardwaj and I. Bala, On weak statistical con- vergence, Int. J. Math. Math. Sci. 2007, Art. ID 38530, β, f 1 α, f 9 pp. Then x ∈ Sθ (b) for 2 < β ≤ 1 but x ∈/ Sθ (b) for 0 < 1 α ≤ 2 . [6] V. K. Bhardwaj and S. Dhawan, Density by moduli From Theorem 5 yields the following corollary. and lacunary statistical convergence, Abstr. Appl. Anal. 2016, Art. ID 9365037, 11 pp. Corollary 1 If a sequence is f −lacunary statistically [7] V. K Bhardwaj and S. Gupta, On some generaliza- bounded of order α, then it is f −lacunary statistically tions of statistical boundedness, J. Inequal. Appl. 2014, bounded. 2014:12. Theorem 6 [8] V. K. Bhardwaj and S. Dhawan, f −statistical conver- i) Sα, f (b) is not symmetric, θ gence of orderα and strong Cesàro summability of α, f ii) θ (b) is normal and hence monotone, order α with respect to a modulus. J. Inequal. Appl. α, f iii) Sθ (b) is a sequence algebra. 2015, 2015:332, 14.

38 REFERENCES [9] V. K. Bhardwaj ; S. Gupta ; S. A. Mohiuddine and (1985), 301-313. A. Kılıçman, On lacunary statistical boundedness. J. [23] J. Fridy and C. Orhan, Lacunary statistical conver- Inequal. Appl. 2014, 2014:311, 11 pp. gence, Pacific J. Math. 160 (1993), 43-51. [10] H. Çakallı, Lacunary statistical convergence in topo- [24] J.A. Fridy and C. Orhan, Statistical limit superior and logical groups, Indian J. Pure Appl. Math. 26(2) (1995) limit inferior, Proc. Amer. Math. Soc. 125(12) (1997), 113–119. 3625–3631. [11] H. Çakallı, C. G. Aras and A. Sönmez, Lacunary sta- [25] A. D. Gadjiev and C. Orhan, Some approximation tistical ward continuity, AIP Conf. Proc. 1676, 020042 theorems via statistical convergence, Rocky Mountain (2015); http://dx.doi.org/10.1063/1.4930468. J. Math. 32(1) (2002), 129-138. [12] H. Çakallı and H. Kaplan, A variation on lacunary [26] M. Mursaleen, λ− statistical convergence, Math. Slo- statistical quasi Cauchy sequences, Commun. Fac. Sci. vaca, 50(1) (2000), 111 -115. Univ. Ank. Sér. A1 Math. Stat. 66(2) (2017) 71–79. [27] E. Sava¸s, Asymptotically J−lacunary statistical [13] A. Caserta, G. Di Maio and L. D. R. Kocinac,ˇ Sta- equivalent of order α for sequence of sets, J. Non- tistical convergence in function spaces, Abstr. Appl. linear Sci. Appl. 10(6) (2017), 2860–2867. Anal. 2011, Art. ID 420419, 11 pp. [28] E. Sava¸sGeneralized asymptotically I−lacunary sta- [14] R. Çolak, Statistical convergence of order α, Modern tistical equivalent of order α for sequences of sets, Methods in Analysis and Its Applications, New Delhi, Filomat 31(6) (2017), 1507–1514. India: Anamaya Pub, 2010: 121–129. [29] E. Sava¸s Iλ −statistically convergent functions of or- [15] R. Çolak, On λ−statistical convergence, Conference der α, Filomat 31(2) (2017), 523–528. on Summability and Applications, May 12-13, 2011, Istanbul Turkey. [30] E. Sava¸sOn I−lacunary statistical convergence of weight g of sequences of sets. Filomat 31 (2017), no. [16] J. S. Connor, The statistical and strong p−Cesaro 16, 5315–5322 convergence of sequences, Analysis 8 (1988), 47-63. [31] I. J. Schoenberg, The integrability of certain func- [17] P. Das and E. Savas, On I−statistical and I−lacunary tions and related summability methods, Amer. Math. statistical convergence of order α, Bull. Iranian Math. Monthly 66 (1959), 361-375. Soc. 40 (2014), no. 2, 459–472. [32] H. ¸Sengüland M. Et, On lacunary statistical conver- [18] M. Et and H. ¸Sengül,Some Cesaro-type summability gence of order α, Acta Math. Sci. Ser. B Engl. Ed. spaces of order α and lacunary statistical convergence 34(2) (2014), 473–482. of order α, Filomat 28(8) (2014), 1593–1602. [33] H. ¸Sengüland M. Et, f −Lacunary Statistical Conver- [19] M. Et and S.A. Mohiuddine, ¸Sengül,Hacer. On la- gence and Strong f −Lacunary Summability of Order cunary statistical boundedness of order α, Facta Univ. α, Filomat (Accepted). Ser. Math. Inform. 31 (2016), no. 3, 707–716. [34] T. Salat, On statistically convergent sequences of real [20] H. Fast, Sur la convergence statistique, Colloq. Math. numbers, Math. Slovaca 30 (1980), 139-150. 2 (1951), 241-244. [35] H. Steinhaus, Sur la convergence ordinaire et la con- [21] A. R. Freedman ; J. J. Sember and M. Raphael, Some vergence asymptotique, Colloquium Mathematicum 2 Cesaro-type summability spaces, Proc. Lond. Math. (1951),73-74. Soc. 37(3) (1978), 508-520. [36] A. Zygmund, Trigonometric Series, Cambridge Uni- [22] J. Fridy, On statistical convergence, Analysis 5 versity Press, Cambridge, UK, 1979.

39 5Th INTERNATIONAL CONFERENCE ON RECENT ADVANCES IN PURE AND APPLIED MATHEMATICS (ICRAPAM 2018) JULY 23-27 2018 at Karadeniz Technical University Prof. Dr. Osman Turan Convention Center in Trabzon, TURKEY ICRAPAM CONFERENCE PROCEEDING

Aczél Type Inequalities for Hilbert Space Operators

Ulas Yamancı1, Mehmet Gürdal∗2 1Süleyman Demirel University, Department of Statistics, Turkey 2Süleyman Demirel University, Department of Mathematics, Turkey

Keywords Abstract: Some Aczél type inequalities for Hilbert space operators are established. Keywords Aczél-type Several particular cases of interest are given as well. inequality, Keywords selfadjoint operator, Keywords Hilbert space operator

n n 1. Introduction λ1 λ1 λ2 λ2 numbers such that a1 − ∑ai > 0 and b1 − ∑bi > 0. i=2 i=2 In 1956, the Aczél’s inequality was given by Aczél [1] as Then following, which has some applications in mathematical ! 1 ! 1 analysis and in the theory of functional equations in n λ1 n λ2 n λ1 λ1 λ2 λ2 non-Euclidean geometry. During the last decades several a1 − ∑ai b1 − ∑bi ≥ a1b1 − ∑aibi, i=2 i=2 i=2 interesting generalization of this significant inequality (3) were obtained in [5, 6]. which is called as Aczél-Vasic-Pe´ cariˇ c´ inequality.

+ Vasic´ and Pecariˇ c´ [8] obtained a further extension of the Theorem A. Let n ∈ N , n ≥ 2, and let ai,bi n Aczél inequality (1). Tian [5] also obtained the reversed 2 version of inequality which were obtained by Vasic´ and (i = 1,2,...,n) be real numbers such that a1 − ∑ai > 0 i=2 Pecariˇ c.´ n 2 2 and b1 − ∑bi > 0. Then i=2 Bjelica [2] gave a new interesting Aczél-type inequality as following. 2 n ! n ! n ! a2 − a2 b2 − b2 ≤ a b − a b . + 1 ∑ i 1 ∑ i 1 1 ∑ i i (1) Theorem D. Let n ∈ N , n ≥ 2, let 0 < λ ≤ 2, and let i=2 i=2 i=2 ai,bi (i = 1,2,...,n) be positive real numbers such that n n In 1959, Popoviciu [4] established an extension of the λ λ λ λ a1 − ∑ai > 0 and b1 − ∑bi > 0. Then inequality (1) in the following theorem. i=2 i=2

1 1 + 1 n ! λ n ! λ n Theorem B. Let n ∈ N , n ≥ 2, let λ1 > 1, λ2 > 1, + λ λ λ λ λ1 a1 − ∑ai b1 − ∑bi ≤ a1b1 − ∑aibi, 1 i=2 i=2 i=2 = 1, and let ai,bi (i = 1,2,...,n) be nonnegative real λ2 n n which is called as Aczél-Bjelica inequality. λ1 λ1 λ2 λ2 numbers such that a1 − ∑ai > 0 and b1 − ∑bi > 0. i=2 i=2 Let A be a selfadjoint linear operator on a complex Hilbert Then space H. The Gelfand map establishes a ∗-isometrically 1 1 isomorphism Φ between the set C(Sp(A)) of all continuous n ! λ1 n ! λ2 n aλ1 − aλ1 bλ2 − bλ2 ≤ a b − a b , functions defined on the spectrum of A, denoted Sp(A), 1 ∑ i 1 ∑ i 1 1 ∑ i i ∗ ∗ i=2 i=2 i=2 an the C -algebra C (A) generated by A and the identity (2) operator 1H on H as follows (see for instance [3]): which is called as Aczel-Popoviciu inequality. For any f ,g ∈ C(Sp(A))and any α,β ∈ C we have In 1982, Vasic´ and Pecariˇ c´ [7] gave the following reversed (i) Φ(α f + βg) = αΦ( f ) + βΦ(g); version of Aczél-Popoviciu inequality (2). (ii) Φ( f g) = Φ( f )Φ(g) and Φ( f ) = Φ( f )∗ ; (iii) ||Φ( f )|| = || f || := sup | f (t)|; + t∈Sp(A) Theorem C. Let n ∈ N , n ≥ 2, let λ1 < 1 (λ1 6= 0), 1 1 (iv) Φ( f0) = 1H and Φ( f1) = A, where f0 (t) = 1 and + = 1, and let ai,bi (i = 1,2,...,n) be positive real f (t) = t, for t ∈ Sp(A). λ1 λ2 1

∗ Corresponding author: [email protected] 40 /

With this notation we define for any unit vector x ∈ H and n ∈ J. Applying the func- tional calculus again to the self-adjoint operator B we have f (A) := Φ( f ) for all f ∈ C(Sp(A)) D E [ f (A)g(A)]2 x,x − g2(B) f 2(A)x,x − f 2(B) g2(A)x,x and it is called the continuous functional calculus for a +[ f (B)g(B)]2 selfadjoint operator A. D 2 E If A is a selfadjoint operator and f is a real valued contin- ≤ [ f (A)g(A) − f (B)g(B)] x,x uous function on Sp(A), then f (t) ≥ 0 for any t ∈ Sp(A) implies that f (A) ≥ 0 on H. Therefore, if f and g are and then real valued functions on Sp(A) then the following basic D 2 E 2 2 property holds: [ f (A)g(A)] x,x − g (B)y,y f (A)x,x D E − f 2(B)y,y g2(A)x,x + [ f (B)g(B)]2 y,y D 2 E f (t) ≥ g(t) for any t ∈ Sp(A) implies that f (A) ≥ g(A) ≤ [ f (A)g(A) − f (B)g(B)] x,x in the operator order of B(H). for any unit vector x,y ∈ H . In this paper we obtain some inequalities analogue to (1) for operators in the real space B (H ) of all self-adjoint We obtain the following result when to replace B by A and operators on H . y by x.

2. Main Results Corollary 2.2. Let f ,g be continuous functions defined on an interval J and f ,g ≥ 0. Then The main result is as following: 2 f 4(A)x,x ≤ f 2(A)x,x Theorem 2.1. Let f ,g be continuous functions defined on an interval J and f ,g ≥ 0. Then we have for any self-adjoint operator A ∈ B (H ) with spectra contained in J and all unit vectors x ∈ H . D 2 E 2 2 [ f (A)g(A)] x,x − g (B)y,y f (A)x,x References D E − f 2(B)y,y g2(A)x,x + [ f (B)g(B)]2 y,y [1] Aczél, J., 1956. Some general methods in the theory of D 2 E ≤ [ f (A)g(A) − f (B)g(B)] x,x functional equations in one variable. New applications of functional equations, Uspehi. Mat. Nauk (N. S.), for any self-adjoint operators A,B ∈ B (H ) with spectra 11(3), 3-68. contained in J and all unit vectors x,y ∈ . H [2] Bjelica, M., 1990. On inequalities for indefinite form. Anal. Numer. Theor. Approx., 19(2), 105-109. Proof. Let a1,a2,b1,b2 be positive scalars. Then using inequality (1), we have [3] Furuta, T., Micic Hot, J., Pecariˇ c,´ J., Seo, Y., 2005. Mond-Pecariˇ c´ Method in Operator Inequalities. In- 2 2
2 2
2 a1 − a2 b1 − b2 ≤ (a1b1 − a2b2) . equalities for Bounded Selfadjoint Operators on a Hilbert Space, Element, Zagreb. Suppose that m,n ∈ J. Since f (m) ≥ 0, g(m) ≥ 0 for all [4] Popoviciu, T., 1959. On an inequality. Gaz. Mat. Fiz. m ∈ J, putting a1 = f (m), a2 = f (n), b1 = g(m), b2 = g(n) in the inequality above we have Ser. A, 11(64), 451-461. [5] Tian, J., 2012. Reversed version of a generalized 2 2 2 2 [ f (m)g(m)] − [ f (m)g(n)] − [ f (n)g(m)] + [ f (n)g(n)] Aczél’s inequality and its application, J. Inequal. Appl., ≤ [ f (m)g(m) − f (n)g(n)]2 (4) 2012: 202. [6] Tian, J., Ha, M.H., 2018. Properties and refinements for all m,n ∈ J. Let A be a self-adjoint operator. Then, by Aczél-type inequalities. J. Math. Inequal., 12(1), 175- using functional calculus for operator A to inequality (4) 189. we have [7] Vasic,´ P.M., Pecariˇ c,´ J.E., 1982. On Hölder and some [ f (A)g(A)]2 − [ f (A)g(n)]2 − [ f (n)g(A)]2 + [ f (n)g(n)]2 related inequalities. Mathematica Rev. D’Anal. Num. Th. L’Approx., 25, 95-103. ≤ [ f (A)g(A) − f (n)g(n)]2 , [8] Vasic,´ P.M., Pecariˇ c,´ J.E., 1979. On the Jensen inequal- and hence ity for monotone functions. An. Univ. Timi¸soaraSer. ¸St.Matematice, 17(1), 95-104. D E [ f (A)g(A)]2 x,x − g2(n) f 2(A)x,x − f 2(n) g2(A)x,x +[ f (n)g(n)]2 D E ≤ [ f (A)g(A) − f (n)g(n)]2 x,x

41 5Th INTERNATIONAL CONFERENCE ON RECENT ADVANCES IN PURE AND APPLIED MATHEMATICS (ICRAPAM 2018) JULY 23-27 2018 at Karadeniz Technical University Prof. Dr. Osman Turan Convention Center in Trabzon, TURKEY ICRAPAM CONFERENCE PROCEEDING

Coefficient bound estimates for beta-starlike functions of order alpha

Nizami Mustafa∗1, Veysel Nezir1 1Kafkas University, Faculty of Science and Letters, Department of Mathematics, Kars, Turkey

Keywords Abstract: In this paper, we investigate coefficients problems for the beta starlike functions Starlike function, of order alpha, which is analytic and univalent in the open unit disk in the complex plane. Convex function, We give the sharp estimates for the first three coefficients of the functions belonging to Beta-starlike function of order this class. The main results obtained here for the coefficients bounds estimates extend the alpha, results obtained by Thomas recently. Coefficient problem

1. Introduction and preliminaries (see for example [4, 5, 11]). Thus, Mβ is a natural subclass ∗ of S. It is clear that M0 = S and M1 = C. Let A be the class of analytic functions f (z) in the open Furthermore, for α,β ∈ R a function f ∈ A is called β − unit disk U = {z ∈ C : |z| < 1} in the complex plane of convex function of order α (α ∈ R) and it is denoted f ∈ the form Mβ (α), if the following condition is satisfied 2 3 n f (z) = z + a2z + a3z + ··· + anz + ···   z f 00 (z) z f 0 (z) ∞ ℜ β 1 + + (1 − β) > α, z ∈ U. n f 0 (z) f (z) = z + ∑ anz ,an ∈ C, (1) n=2 In the special case, for α = 0 we have Mβ (0) = Mβ . normalized by f (0) = 0 = f 0 (0) − 1. We denote by S the Inspired by the mentioned above studies, we define the subclass of A consisting of functions which are also uni- following subclass of analytic functions. valent in U. Some of the important and well-investigated subclasses of the univalent function class S are the classes Definition 1.1. Let α, β ∈ R. A function f ∈ A given S∗ and C, respectively, starlike and convex in the open unit by (1) is said to be in the class Mβ (α) if the following disk U. It is well-known that a function f ∈ A belongs to condition is satisfied S∗ and C if and only if ( 00 β  0 1−β ) n  z f 0(z)  o z f (z) z f (z) S∗ = f ∈ A : ℜ > 0, z ∈ U and ℜ 1 + > α, z ∈ U. (3) f (z) f 0 (z) f (z) n  z f 00(z)  o C = f ∈ A : ℜ 1 + f 0(z) > 0, z ∈ U , respectively. It is easy to verify that C ⊂ S∗ ⊂ S. For We will call the class Mβ (α) of the analytic functions details on these classes, one could refer to the monograph β-starlike functions of order α. by Goodman [3] and by Srivastava and Owa [9]. Moreover, a function f ∈ A is called starlike and convex The class Mβ (α) and similar subclasses of analytic func- function of order α ∈ R in the open unit disk U and these tions have been studied by many mathematicians (see for are denoted S∗ (α) and C (α), respectively, if and only if example [2, 4, 5, 11] and references therein). In contrast, the defined above class Mβ (α) has been less well studied. In the special case very recently the class Mβ (0) = Mβ z f 0 (z)  z f 00 (z) ℜ > α and ℜ 1 + > α, z ∈ U. (2) was examined by Thomas [10]. f (z) f 0 (z) Note that M0 (α) = S∗ (α) and M1 (α) = C (α). The pres- ∗ ∗ ence of powers in (3) obviously creates difficulties, and In the special case, for α = 0 we have S (0) = S and is probably the reason why relatively little appears to be C ( ) = C 0 , respectively. known about Mβ (α). However, as in the case of M (α), For β ∈ R, the class M of β − convex functions in U β β functions in Mβ (α) are also contained in S∗ [6], again defined by providing a natural subclass of S.   z f 00 (z) z f 0 (z) In this paper, we give a series of sharp inequalities involv- ℜ β 1 + + (1 − β) > 0, z ∈ U, f 0 (z) f (z) ing the first three coefficients of the functions belonging to the class Mβ (α). The main results obtained here for the is well-known, and contains a great many interesting prop- coefficients bounds estimates extend the results obtained ∗ erties, the most basic being that for all β ∈ R, Mβ ⊂ S by Thomas [10] very recently.

∗ Corresponding author: [email protected] 42 /  √  To prove our main results, we shall require the following for β ≤ 7 + 57 /2. well-known lemmas (see e.g. [1, 7, 8]). In these following  √  lemmas, we denote by P the set of functions p(z) ana- In the case β ≥ 7 + 57 /2, the equality (6) we write ∞ lytic in U of the form p(z) = 1 + ∑n=1 pnz, and satisfying as follows ℜp(z) > 0 for z ∈ U.   1 − α λ 2 a3 = p2 − p1 , (10) Lemma 1.2. Let p ∈ P. Then |pn| ≤ 2 for all n = 1,2,3,... 2(1 + 2β) 2 and
2  where λ = (1 − α) β 2 − 7β − 2 /(1 + β) . It is clear λ 2 2, if λ ∈ [0,2],  √  p2 − p ≤ max{2,2|λ − 1|} = 2 1 2|λ − 1|, elsewhere. that λ ≥ 0 for β ≥ 7 + 57 /2 and α ∈ [0,1) . Also, 2 2 3 we can easily see that β − 7β − 2 ≤ 2(1 + β) or Lemma 1.3. Let p ∈ P. Then, p3 − 2Bp1 p2 + Dp ≤ 2 1 2 − −
/( + )2 ≤ ≥ if B ∈ [0,1] and B(2B − 1) ≤ D ≤ B. β 7β 2 1 β 2 for β 0. It follows that (1 − α)β 2 − 7β − 2
/(1 + β)2 ≤ 2, since 0 < 1−α ≤ 1; 2. Coefficient bound estimates for the function class that is, λ ≤ 2. Mβ (α) Thus, using Lemma 1.2, we obtain the second inequality for |a3|. In this section, our main aim is to give sharp estimates for Similarly to proof of the inequalities for |a3|, we obtain the first three coefficients of the functions belonging to the first inequality for |a4|, and using Lemma 1.3 we obtain class Mβ (α). second inequality for |a4|. Theorem 2.1. Let the function f (z) given by (1) belong Now, we need to show that the inequalities obtained in the to the class Mβ (α), β ≥ 0, α ∈ [0,1). Then, theorem are sharp. Really if the function f (z) in (4) is   chosen so that  (1−α)(2+7β−β 2)  1−α 2(1 − α) 1+2β 1 + 2 , if β ∈ [0,β0], |a2| ≤ ,|a3| ≤ (1+β)  00 β  0 1−β 1 + β  1−α , ≥ , z f (z) z f (z) 1 + (1 − 2α)z 1+2β if β β0 1 + = , f 0 (z) f (z) 1 − z   (1−α)(−4β 2+19β+3)  1 + +  2(1−α) (1+β)(1+2β)  2 , if ∈ [0, ], then inequality obtained in the theorem for |a2| and the first  3(1+3β)  (1−α) (8β 4−47β 3+154β 2+23β+6)  β β1 |a4| ≤ 3(1+β)3(1+2β) inequalities for |a3| and |a4| are sharp; that is, equality is   2(1−α) |a | 3(1+3β) , if β ≥ β1, attained in the inequality for 2 , and the first inequalities  √  for |a3| and |a4|. The second estimate for |a3| is sharp if where β0 = 7 + 57 /2 and β1 = 6.794. All the inequal- we chose ities obtained here are sharp.  z f 00 (z)β z f 0 (z)1−β 1 + (1 − 2α)z2 1 + = . Proof. Let f ∈ Mβ (α), β ≥ 0, α ∈ [0,1). Then, f 0 (z) f (z) 1 − z2  00 β  0 1−β z f (z) z f (z) The second estimate for |a | is sharp when 1 + = α + (1 − α) p(z), (4) 4 f 0 (z) f (z)  z f 00 (z)β z f 0 (z)1−β 1 + (1 − 2α)z3 where p ∈ P. Comparing the coefficients of the like power 1 + = . f 0 (z) f (z) 1 − z3 of z in both sides of the above equality, we can write 1 − α Thus, the proof of Theorem 2.1 is completed. a2 = p1, (5) By setting = 0 in Theorem 2.1, we arrive at the follow- 1 + β α ing corollary which confirms the results obtained in [10, Theorem 3.1]. 2
(1 − α) −β 2 + 7β + 2 1 − α a = p2 + p , (6) Corollary 2.2. Let the function f (z) given by (1) belong 3 2 1 2(1 + 2β) 2 4(1 + β) (1 + 2β) to the class Mβ , β ≥ 0. Then, 3 4 3 2 (1−α) (8β −47β +154β +23β+6) 3 a4 = p + 36(1+β)3(1+2β)(1+3β) 1 ( 3(1+3β) 2 2 (7) 2 , if β ∈ [0,β0], (1−α) (−4β +19β+3) − (1+β)2(1+2β) p p + 1 α p . |a2| ≤ ,|a3| ≤ 6(1+β)(1+2β)(1+3β) 1 2 3(1+3β) 3 1 + β 1 1+2β , if β > β0, By applying the inequality |p1| ≤ 2, from (5) we have

2(1 − α)  2(2β 4+7β 3+292β 2+113β+18) |a2| ≤ . (8)  3 , if β ∈ [0,β1], 1 + β |a4| ≤ 9(1+β) (1+2β)(1+3β) 2  , if β > β1, Since the coefficients of p2 and p in the equality (6) are 3(1+3β) 1 2 √   √ nonnegative when β ∈ [0,β0], where β0 = 7 + 57 /2,   where β0 = 7 + 57 /2 and β1 = 6.794. All the inequal- by applying the inequalities |pn| ≤ 2, n = 1,2, we obtain ities are sharp. " 2
# 1 − α (1 − α) 2 + 7β − β By setting β = 0 and β = 1 in Theorem 2.1, we obtain the |a3| ≤ 1 + (9) 1 + 2β (1 + β)2 following results, respectively. 43 /

Corollary 2.3. Let the function f (z) given by (1) belong [3] Goodman, A. W. 1983. Univalent Functions. Volume to the class S∗ (α), α ∈ [0,1). Then, I, Polygonal. Mariner Comp., Washington, 246p. [4] Kulshrestha, P. K. 1974. Coefficients for alpha-convex |a2| ≤ 2(1 − α), |a3| ≤ (1 − α)(3 − 2α), univalent functions. Bull. Amer. Math. Soc., 80, 341- h i 2(1 − α) 2(1 − α)2 + 4 − 3α 342. |a | ≤ . 4 3 [5] Miller, S. S., Mocanu, P. T., Reade, M. O. 1973. All All the estimates are sharp. α-convex functions are univalent and starlike. Proc. Amer. Math., 37, 553-554. Corollary 2.4. Let the function f (z) given by (1) belong to the class C (α), α ∈ [0,1). Then, [6] Lewandowski, Z., Miller, S., Zlotkiewicz, E. J. 1974. Gamma-starlike functions. Ann. Univ. Mariae Curie- (1 − α)(3 − 2α) Sklodowska Sect. A, 28, 32-36. |a2| ≤ 1 − α, |a3| ≤ , 3 [7] Libera, R. J., Zlotkiewicz, E. J. 1982. Early coef- h i (1 − α) 2(1 − α)2 + 4 − 3α ficients of the inverse of a regular convex functions. |a | ≤ . Proc. Amer. Math. Soc., 85, no. 2, 225-230. 4 6 [8] Pommerenke, C. H. 1975. Univalent Functions. Van- All the estimates are sharp. denhoeck and Rupercht, Göttingen. Remark 2.5. By choose α = 0 in Corollary 2.3 and 2.3, we [9] Srivastava, H. M., Owa, S. 1992. Current Topics in An- can find coefficient estimates for the function subclasses alytic Function Theory. World Scientific, Singapore. S∗ and C, respectively. [10] Thomas, D. K. 2018. On the coefficients of gamma- References starlike functions. J. Korean Math. Soc., 55, no. 1, 175-184. [1] Ali, R. M. 2003. Coefficients of the inverse of strongly starlike functions. Bull. Malays. Ath. Sci. Soc. (2), 26, [11] Todorov, P. G. 1987. Explicit formulas for the coef- no. 1, 63-71. ficients of α-convex functions, . Canad. J. Math., 39, no. 4, 769-783. [2] Frasin, B. A., Vijaya, K., Hasthuri, M. 2016. TWMS J. Pure Appl. Math., 7, no. 2, 185-199.

44 5Th INTERNATIONAL CONFERENCE ON RECENT ADVANCES IN PURE AND APPLIED MATHEMATICS (ICRAPAM 2018) JULY 23-27 2018 at Karadeniz Technical University Prof. Dr. Osman Turan Convention Center in Trabzon, TURKEY ICRAPAM CONFERENCE PROCEEDING

On the fixed point property for a degenerate Lorentz-Marcinkiewicz Space

Veysel Nezir∗1, Nizami Mustafa1 1Kafkas University, Faculty of Science and Letters, Department of Mathematics, Kars, Turkey

Keywords Abstract: Recently, Nezir constructed an equivalent renorming of `1 and witnessed 1 Nonexpansive mapping, that it produced a degenerate ` -analog Lorentz-Marcinkiewicz space `δ,1, where the Nonreflexive Banach space, weight sequence = ( ) = (2,1,1,1,···) is a decreasing positive sequence in δ δn n∈N Fixed point property, ∞ 1 ` \c0, rather than in c0\` (the usual Lorentz situation). Then, he also obtained its Closed, bounded, convex set, 0 isometrically isomorphic predual `δ,∞ and dual `δ,∞, corresponding degenerate c0-analog Lorentz-Marcinkiewicz spaces and `∞-analog Lorentz-Marcinkiewicz spaces, respectively. In his study he investi- gated fixed point properties of the resulting spaces. In this study, we generalize his work by constructing another equivalent norm on `1 and obtaining his generalized 1 degenerate ` -analog Lorentz-Marcinkiewicz space `δ,1, where the weight sequence = ( ) = ( + , , , ,···), for ≥ > 0. Then, we show that ` fails to have δ δn n∈N β α α α α α β δ,1 the fixed point property (fpp) for nonexpansive mappings since they contain an asymp- totically isometric copy of `1 but we also show that there exists an infinite dimensional subspace of `δ,1 with fpp for nonexpansive mappings.

1. Introduction and Preliminaries By our recent paper in print [14], we gave positive an- swer for this question under affinity condition by a very There is strong relationship between the concept of Re- recent accepted paper. Unconditionally resolving this prob- flexive Banach spaces and Banach spaces having the fixed lem is still a research topic, and the reason for the diffi- point property. It has been questioned whether the two con- culty of solving is that c does not offer adequate tools cepts are equivalent unconditionally or depending on some 0 for researchers; on the contrary, the `1 space provides pos- conditions. For example, it was showed that under the con- sibilities facilitating the research, such as having Schur ditions of some geometric properties for Banach spaces property or having weak Opial property, although it has such as uniform convexity or normal structure, reflexiv- many common properties with c . Thus, for the aim of ity implies fixed point property. Less contingent relations 0 solving the aforementioned precious question, c -analogue have become a matter of curiosity and the role of the norm 0 of P. K. Lin’s result, or to construct bridges that can yield of the Banach space in the complete equivalence of the a solution method for that, due to possibility of offering two concepts has recently become the center of interest. more tools, researchers have showed interests in working It has been observed that most non-reflexive classical Ba- 1 1 on ` -like and c0-like spaces. nach spaces such as ` and c0 fail the fixed point property and it is still and has been an open question for over 50 One of Banach spaces offering the closest features to `1’s is 1 years whether or not all nonreflexive Banach spaces can ` - analogue Lorentz-Marcinkiewicz space lw,1 that I stud- be renormed to have the fixed point property. In 2008, ied in the Ph.D. thesis of the firs author [13], supervised by P. K. Lin gave the first example of a non-reflexive space Lennard, and its fixed point theory oriented properties were which can be renormed to have the fixed point property studied but it has not been proven whether there are com- [10]. He verified this fact by renorming `1. Banach spaces pletely similar features even in this space. For example, `1 containing nice copies (asymptotically isometric copies) of has the weak fixed point property which means every non- 1 ` or c0 cannot have the fixed point property. Futhermore, expansive self mapping defined on any weakly compact if a Banach space is a Banach latis or has an uncontional subset has a fixed point while it fails the fixed property basis then it is non-reflexive if and only if it contains either meaning there exist a closed, bounded and convex subset 1 an isomorphic copy of ` or c0. Because these two non- and a fixed point-free nonexpansive self mapping defined 1 reflexive Banach spaces ` and c0 share many common on that set; and in the literature, it was showed that there properties and they are the fundamental examples of non- exists an infinite dimensional subspace of `1 constructed reflexive Banach spaces, in order to investigate whether or by non-weakly* compact, closed, bounded and convex not non-reflexive Banach spaces can be renormed to have subsets that has the fixed point property [6] whereas any the fixed point property, investigating the question on the infinite dimensional subspace of c0 fails the fixed point second example c0 has become one of the most important property. However, the correponding questions of the last subjects for the researchers. two results for `1- analogue Lorentz-Marcinkiewicz space

∗ Corresponding author: [email protected] 45 /

0 lw,1 and c0-analogue Lorentz-Marcinkiewicz space lw,∞; This space is non-separable and an analogue of l∞ space. respectively, are still open. With our most recent work, which is under review, we have found a Banach space Definition 1.4. which is a degenerate Lorentz-Marcinkiewicz space and ( n ? ) ∑ j=1 x j some questions focusing on fixed point properties of the l0 := x = (x ) ∈ c limsup = 0 . w,∞ n n∈N 0 n w space itself, its predual and its dual have been solved. n−→∞ ∑ j=1 j In this paper, we partically generalize our recent work This is a separable subspace of l and an analogue of c and construct another equivalent norm on `1, then we w,∞ 0 space. obtain our generalized degenerate `1-analog Lorentz- Marcinkiewicz space `δ,1, where the weight sequence Definition 1.5. = ( ) = ( + , , , ,···), for ≥ > 0. Then, δ δn n∈N β α α α α α β ( ) ` ∞ we show that δ,1 fails to have the fixed point property l x x c x w x ? w,1 := = ( n)n∈N ∈ 0 k kw,1 := ∑ j j < ∞ . (fpp) for nonexpansive mappings since they contain an j=1 asymptotically isometric copy of `1 but we also show that there exists an infinite dimensional subspace of `δ,1 with This is a separable subspace of lw,∞ and an analogue of l1 0 ? ∼ ? ∼ fpp for nonexpansive mappings. space with the following facts: (lw,∞) = lw,1 and (lw,1) = ∼ Now, we give preliminaries including our recent paper’s lw,∞ where the star denotes the dual of a space while = results below that would lead readers understand similar denotes isometrically isomorphic. results can be given for our Banach space as well. More information about Lorentz spaces can be seen in Definition 1.1. Let (X,k · k) be a Banach space and C be [11]. a non-empty closed, bounded, convex subset. We also need the following theorems and definitions to 1. If T : C −→ C is a mapping such that for all λ ∈ obtain our results.
[0,1] and for all x,y ∈ C, T (1 − λ)x + λ y = (1 − Theorem 1.6. [9] Let X be a Banach space. If X has λ)T(x) + λ T(y) . then T is said to be an affine map- an unconditional√ basis (en) with unconditional constant ping. 33−3 λ < 2 , then X has the w-fpp. 2. If T :C −→C is a mapping such that kT(x)−T(y)k ≤ Theorem 1.7. [8] If X is a Banach space containing an kx − yk , f or all x,y ∈ C then T is said to be a isomorphic copy of c , then X fails the fixed point property nonexpansive mapping. 0 for affine strongly asymptotically nonexpansive mappings. Also, if for every nonexpansive mapping T : C −→ C, there exists x ∈C with T(x) = x, then X is said to have Definition 1.8. [3] We call a Banach space (X,k · k) con- 1 the fixed point property for nonexpansive mappings tains an ai copy of ` if there exist a sequence (xn)n in X [FPP(ne)]. and a null sequence (εn)n in (0,1) so that

3. If T : C −→C is a mapping such that ∃{αn,m : n,m ∈ ∞ ∞ ∞

N,n ≥ m ≥ 0} ⊆ [1,∞) such that [∀x,y ∈ K and ∀n ≥ ∑ (1 − εn)|an| ≤ ∑ anxn ≤ ∑ |an| , n n m m n=1 n=1 n=1 m, kT x − T yk ≤ αn,mkT x − T yk], [αn,m → 1 as n ≥ m → ∞], and [α → 1 as n → ∞ ,∀m] then T is n,m for all (a ) ∈ `1. said to be a strongly asymptotically nonexpansive [8]. n n Definition 1.2. We should note that in the first author’s Definition 1.9. [3] We call a Banach space (X,k · k) con- Ph.D. thesis [13], written under supervisor of Chris tains an ai copy of c0 if there exist a sequence (xn)n in X Lennard, we studied the usual Lorentz-Marcienkiewicz and a null sequence (εn)n in (0,1) so that spaces and their fixed point properties; hence, we can give ∞ their definitions below to understand how different the sup(1 − εn)|an| ≤ ∑ anxn ≤ sup|an| , degenerate ones are. n n=1 n 1 + Let w ∈ (c0 \ ` ) , w1 = 1 and (wn)n∈N be decreasing; for all (an)n ∈ c0. that is, consider a scalar sequence given by w = (wn)n∈N, wn > 0,∀n ∈ N such that 1 = w1 ≥ w2 ≥ w3 ≥ ··· ≥ Theorem 1.10. [3] If a Banach space (X,k · k) contains wn ≥ wn+1 ≥ ...,∀n ∈ N with wn −→ 0 as n −→ ∞ and an ai copy of `1 or an ai copy of c , then X fails FPP(n.e). ∞ 0 ∑n=1 wn = ∞. This sequence is called a weight sequence. 1 For example, wn = n ,∀n ∈ N. 1.1. Results of a recent paper: Fixed point properties Definition 1.3. for a degenerate Lorentz-Marcinkiewicz Space ( n ? ) ∑ x j The followings are the results of the first author’s re- l x x c x j=1 w,∞ := = ( n)n∈N ∈ 0 k kw,∞ := sup n < ∞ . cently submitted paper entitled “Fixed point properties n∈N ∑ j=1 w j for a degenerate Lorentz-Marcinkiewicz Space". For all Here, x? represents the decreasing rearrangement of the x = (x ) ∈ `1, we define n n∈N sequence x, which is the sequence of |x| = (|x j|) j∈N, ar- ranged in non-increasing order, followed by infinitely many ∞ ~x~ := kxk + kxk = |x | + sup|x | . |x| 1 ∞ ∑ n n zeros when has only finitely many non-zero terms. n=1 n∈N 46 /

1 1 0 Clearly ~ · ~ is an equivalent norm on ` with kxk1 ≤ Theorem 1.14. ∀δ ∈ c0 \ ` , ∃Z ⊆ lδ,∞ such that Z is an x ≤ kxk , ∀x ∈ `1 0 ~ ~ 2 1 . ai copy of c0 and so l fails the fixed point property for 1 δ,∞ We shall call ~ · ~ the 1
∞-norm on ` . affine, k · k -nonexpansive mappings. 1 ∗ ∗ ∗ ∗ ∗ δ,∞ Note that ∀x ∈ ` , ~x~ = 2x1 +x2 +x3 +x4 +··· where z is the decreasing rearrangement of |z| = (|z |) , ∀z ∈ c . Theorem 1.15. Define n n∈N 0 Let δ1 := 2,δ2 := 1,δ3 := 1,··· ,δn := 1, ∀n ≥ 4. 1 E := {u = (s1 δ1,s2 δ2,s3 δ3,...) |s ∈ c0 ,1 = s1 ≥ s2 ≥ ··· ≥ 0 } We see that (` ,~ · ~) is a (degenerate) Lorentz space `δ,1, where the weight sequence = ( ) is a decreasing δ δn n∈N . ∞ 1 positive sequence in ` \c0, rather than in c0\` (the usual 0 Then, E ⊆ lδ,∞ is a convex, closed, bounded set, and ∃ T : Lorentz situation). E → E s.t. T is fixed point free, k · k -nonexpansive, 0 δ,∞ This suggests that `δ,∞ = (c0,k · k) is an isometric predual affine mapping. 1 of (` ,~ · ~) where for all z ∈ c0, Theorem 1.16. Let Z be any closed, non-reflexive vector n 0 ∗ subspace of lδ,∞. Then, Z contains an isomorphic copy ∑ z j n j=1 1 ∗ of c0 and so (Z,k · kδ,∞) fails the fixed point property for kzk := sup n = sup ∑ z j . strongly asymptotically nonexpansive maps. n∈ n∈ n + 1 N ∑ δ j N j=1 j=1 1 Theorem 1.17. `δ,1 contains an ai copy of ` and so it fails the fixed point property for · -nonexpansive mappings. But there is a way to re-write kzk without using decreasing ~ ~ rearrangements of |z|. This may help with calculations Theorem 1.18. `δ,1 fails fpp for affine ~ · ~-nonexpansive involving this norm. mappings Fix z ∈ c0, arbitrary. Lemma 1.19. `δ,1 has the weak fixed point property. n 1 ∗ kzk = sup z . 1 ∗ ∑ j Lemma 1.20. ` , = (` ,~ · ~) has the weak fixed point n∈ n + 1 δ 1 N j=1 property for nonexpansive mappings with respect to the 0 n predual ` = (c0,k · k). ∗ δ,∞ Note that ∀n ∈ N, ∑ z j = sup ∑ |zi|, where #(K) is j=1 K⊆N i∈K 1 ∗ #(K)=n Proof. In order to prove `δ,1 = (` ,~ · ~) has the weak the number of elements in K for all finite subsets K ⊆ N. fixed point property for nonexpansive mappings, we use Thus, the strategy given in [5, 7, Theorem 8.9, Theorem 4.3; respectively]. There one can see that proof depends on 1 1 z z Goebel and Kuczumow’s lemma [6] for ` with its usual k k = sup sup ∑ | i| 1 n∈N n + 1 K⊆N i∈K norm. Their lemma says that if {xn} is a sequence in l #(K)=n converging to x in weak-star topology, then for any y ∈ l1, 1 = sup sup |zi|. ∑ r (y) = r (x) + ky − xk where r (y) = limsupkxn − yk . n∈N K⊆N #(K) + 1 i∈K 1 1 #(K)=n n

Now we note that `δ,1 is isometrically isomorphic to sub- Hence, for all z ∈ c0, space 1 X := (x∗,x∗,x∗,x∗,x∗,x∗,x∗,···) : x = (x ) ∈ (`1,k · k ) . kzk = sup |z |. (1) 1 1 2 3 4 5 6 n n∈N 1 ∑ i Then, clearly we obtain Goebel and Kuczumow’s result /06=K⊆N #(K) + 1 i∈K #(K)<∞ above for the equivalent norm ~ · ~ and noting that B :=  1 ∗ B 1 := x ∈ ` : ~x~ ≤ 1 is a weak -compact set by ∞ (` ,~·~) Also, note that formula 3 can be extended to ` : ∀w = the Banach-Alaoglu Theorem [7, Proposition 6.13] which (w ) ∈ `∞, we define i i∈N concludes our proof. Note that the above result clearly says that every · - 1 ~ ~ kwk := sup ∑ |wi|. (2) nonexpansive mapping T : B −→ B has a fixed point. /06=K⊆N #(K) + 1 i∈K #(K)<∞ 2  Example 1.21. Fix b ∈ 0, 3 . Define the sequence ( fn)n∈ in c0 by setting f1 := be1, f2 := be2 and fn := en, Lemma 1.11. Dual space of (`1,~ · ~) is isometrically N ∞ 1 ∗ ∞ for all integers n ≥ 3. Next, define the closed, bounded, isomorphic to (` ,k · k); i.e., (` ,~ · ~) =∼ (` ,k · k). 1 convex subset E = Eb of ` by ∞ Remark 1.12. B ∞ = {x ∈ ` : kxk ≤ 1} has the (` ,k·k∞) ( ∞ ∞ ) fixed point property for nonexpansive mappings since it E := ∑ tn fn : each tn ≥ 0 and ∑ tn = 1 . is a hyperconvex metric space and hyperconvex metric n=1 n=1 spaces have fpp(ne) by Soardi [15]. Theorem 1.22. The set E defined as in the example 0 Theorem 1.13. Let X := lδ,∞ and let k · k = k · kδ,∞ as we above has the fixed point property for ~.~-nonexpansive discussed in above notes. Then, Banach space (X,k · k) mappings where the norm ~·~ on `1 is given as below: 1 has the weak fixed point property. ~x~ = kxk1 + kxk∞, ∀x ∈ ` . 47 /

2. Main Results: Generalizing the idea Proof. First of all, considering canonical basis (e ) , n n∈N 1 1 define the sequence (Qn)n∈ in ` by Q1 := e1, 1 N β+α Let ≥ > 0. For all x = (x ) ∈ ` , we define 1 1 β α n n∈N Q2 := β+2α (e2 + e3), Q3 := β+3α (e4 + e5 + e6), Q4 := ∞ 1 β+4α (e7 +e8 +e9 +e10),··· and note that ~Qn~ = 1, ∀n ∈ ~x~ := αkxk1 + βkxk = α |xn| + β sup|xn| . 1 ∞ ∑ and that for any x := (ξn) ∈ ` , there is unique rep- n=1 n∈N N n∈N resentation of x with scalars µ1 = (β + α)ξ1, µ2 = (β + 1 2α)ξ , µ = (β + 3α)ξ , ··· µ = (β + nα)ξ , ∀n ≥ 4 Clearly ~ · ~ is an equivalent norm on ` with αkxk1 ≤ 2 3 3 n n 1 ∞ ~x~ ≤ (β + α)kxk1, ∀x ∈ ` . such that x = ∑ µnQn. We shall call ~ · ~ the 1 ∞-norm on `1. n=1
1∗ 1  β+α ∗ ∗ ∗ ∗  Now, consider any f ∈ ` . Then since f is lin- Note that ∀x ∈ ` , ~x~ = α α x1 + x2 + x3 + x4 + ··· ∞ ∞ ∗ ear and bounded, f (x) = ∑ µn f (Qn) = ∑ µnγn, γn := where z is the decreasing rearrangement of |z| = n=1 n=1 (|z |) , ∀z ∈ c . n n∈N 0 f (Qn), ∀n ∈ N. Let δ1 := (β + α),δ2 := α,δ3 := α,··· ,δn := α, ∀n ≥ 4. So ( ) ∈ `∞. γn n∈N 1 1 We see that (` ,~ · ~) is a (degenerate) Lorentz space `δ,1, Also, sup β+#(K)α ∑ |γi| ≤ sup|γk| ≤ k f k∗ = where the weight sequence = ( ) is a decreasing /06=K⊆N i∈K k∈ δ δn n∈N N positive sequence in `∞\c , rather than in c \`1 (the usual #(K)<∞ 0 0 sup | f (x)|. ♥ where k · k denotes the operator norm. Lorentz situation). ∗ x∈`1 0 This suggests that ` = (c0,k · k) is an isometric predual ~x~=1 δ,∞ ∞ 1 Furthermore, if (γn)n∈ ∈ ` is given arbitrarily, we can of (` ,~ · ~) where for all z ∈ c0, N ∗ construct linear functionals in `1 with the elements of `∞ n as follows: ∗ ∞ ∑ z j n j=1 1 ∗ g(x) = ∑ µkγk. kzk := sup n = sup ∑ z j . n∈ n∈ β + nα k=1 N ∑ δ j N j=1 j=1 Showing linearity is no problem and for the boundedness of the linear functional g, we will use the inequality 5.2(5) But there is a way to re-write kzk without using decreasing in [12, page 52]. Then we have rearrangements of |z|. This may help with calculations involving this norm. ∞ |g(x)| ≤ |µ γ | Fix z ∈ c , arbitrary. ∑ k k 0 k=1 ∞ 1 n 1 ∗ ∗ ≤ sup |γi| µ kzk = sup ∑ z j . β + #(K)α ∑ ∑ k n∈ β + nα /06=K⊆N i∈K k=1 N j=1 #(K)<∞

n 1 Note that ∀n ∈ , z∗ = sup |z |, where #(K) is ≤ sup ∑ |γi|~x~ N ∑ j ∑ i /06=K⊆ β + #(K)α j=1 K⊆ i∈K N i∈K N #(K)<∞ #(K)=n the number of elements in K for all finite subsets K ⊆ . N Thus, indeed the functional g is bounded and by the linear- Thus, ∗ ity, g ∈ `1 . Also, the above inequality says that for any ∗ 1 f ∈ `1 and for any x ∈ `1, kzk = sup sup ∑ |zi| 1 β + nα k f k∗ = sup | f (x)| ≤ sup β+#(K)α ∑ |γi| = n∈N K⊆N i∈K i∈K #(K)=n ~x~=1 /06=K⊆N #(K)<∞ 1 k(γk) k. ♥♥ = sup sup ∑ |zi|. k∈N n∈N K⊆N β + #(K)α i∈K Hence, ♥ together with ♥♥ tells us that the norm is pre- #(K)=n served; i.e., k f k∗ = k(γk)k∈ k. Therefore, the isorphism N ∗ between the two given normed spaces `1 and `∞ is a fact. Hence, for all z ∈ c0, ∗ So taking an element out of `1 is in certain sense the same 1 as speaking about an element out of `∞. kzk = sup ∑ |zi|. (3) /06=K⊆ β + #(K)α 0 N i∈K Remark 2.2. Let X := lδ,∞ and let k·k = k·kδ,∞ as we dis- #(K)<∞ cussed in above notes. Then, clearly due to Lin’s theorem Also, note that formula 3 can be extended to `∞: ∀w = about unconditionality constant√ of the space’s uncondi- ∞ tional basis 1.6, which is 1 < 33−3 for our space, and so (wi)i∈ ∈ ` , we define 2 N Banach space (X,k · k) has the weak fixed point property. 1 kwk := sup |wi|. (4) ∑ 2.1. Fixed Point Properties for `δ,1. /06=K⊆N β + #(K)α i∈K #(K)<∞ As we discussed in Section 2, recall that `1 with equiva- 1 Lemma 2.1. Dual space of (` ,~ · ~) is isometrically iso- lent norm ~ · ~ is a (degenerate) Lorentz space `δ,1 with ∗ morphic to (`∞,k · k); i.e., (`1, · ) ∼ (`∞,k · k). the weight sequence = ( ) = ( + , , , ,···) ∈ ~ ~ = δ δn n∈N β α α α α 48 /

∞ c \ `1 such that for all x = (x ) ∈ `1, x = x∗ Theorem 2.5. The set E defined as in the example above 0 n n∈N ~ ~ ∑ δn n n=1 has the fixed point property for ~.~-nonexpansive map- where x = (x∗) is decreasing rearrangement of x. 1 n n∈N pings where the norm ~·~ on ` is given as below: ~x~ = Now we show that `δ,1 contains an asymptotically iso- 1 kxk1 + kxk∞, ∀x ∈ ` . metric (ai) copy of `1 and so it fails to have fpp for non- expansive mappings but in the next subsection, we will show that there exists a large class of nonweakly∗ compact, Proof. We will be using the proof steps of Goebel and 1 Kuczumow given in detailed as in Everest’s Ph.D. thesis closed, bounded and convex subsets of `δ,1 = (` ,~ · ~) with fpp(ne) using the ideas of Goebel and Kuczumow [4], written under supervision of Lennard. Let T : E → E [6] where they show that there exists a large class of be a nonexpansive mapping. Then, there exists a sequence   nonweakly∗ compact, closed, bounded and convex sub- x(n) ∈ E such that Tx(n) − x(n) −→ 0 and so 1 n∈N n sets of (` ,k · k1) with fpp(ne).   (n) (n)   Tx − x −→ 0. Without loss of generality, passing 2.1.1. ` contains an ai copy of `1. 1 n δ,1 to a subsequence if necessary, there exists z ∈ `1 such that x(n) converges to z in weak∗ topology. Then, by the proof Theorem 2.3. ` contains an ai copy of `1 and so it fails δ,1 of Lemma 1.20, we can define a function s : `1 −→ [0,∞) the fixed point property for · -nonexpansive mappings. ~ ~ by Proof. Consider the sequence (Q ) constructed as in n n∈N Lemma 2.1 using canonical basis (e ) given by Q := s(y) = limsup x(n) − y , ∀y ∈ `1 n n∈N 1 1 1 1 n e1, Q2 := (e2 + e3), Q3 := (e4 + e5 + e6),   β+α β+2α β+3α   Q := 1 (e +e +e +e ),··· and note that ~Q ~ =   3 β+4α 7 8 9 10 n and so 1, ∀n ∈ N. Then for all t = (t ) ∈ `1, 1 n n∈N s(y) = s(z) + ~y − z~ , ∀y ∈ ` . t 1 e +t 1 (e + e ) ∞ 1 β+α 1 2 β+2α 2 3 +t 1 (e + e + e ) Next, define ∑ tnQn =  3 β+3α 4 5 6  n=1   +t 1 (e + e + e + e ) + ···     4 β+4α 7 8 9 10  ( )     ∗ ∞ ∞   " α 2α 3α #  w    β+α |t1| + β+2α |t2| + β+3α |t3|  W := E = ∑ tn fn : each tn ≥ 0 and ∑ tn ≤ 1  4α  n=1 n=1 + β+4α |t4| + ··· = " β W β W β # |t1| |t2| |t3| β+α β+2α β+3α Case 1: z ∈ E + W β . |t4| + ··· β+4α Then, we have s(Tz) = s(z) + ~Tz − z~ and ≤ |t1| + |t2| + |t3| + |t4| + ··· and s(Tz) = limsup Tz − x(n) n   ∞ ∞  (n) t Q ≥ (1 − ε )|t | ≤ limsupTz − T x ∑ n n ∑ n n n n=1  n=1      (n)  (n)   +limsup x − T x  where ε := α , ∀n∈ . n n β+nα  N    (n)  ≤ limsup z− x  2.1.2. An infinite-dimensional subspace of ` = n δ,1   1 = s(z).   (` ,~ · ~) with fpp(ne)  

In this section, we will show that there exists a large class Therefore, ~z − Tz~ ≤ 0 and so Tz = z. of nonweakly∗ compact, closed, bounded and convex sub- 1 Case 2: z ∈ W \ E. sets of `δ,1 = (` ,~ · ~) with fpp(ne) using the ideas of Goebel and Kuczumow [6] where they show that there ex- ∞ ∞ Then, z is of the form ∑ γn fn such that ∑ γn < ists a large class of nonweakly∗ compact, closed, bounded n=1 n=1 1 1 and γn ≥ 0, ∀n ∈ . and convex subsets of (` ,k · k1 with fpp(ne). N

Example 2.4. Fix b ∈ (0,1). Define the sequence ( fn)n∈N ∞ in c0 by setting f1 := be1, f2 := be2 and fn := en, for all Define δ := 1 − ∑ γn and next define integers n ≥ 3. Next, define the closed, bounded, convex n=1 1 ∞ subset E = Eb of ` by hλ := (γ1 + λδ) f1 + (γ2 + (1 − λ)δ) f2 + ∑ γn fn. n=3 ( ∞ ∞ ) E := ∑ tn fn : each tn ≥ 0 and ∑ tn = 1 . n=1 n=1 We want hλ to be in E, so we restrict values of λ to be in 49 /

 γ1 γ2  −1 − δ , δ + 1 , then Then, ∞ ~y − z~ = βb|t1 − γ1| + αb ∑ |tk − γk| ~hλ − z~ = ~λδ f1 + (1 − λ)δ f2~ k=1 = ~(λδb,(1 − λ)δb,0,0,···)~ ∞ +α(1 − b) ∑ |tk − γk| = bδβ max{|λ|,|1 − λ|} k=3 +bδα|λ| + bδα|1 − λ| ≥ αbδ + βb|t1 − γ1|  b ( + ) ∞  δ β α  −2bδαλ +α(1 − b) ∑ |tk − γk|  k=3   γ1
 −bδβλ if λ ∈ − δ , 0 ,  ≥ αbδ + βb|t1 − γ1| + α(1 − b)|δ    bδ(β + α) −(t1 − γ1) − (t2 − γ2)|   1
= max −bδβλ if λ ∈ 0, 2 , ≥ αbδ + βb|t1 − γ1| + α(1 − b)δ   −2α(1 − b)|t1 − γ1|  1   bδα + bδβλ if λ ∈ , 1 ,  2 = αδ − (2α(1 − b) − βb)|t1 − γ1|.    b (2 + ) δ  δλ α β Subcase 2.1.1: Assume 2 ≥ |t1 − γ1|.  γ2  −bδα if λ ∈ 1, δ + 1 Then clearly the last inequality from above says that b(2α+β)δ ~y − z~ ≥ 2 . δ Subcase 2.1.2: Assume 2 < |t1 − γ1|. Define Then ~y − z~ ≥ αbδ + βb|t − γ | Γ := min ~hλ − z~. 1 1 ∈ − − γ1 , γ2 + ∞ λ [ 1 δ δ 1] +α(1 − b) ∑ |tk − γk| k=3 Therefore, ~h − z~ is minimized when λ ∈ [0,1] with b(2α + β)δ λ ≥ . unique minimizer such that its minimum value would be 2 Γ = (2α+β)bδ . 2 Subcase 2.2: |t2 − γ2| ≥ |t1 − γ1| and b|t2 − γ2| ≥ |tk − ∞ ∞ γk|, ∀k ≥ 3. Now fix y ∈ E of the form ∑ tn fn such that ∑ tn = 1 n=1 n=1 Then, with tn ≥ 0, ∀n ∈ N. ∞ Then, ~y − z~ = βb|t2 − γ2| + αb ∑ |tk − γk| k=1 ∞ ∞ ∞ +α(1 − b) ∑ |tk − γk| k=3 ~y − z~ = tk fk − γk fk ∑ ∑ ∞ k=1 k=1    ≥ αbδ + βb|t2 − γ2| + α(1 − b) |tk − γk|  (t − γ )be + (t− γ )be + (t − γ )e ∑ =  1 1 1 2 2 2 3 3 3 k=3  +(t − γ )e + ···  4 4 4  ≥ αbδ + βb|t2 − γ2| + α(1 − b)|δ   ( + )b|t − | + b|t − |     α β 1 γ1 α 2 γ2   −(t1 − γ1) − (t2 − γ2)|   +α|t − γ | + α|t − γ |    3 3 4 4  ≥ αbδ + βb|t − γ | + α(1 − b)δ   2 2  +α|t5 − γ5| + ··· ,    −2α(1 − b)|t2 − γ2|  (α + β)b|t2 − γ2| + αb|t1 − γ1|    = αδ − (2α(1 − b) − βb)|t − γ |.  +α|t3 − γ3| + α|t4 − γ4|  2 2    +α|t5 − γ5| + ··· ,  δ   Subcase 2.2.1: Assume ≥ |t2 − γ2|.  (α + β)|t3 − γ3| + αb|t1 − γ1|  2   Then clearly the last inequality from above says that  +αb|t2 − γ2| + α|t4 − γ4|  = max ~y − z~ ≥ b(2α+β)δ . +α|t5 − γ5| + ··· , 2   δ  (α + β)|t4 − γ4| + αb|t1 − γ1|  Subcase 2.2.2: Assume 2 < |t2 − γ2|.    +αb|t2 − γ2| + α|t3 − γ3|  Then    +α|t5 − γ5| + ··· ,    ~y − z~ ≥ αbδ + βb|t2 − γ2|  (α + β)|t5 − γ5| + αb|t1 − γ1|    ∞  +αb|t2 − γ2| + α|t3 − γ3|    +α(1 − b) ∑ |tk − γk|  +α|t4 − γ4| + α|t6 − γ6| + ··· ,  k=3    ············  b(2α + β)δ ≥ . 2

Subcase 2.1: |t1 − γ1| ≥ |t2 − γ2| and b|t1 − γ1| ≥ |tk − Subcase 2.3: |t3 − γ3| ≥ b|t1 − γ1|, |t3 − γ3| ≥ b|t2 − γ2|, γk|, ∀k ≥ 3. and |t3 − γ3| ≥ |tk − γk|, ∀k ≥ 4. 50 /

Then, Thus, we continue in this way and see that ~y − z~ ≥ b(2α+β)δ ∞ 2 from all cases. ~y − z~ = β|t3 − γ3| + αb ∑ |tk − γk| Therefore, when λ is choosen to be in [0,1], for any y ∈ E k=1 and for z ∈W \E, ~y − z~ ≥ Γ such that there exists unique ∞ λ ∈ [0,1] with ~h − z~=Γ. Now, we can see that for +α(1 − b) |t − γ | 0 λ0 ∑ k k h = h , k=3 λ0 ∞ (n) ≥ bαδ + β|t3 − γ3| + α(1 − b) ∑ |tk − γk| s(Th) = limsup Th − x k=3 n   ≥ bαδ + β|t − γ | + α(1 − b)|δ  (n) 3 3 ≤ limsupTh − T x n −(t1 − γ1) − (t2 − γ2)|    (n)  (n) ≥ bαδ + β|t3 − γ3| + α(1 − b)δ +limsup x − T x  n 2(1 − b)   − |t − |  (n)  α 3 γ3 ≤ limsup h− x  b n 2α(1 − b) − bβ   = αδ − |t − γ |. = s(h).   b 3 3   Also, s(Th) = s(z) + ~z − Th~ and s(h) = s(z) + ~z − h~. Subcase 2.3.1: Assume bδ ≥ |t − |. 2 3 γ3 Hence, since Th ∈ E and Then clearly the last inequality from above says that b(2α+β)δ ~y − z~ ≥ 2 . ~z − Th~ ≤ ~z − h~ = Γ =⇒ Th = h Subcase 2.3.2: Assume bδ < |t − γ |. 2 3 3 (since the minimizer hλ0 is unique) Then Therefore, E has FPP (n.e.) as desired. ~y − z~ ≥ bαδ + β|t3 − γ3| ∞ Remark 2.6. Generalizing the sets as Goebel & Kuczumow +α(1 − b) ∑ |tk − γk| k=3 [6] or Everest [4] did, one can obtain larger classes with b(2α + β)δ fixed point property for nonexpansive mappings which ≥ . 2 could be considered as a future project for the other re- searchers in the field and for us as well. Subcase 2.4: |t4 − γ4| ≥ b|t1 − γ1|, |t4 − γ4| ≥ b|t2 − γ2|, and |t4 − γ4| ≥ |tk − γk|, ∀k ≥ 5 and for k = 3. Acknowledgment Then, The author is grateful to Chris Lennard for his valuable ∞ comments and helpful discussions on the subject. ~y − z~ = β|t4 − γ4| + bα ∑ |tk − γk| k=1 ∞ References +α(1 − b) ∑ |tk − γk| k=3 [1] Brouwer, L. E. J. 1911. Über abbildung von man- ∞ nigfaltigkeiten. Mathematische Annalen, 71, no. 1, ≥ bαδ + β|t4 − γ4| + α(1 − b) ∑ |tk − γk| 97–115. k=3 [2] Domínguez Benavides, T. 2009. A renorming of some ≥ bαδ + β|t4 − γ4| + α(1 − b)|δ nonseparable Banach spaces with the fixed point prop- −(t − γ ) − (t − γ )| 1 1 2 2 erty. Intern. J. Math. Anal. Appl., 350, no. 2, 525–530. ≥ bαδ + β|t4 − γ4| + α(1 − b)δ [3] Dowling, P. N., Lennard, C.J., Turett, B. 2001. 2(1 − b) 1 − α|t4 − γ4| Renormings of l and c0 and fixed point properties. b In: Handbook of Metric Fixed Point Theory, Springer 2α(1 − b) − bβ = δ − |t − γ |. Netherlands, 269–297. b 4 4 [4] Everest, T. 2013. Fixed points of nonexpansive maps bδ 1 Subcase 2.4.1: Assume 2 ≥ |t4 − γ4|. on closed, bounded, convex sets in l . Ph.D., Univer- Then clearly the last inequality from above says that sity of Pittsburgh, Pittsburgh, PA, USA. y − z ≥ b(2α+β)δ . ~ ~ 2 [5] Goebel, K., Kirk, W. A. 1990. Topics in metric fixed bδ Subcase 2.4.2: Assume 2 < |t4 − γ4|. point theory. Cambridge University Press. Then [6] Goebel, K., Kuczumow, T. 1979. Irregular convex sets ~y − z~ ≥ bδ + |t4 − γ4| with fixed-point property for nonexpansive mappings. ∞ Colloq. Math., 40, no. 2, 259–264. +(1 − b) ∑ |tk − γk| k=3 [7] Khamsi, M. A., Kirk, W. A. 2011. An introduction b(2α + β)δ to metric spaces and fixed point theory. John Wiley & ≥ . 2 Sons. 51 /

[8] Lennard, C. J., Nezir, V. 2017. Semi-strongly asymp- und ihrer Grenzgebiete, 92, Springer-Verlag. totically non-expansive mappings and their applica- [12] Lorentz, G. G. 1950. Some new functional spaces. tions on fixed point theory. Hacet. J. Math. Stat., 46, Ann. Math., 37-55. no. 4, 613–620. [13] Nezir, V. 2012. Fixed point properties for c0-like [9] Lin, P. K. 1985. Unconditional bases and fixed points spaces. Ph.D., University of Pittsburgh, Pittsburgh, PA, of nonexpansive mappings. Pac. J. Math., 116, no. 1, USA. 69–76. [14] Nezir, V., Mustafa, N. 2018. c can be renormed to [10] Lin, P. K. 2008. There is an equivalent norm on ` 0 1 have the fixed point property for affine nonexpansive that has the fixed point property. Nonlinear Analysis: mappings. Filomat, to appear. Theory, Methods & Applications, 68, no. 8, 2303- 2308. [15] Soardi, P. M. 1979. Existence of fixed points of non- expansive mappings in certain Banach lattices. Proc. [11] Lindenstrauss, J., Tzafriri, L. 1977. Classical Banach Amer. Math. Soc., 73, 25–29. spaces I: sequence spaces, Ergebnisse der Mathematik

52 5Th INTERNATIONAL CONFERENCE ON RECENT ADVANCES IN PURE AND APPLIED MATHEMATICS (ICRAPAM 2018) JULY 23-27 2018 at Karadeniz Technical University Prof. Dr. Osman Turan Convention Center in Trabzon, TURKEY ICRAPAM CONFERENCE PROCEEDING

On the coefficient bounds of certain subclasses of analytic functions of complex order

Nizami Mustafa1, Tarkan Öztürk∗1 1Kafkas University, Faculty of Science and Letters, Department of Mathematics, Kars, Turkey

Keywords Abstract: In this paper, we introduce and examine a new subclass S∗C (β,α;τ) of Analytic function, analytic functions of complex order in the open unit disk. Here, we obtain upper bound Coefficient bound, estimates for the first three coefficients for the functions belonging to this class. Starlike function, Convex function, Analytic function of complex order

1. Introduction and preliminaries

 0 2 00  Let A be the class of analytic functions f (z) in the open z f (z)) + βz f (z) ℜ 0 > α, α ∈ [0,1), β ∈ [0,1], z ∈U. unit disk U = {z ∈ C : |z| < 1}, normalized by f (0) = βz f (z) + (1 − β) f (z) 0 = f 0(0) − 1 of the form

∞ Also, we will denote TS∗C (β,α) = T ∩ S∗C (β,α). Note f (z) = z + a z2 + a z3 + ··· + a zn + ··· = z + a zn,a ∈ . 2 3 n ∑ n n C that the class TS∗C ( ,0) has been examined by Altınta¸s n=2 β (1) et all. [1, 2]. Also, let us define by T the subclass of all functions f (z) In special case, for β = 0 and β = 1, respectively, we have in A of the form S∗C (0,α) = S∗ (α) and S∗C (1,α) = C (α), in terms of ∗ ∞ the simpler classes S (α) and C (α), defined by (2) and 2 3 n n f (z) = z − a2z − a3z − · · · − anz − · · · = z − ∑ anz ,an ≥ 0. (3), respectively. n=2 (2) We define a subclass of analytic functions as follows. It is well-known that a function f : C → C is said to be univalent if the following condition is satisfied: z1 = Definition 1.1. A function f ∈ S given by (1) is said to ∗ ∗ z2 if f (z1) = f (z2) or f (z1) 6= f (z2) if z1 6= z2. We denote be in the class S C (β,α;τ), α ∈ [0,1), β ≥ 0,τ ∈ C = by S the subclass of A consisting of functions which are C − {0} if the following condition is satisfied also univalent in U. ∗ Also, we will denote by S (α) and C (α) the subclasses  1  z f 0 (z) + βz2 f 00 (z)  ℜ 1 + − 1 > α, α ∈ [0,1), z ∈ U. (5) of S that are, respectively, starlike and convex functions of τ βz f 0 (z) + (1 − β) f (z) order alpha in the open unit disk U. By definition (see for details [3,4,7]) In special case, we have S∗C (β,α;1) = S∗C (β,α) for τ = 1.  z f 0(z)  S∗ (α) = f ∈ S : ℜ > α, z ∈ U , (3) f (z) The object of the present paper is to obtained upper bound estimates for the first three coefficients for the functions and belonging to the class S∗C (β,α;τ), α ∈ [0,1), β ≥ 0, ∗ τ ∈ C . To prove our main results, we need to require the following   z f 00 (z)  C (α) = f ∈ S : ℜ 1 + > α, z ∈ U . (4) lemma. f 0 (z)

Lemma 1.2. [5] If p ∈ P, then the estimates |pn| ≤ 2,n = For details on these classes, one could refer to the mono- 1,2,3,... are sharp, where P is the family of all functions p, graph by Goodman [4]. analytic in U for which p(0) = 1 and ℜ(p(z)) > 0 (z ∈ U), An interesting unification of the functions classes and S∗ (α) and C (α) is provided by the class S∗C (β,α) of functions f ∈ S, which also satisfies the following condi- 2 tion: p(z) = 1 + p1z + p2z + ··· , z ∈ U. (6)

∗ Corresponding author: [email protected] 53 /

2 2. Upper bound estimates for the coefficients Since coefficients of p2 and p1 are positive, using triangle inequality and applying the inequalities |pn| ≤ 2, n = 1,2, In this section, we will obtain upper bound estimates for from (13) we obtain the first three coefficients of the functions belonging to the ∗ ∗ class S C (β,α;τ), α ∈ [0,1), β ≥ 0,τ ∈ C . |τ|(1 − α)(2|τ| + 1) |a | ≤ . (16) The following theorem is on upper bound estimates for the 3 1 + 2β coefficients of the functions belonging to this class. Similarly, from (14), we have Theorem 2.1. Let the function f (z) given by (1) be in the ∗ ∗ class S C (β,α;τ),α ∈ [0,1), β ≥ 0, τ ∈ C . Then, 2(1−α)|τ| |τ|(1−α)(2|τ|+1)  2 2  |a2| ≤ and |a3| ≤ . 2|τ|(1 − α) 2|τ| (1 − α) + 3|τ|(1 − α) + 1 1+β 1+2β |a | ≤ . Also, 4 3(1 + 3β) (17)

 2 2  Thus, from (15)-(17) the proof of inequalities in theorem 2|τ|(1 − α) 2|τ| (1 − α) + 3|τ|(1 − α) + 1 is completed. |a | ≤ . 4 3(1 + 3β) To see that inequalities obtained in the theorem are sharp, we note that equality is attained in the inequalities, when All inequalities obtained here are sharp. p1 = p2 = p3 = 2. ∗ ∗ Proof. Let f ∈ S C (β,α;τ), α ∈ [0,1),β ≥ 0,τ ∈ C . It Moreover, we can easily show that extremal function is the follows that particular solution of the following linear homogeneous differential equation 1  z f 0 (z) + βz2 f 00 (z)  1 + − 1 = α + (1 − α) p(z),z ∈ U, (7) τ βz f 0 (z) + (1 − β) f (z) 2 00 0 +∞ n β (1 − z)z y + [1 − β − (1 − β + 2βτ (1 − α))z]zy where function p(z) = 1 + ∑n=1 pnz is in the class P. The equation (7), we can write as follows: −(1 − β)[1 + (2τ (1 − α) − 1)z]y = 0.

+∞ ( +∞ ) +∞ n n n Thus, the proof of Theorem 2.1 is completed. ∑ (n − 1)[1 + (n − 1)β]anz = τ z + ∑ [1 + (n − 1)β]anz ∑ (1 − α) pnz . n=2 n=2 n=1 Setting τ = 1 in Theorem 2.1, we can readily deduce the following result. Therefore, Corollary 2.2. Let the function f (z) given by (1) be in the 2 3 4 ∗ (1 + β)a2z + 2(1 + 2β)a3z + 3(1 + 3β)a4z + ... = class S C (β), β ≥ 0. Then,  p z2 + [(1 + β)a p + p ]z3  τ (1 − α) 1 2 1 2 . +[(1 + 2β)a p + (1 + β)a p + p ]z4 + ... 2(1 − α) 3(1 − α) 3 1 2 2 3 |a | ≤ , |a | ≤ , (8) 2 1 + β 3 1 + 2β Comparing the coefficients of the like power of z in both 2(1 − α)[(1 − α)(5 − 2α) + 1] sides of (8), we have |a | ≤ . 4 3(1 + 3β) (1 + β)a2 = τ (1 − α) p1, (9) Setting β = 0 and β = 1 in Corollary 2.2, we can readily deduce the following results, respectively.

2(1 + 2β)a3 = τ (1 − α)[(1 + β)a2 p1 + p2], (10) Corollary 2.3. Let the function f (z) given by (1) be in the class S∗. Then, 3(1 + 3β)a4 = τ (1 − α)[(1 + 2β)a3 p1 + (1 + β)a2 p2 + p3]. (11) |an| ≤ n, n = 2,3,4. From these, we get Corollary 2.4. Let the function f (z) given by (1) be in the τ (1 − α) class C. Then, a = p , (12) 2 1 + β 1 |an| ≤ 1, n = 2,3,4. τ (1 − α) τ2 (1 − α)2 a = p + p2, (13) Remark 2.5. As you can see, results obtained in Corollary 3 ( + ) 2 ( + ) 1 2 1 2β 2 1 2β 2.3 and 2.4 verifies results obtained in Corollary 2.2 and 2.3 in [6] , respectively, for α = 0. 2 2 3 3 τ (1 − α) τ (1 − α) τ (1 − α) 2 3 Remark 2.6. Using this work, one can examine a3 − µa a4 = p3 + p1 p2 + p1. 2 3(1 + 3β) 2(1 + 3β) 3(1 + 3β) the Fekete - Szegö problem for the coefficients of the (14) function class S∗C (β,α;τ). Also, using this work one can Since |p1| ≤ 2, from (12), we obtain 2 find H2 (2) = a2a4 −a3 second Hankel determinant for the functions belonging in the class S∗C (β,α;τ). Then, one 2(1 − α)|τ| 2 |a2| ≤ . (15) would find the upper bound estimate for the a2a4 − a . 1 + β 3 54 /

References New York.

[1] Altınta¸s,O., Irmak, H., Srivastava , H. M. 1995. Frac- [4] Goodman, A. W. 1983. Univalent Functions. Volume I, Polygonal. Mariner Comp., Washington, 246p. tional calculus and certain starlike functions with neg- ative coefficients. Computers & Mathematics with Ap- [5] Pommerenke, C. H. 1975. Univalent Functions. Van- plications, 30, no.2, 9-16. denhoeck and Rupercht, Göttingen. [2] Altınta¸s,O., Özkan , Ö., Srivastava , H. M. 2004. [6] Mustafa, N., Öztürk, T. 2018. On the coefficient Neighbourhoods of a Certain Family of Multivalent bounds of certain subclasses of analytic functions of Functions with Negative Coefficients. Computers & complex order. Journal of Scientific and Engineering Mathematics with Applications, 47, 1667-1672 Research, 5, no. 6, 133-136. [3] Duren, P. L. 2016. Univalent Functions, Grundlehren [7] Srivastava, H. M., Owa, S. 1992. Current Topics in An- der Mathematischen Wissenshaften. Springer-Verlag, alytic Function Theory. World Scientific, Singapore.

55 5Th INTERNATIONAL CONFERENCE ON RECENT ADVANCES IN PURE AND APPLIED MATHEMATICS (ICRAPAM 2018) JULY 23-27 2018 at Karadeniz Technical University Prof. Dr. Osman Turan Convention Center in Trabzon, TURKEY ICRAPAM CONFERENCE PROCEEDING

Korovkin Theory for Extraordinary Test Functions via Power Series Method

Tugba˘ Yurdakadim∗1, Emre Ta¸s2 1Hitit University, Science and Arts Faculty, Department of Mathematics, Turkey 2Ahi Evran University, Science and Arts Faculty, Department of Mathematics, Turkey

Keywords Abstract: Korovkin subsets are the subsets of functions satisfying the same property as Power series method, {1,x,x2} and have been studied by many authors. In this paper we introduce Korovkin Korovkin subset, subsets in the sense of power series method for T and present a characterization theorem Abstract Korovkin theory that a subset of C0(X) is Korovkin subset in the sense of power series method for T.

1 1. Introduction So the sequence x = (x j) is convergent to 2 in the sense of power series method but it is obvious that x is not con- In the development of the theory of approximation by vergent in the ordinary sense. Note that the power series positive linear operators, the Korovkin theory has a big method is regular if and only if importance. The classical Korovkin type theorems provide conditions for whether a given sequence of positive linear j p jt operators converge to the identity operator in the space of lim = 0, f or each j ∈ N (1) t→R− p(t) continuous functions on a compact interval [2, 5, 8]. This theory has close connections with real analysis, functional holds [6]. Throughout the paper we assume that power analysis and summability theory. In approximation theory, series method is regular. in order to correct the lack of convergence, summability Definition 1. Let f : E → R be a real function on a topo- methods are used since it is well known that they provide logical space E. The set a nonconvergent sequence to converge [3, 11, 15]. On the other hand, Korovkin-type theorems have also been supp( f ) := {x : f (x) 6= 0} extended in various directions with different aims such as finding other subsets of functions, called Korovkin subsets, is called the support of f . satisfying the same property as {1,x,x2} ; establishing the Let C(E) be the set of all continuous functions on E. If same results in other function spaces or in abstract Banach E is locally compact, we will denote by C (E) the set of spaces [1, 4, 7, 10, 13, 14]. In the present paper, we define c all f ∈ C(E) with compact support supp( f ). A function Korovkin subsets in the sense of power series method and f ∈ C(E) lies in C (E) just if there is some compact subset obtain a characterization theorem for a subset of C (X) to c 0 of E in the complement of which f is identically zero. We be a Korovkin subset in the sense of power series method. denote by C (E) and C (E) all bounded , continuous real First of all, we recall some basic definitions and notations b 0 functions on E and the closure of C (E) with respect to the used in the paper. c usual sup-norm. Let (p ) be real sequence with p > 0 and p , p , p ,... ≥ j 0 1 2 3 Clearly 0, and such that the corresponding power series p(t) := ∞ j ∑ p jt has radius of convergence R with 0 < R ≤ ∞. If, j=0 Cc(E) ⊂ C0(E) ⊂ Cb(E) ⊂ C(E) for all t ∈ (0,R),

∞ since an f ∈ Cc(E) is bounded on its compact support, 1 j lim x j p jt = L hence throughout E. t→R− p(t) ∑ j=0 Recall that positive bounded Radon measure is a positive then we say that x = (x ) is convergent in the sense of linear functional on C0(X). The set of all of the positive j + power series method [9, 12]. bounded Radon measures is denoted by Mb . It is obvious + In order to see that power series method is more effective that every µ ∈ Mb , that is, every positive linear functional than ordinary convergence, let x = (1,0,1,0,...), R = ∞, µ : C0(X) → R is continuous with respect to the norm t 1 given by p(t) = e and for j ≥ 0, p j = . Then it is easy to see j!  that kµk := sup |µ( f )| : f ∈ C0(X), | f | ≤ 1 . 1 ∞ x t j 1 ∞ t2 j 1 et + e−t 1 lim j = lim = lim { } = . t→∞ t ∑ t→∞ t ∑ t→∞ t e j=0 j! e j=0 (2 j)! e 2 2 The following result is known as Urysohn’s lemma.

∗ Corresponding author: [email protected] 56 /

Proposition 1. Let E be a locally compact space and U So one can get be an open neighborhood of the compact subset B. Then |T(g)(y) − v T(g)(z)| = |T(g)(y) − v T(g)(z) − T(g)(z) + T(g)(z)| Cc(E) contains a function ϕ which satisfies j j ≤ |T(g)(z) − T(g)(y)| + |T(g)(z)||v j − 1| 0 ≤ ϕ ≤ 1, ϕ(B) = {1} and supp(ϕ) ⊂ U. ≤ ε + |T(g)(z)||v j − 1|, f or every z ∈ Uv. Definition 2. Let E and F be Banach lattices and consider a positive linear operator T : E → F. A subset M of E is Moreover for every j ≥ v and for every z ∈ Y, we have said to be a Korovkin subset in the sense of power series method of E for T if for every sequence {L j} of positive ∞ linear operators from E into F satisfying 1 j ∞ p jt L j(g)(z) − T(g)(z) 1 j p(t) ∑ (i) sup p(t) ∑ p jt kL jk < ∞ j=0 t∈(0,R) j=0 ∞ ∞ ∞ 1 j 1 j 1 j = µ(g) p jt ϕ j(z) + p jt v jT(g)(z) (ii) lim k p(t) ∑ p jt L j(g)−T(g)k = 0 f or every g ∈ M, p(t) ∑ p(t) ∑ t→R− j=0 j=0 j=0 then ∞ 1 j − T(g)(z) p jt v jϕ j(z) − T(g)(z) ∞ p(t) ∑ 1 j j=0 lim k ∑ p jt L j( f ) − T( f )k = 0 f or every f ∈ E. t→R− p(t) 1 ∞ j=0 j   = ∑ p jt ϕ j(z) µ(g) − v jT(g)(z) p(t) j=0 h 1 ∞ i j 2. Main Results + T(g)(z) ∑ p jt v j − 1 p(t) j=0 In this section, we will present our main result which 1 ∞ ≤ p t jϕ (z) T(g)(y) − v T(g)(z) characterizes a subset of C0(X) is a Korovkin subset in the ∑ j j j p(t) j=0 sense of power series method of C0(X) for a positive linear ∞ operator T. 1 j + T(g)(z) p jt v j − 1 . p(t) ∑ Theorem 1. Let X and Y be locally compact Hausdorff j=0 spaces. Further, assume that X has a countable base Hence using the last inequality we get and Y is metrizable. Given a positive linear operator ∞ T : C0(X) → C0(Y) and a subset M of C0(X), the following 1 p t jL (g)(z) − T(g)(z) statements are equivalent: p(t) ∑ j j (a) M is a Korovkin subset in the sense of power series j=0  ∞ 1 j method of C0(X) for T.  T(g)(z) ∑ p jt v j − 1 ,z ∈/ Uj +  p(t) (b) If µ ∈ M (X) and y ∈ Y satisfying µ(g) = T(g)(y) for  j=0 b  ∞ every g ∈ M, then µ( f ) = T( f )(y) for every f ∈ C0(X). 1 j ≤ p(t) ∑ p jt (ε + |T(g)(z)||v j − 1|) Proof. Assume that µ ∈ M+(X) and y ∈ Y satisfying j=0 b  ∞  1 j µ(g) = T(g)(y) for every g ∈ M. Let us take a decreasing + T(g)(z) ∑ p jt v j − 1 ,z ∈ Uj  p(t) countable base (Uj) of open neighbourhoods of y in Y. j=0 From Proposition 1 if we consider the compact set {y}, we and so choose ϕ j ∈ Cc(Y) such that: 0 ≤ ϕ j ≤ 1, ϕ j(y) = 1 and ∞ also supp(ϕ j) ⊂ Uj. Let us define L j : C0(X) → C0(Y) by 1 j lim k p jt L j(g) − T(g)k = 0. − ∑ t→R p(t) j=0 L j( f ) := µ( f )ϕ j + v jT( f )(1 − ϕ j) f or every f ∈ C0(X) Since M is a Korovkin subset in the sense of power series where v = (v j) is nonnegative, bounded and (|v j − 1|) is method for T, it is obtained that for every f ∈ C0(X), convergent to 0 in the sense of power series method but ∞ not convergent in the ordinary sense. Observe that {L j} is 1 j lim k p jt L j( f ) − T( f )k = 0. a sequence of positive linear operators and also − ∑ t→R p(t) j=0 ∞ ∞ 1 j 1 j  But for every j ≥ 1, L ( f )(y) = µ( f ), then we obtain ∑ p jt kL jk ≤ ∑ p jt kµk + |v j|kTk j p(t) j=0 p(t) j=0 µ( f ) = T( f )(y) for every f ∈ C0(X). This completes the   proof of (b). ≤ kµk + kvkkTk + Conversely assume that if µ ∈ Mb (X) and y ∈ Y satisfy µ(g) = T(g)(y) for every g ∈ M, then µ( f ) = T( f )(y) for ∞ 1 j every f ∈ C0(X). Observe that so sup p(t) ∑ p jt kL jk ≤ H. On the other hand, since t∈(0,R) j=0 + T(g) ∈ C0(Y) for every g ∈ M, for every ε > 0, there exists i f µ ∈ Mb (X) and µ(g) = 0 f or every g ∈ M, then µ = 0. v ∈ N such that (2) Since X has a countable base, every bounded sequence in + T(g)(z) − T(g)(y) ≤ ε, f or every z ∈ Uv. Mb (X) has a vaguely convergent subsequence (See [2]). 57 /

Consider now a sequence {L j} of positive linear operators If we replace T : X → Y with the identity operator from C0(X) into C0(Y) satisfying properties (i) and (ii) IX : X → X, we have following of Definition 2 and suppose that for some f0 ∈ C0(X) the Theorem 2. Let X be a locally compact Hausdorff space ∞ 1 j with a countable base, which is then metrizable as well. sequence { p(t) ∑ p jt L j( f0)} does not converge. So there j=0 Given a subset M of C (X), the following statements are − 0 exist ε0 > 0 and a sequence (tk) which goes to R as k equivalent: tends to infinity and a sequence (yk) in Y such that (i) M is a Korovkin subset in the sense of power series 1 ∞ method of C0(X) for identity operator IX . p t jL ( f )(y )−T( f )(y ) ≥ , f or every k ≥ 1. + ∑ j k j 0 k 0 k ε0 (ii) If µ ∈ Mb (X) and x ∈ X satisfy µ(g) = g(x) for every p(tk) j=0 g ∈ M, then µ( f ) = f (x) for every f ∈ C0(X) i.e. µ = IX . (3) Corollary . We have two cases: (y ) is converging to the point at 1 Under the assumptions of Theorem 1, the k following statements are equivalent: infinity of Y or not. In the first case, since (yk) converges (i) M is a Korovkin subset of C0(X) for T. to the point at infinity of Y we get lim h(yk) = 0 for every k→∞ (ii) M is a Korovkin subset in the sense of power series + h ∈ C0(Y). For every k ≥ 1, define µk ∈ Mb (X) by method of C0(X) for T. 1 ∞ Another noteworthy and useful consequence concerning µ ( f ) := p t jL ( f )(y )( f ∈ C (X)). k p(t ) ∑ j k j k 0 Corollary 1 is that one can obtain all results given in chap- k j=0 ter 6 of [2] for Korovkin subsets in the sense of power From hypothesis, we have series method. ∞ Now we are ready to give an application of Theorem 1. kµ k ≤ sup 1 p t jkL k ≤ H. Since (µ ) is k p(t) ∑ j j k Y → X t∈(0,R) j=0 But first recall that a mapping ϕ : is said to be bounded, we may assume that there exists µ ∈ M+(X) proper if for every compact subset K ∈ X, the inverse im- b age ϕ−1(K) := {y ∈ Y : ϕ(y) ∈ K} is compact in Y where such that µk → µ vaguely (If necessary the sequence µk is replaced with a suitable subsequence). On the other hand X and Y are locally compact Hausdorff spaces. In this case, if g ∈ M, then f oϕ ∈ C0(Y) for every f ∈ C0(X). Corollary 2. Let Y be metrizable locally compact Haus- 1 ∞ j dorff space. If M is a Korovkin subset in the sense of |µk(g)| ≤ ∑ p jtk L j(g)(yk) − T(g)(yk) + T(g)(yk) p(tk) j=0 power series method of C0(X) for IX , then M is a Korovkin 1 ∞ subset in the sense of power series method for any positive j ≤ ∑ p jtk L j(g) − T(g) + T(g)(yk) linear operator T : C0(X) → C0(Y) of the form p(tk) j=0 T( f ) := γ( f oϕ), ( f ∈ C0(X)) which implies µ(g) = lim µk(g) = 0. From (2) we obtain k where γ ∈ Cb(Y), γ ≥ 0 and ϕ : Y → X is a proper mapping. µ( f0) = 0 as well and hence 1 ∞ 3. Concluding Remarks j ∑ p jtk L j( f0)(yk) − T( f0)(yk) p(tk) j=0 Now we can provide some examples of Korovkin subsets in the sense of power series method for identity operator = µk( f0) − T( f0)(yk) → 0. under the light of our Corollary 1 and Corollary 6.7 and This contradicts (3). In the second case the sequence (yk) Proposition 6.8 of [2]. does not converge to the point at infinity of Y. By replacing Given λ1,λ2,λ3 ∈ R, 0 < λ1 < λ2 < λ3 then it with a suitable subsequence, we may assume that it • {e ,e ,e } is a Korovkin subset in the sense of converges to some y ∈ Y. Let us consider λ1 λ2 λ3 λk power series method of C0(X) where eλ (x) := x ∞ k 1 j for every x ∈ X := (0,1] and k = 1,2,3. µk( f ) := ∑ p jtk L j( f )(yk)( f ∈ C0(X)). p(tk) j=0 • {e−λ1 ,e−λ2 ,e−λ3 } is a Korovkin subset in the sense of −λk As in the first case the same reasoning we may assume power series method of C0(X) where e−λk (x) := x + x ∈ X = [ ,+ ) k = , , that there exists µ ∈ Mb (X) such that µk → µ vaguely. for every : 1 ∞ and 1 2 3. Moreover since for every g ∈ M, • { fλ1 , fλ2 , fλ3 } is a Korovkin subset in the sense ∞ 1 j of power series method of C0(X) where fλk (x) := µk(g) − T(g)(yk) ≤ p jt kL j(g) − T(g)k → 0, ∑ k exp(−λkx) for every x ∈ X := [0,+∞) and k = 1,2,3. p(tk) j=0 We can give all of the theorems of this paper for Abel and (g) = T(g)(y) we have µ . So (b) implies Borel convergences since µ( f0) = T( f0)(yk), i.e., 1 ∞ • in the case of R = 1, p(t) = and for j ≥ 0, h 1 j i 1 −t lim p jt L j( f0)(yk) − T( f0)(yk) = 0 k→ ∑ k p j = 1 the power series method coincides with Abel ∞ p(tk) j=0 method which is a sequence-to-function transforma- which contradicts (3). tion, 58 /

t 1 [8] Korovkin, P. P. 1953. On convergence of linear opera- • in the case of R = ∞, p(t) = e and for j ≥ 0, p j = j! tors in the space of continuous functions. Dokl. Akad. the power series method coincides with Borel method. Nauk SSSR 90(1953), 961-964. References [9] Kratz, W., Stadtmüller, U. 1989. Tauberian theorems [1] Altomare, F., Diomede, S. 2001. Contractive Korovkin for Jp-summability. J. Math. Anal. Appl. 139(1989), subsets in weighted spaces of continuous functions. 362-371. Rend. Circ. Mat. Palermo 50(2001), 547-568. [10] Micchelli, C. A. 1975. Convergence of positive linear [2] Altomare, F. 2010. Korovkin-type theorems and ap- operators on C(X). J. Approx. Theory, 13(1975), 305- proximation by positive linear operators. Surveys in 315. Approximation Theory 5.13. [11] Gadjiev, A. D., Orhan, C. 2002. Some approximation [3] Atlihan, Ö. G., Ta¸s,E. 2015. An abstract version of theorems via statistical convergence. Rocky Mountain the Korovkin theorem via A-summation process. Acta. J. Math. 32(2002), 129-138. Math. Hungar. 145(2015), 360-368. [12] Stadtmüller, U., Tali, A. 2011. On certain families of [4] Baskakov, V. A. 1961. On various convergence criteria generalized Nörlund methods and power series meth- for linear positive operators (Russian). Usp. Mat. Nauk. ods. J. Math. Anal. Appl. 238(1999), 44-66. 16(1961), 131-134. [13] Takahasi, S. E. 1990. Bohman-Korovkin-Wulbert op- 2 3 4 [5] Bernstein, F. 1912. Über eine Anwendung der Men- erators on C[0,1] for {1,x,x ,x ,x }. Nihonkai Math. genlehre auf ein der Theorie der säkularen Störungen J. 1(1990), 155-159. herrührendes Problem, Math. Ann., 71(1912), 417- [14] Takahasi, S. E. 1993. Bohman-Korovkin-Wulbert op- 439. erators on normed spaces. J. Approx. Theory 72(1993), [6] Boos, J. 2000. Classical and Modern Methods in 174-184. Summability. Oxford University Press. [15] Ünver, M. 2013. Abel transforms of positive linear [7] Izuchi, K., Takagi, H., Watanabe, S. 1996. Sequen- operators. AIP Conference Proceedings 1558(2013), tial BKW-operators and function algebras. J. Approx. 1148-1151. Theory 85(1996), 185-200.

59 5Th INTERNATIONAL CONFERENCE ON RECENT ADVANCES IN PURE AND APPLIED MATHEMATICS (ICRAPAM 2018) JULY 23-27 2018 at Karadeniz Technical University Prof. Dr. Osman Turan Convention Center in Trabzon, TURKEY ICRAPAM CONFERENCE PROCEEDING

SPACES OF STRONGLY λ - INVARIANT SUMMABLE SEQUENCES

Ekrem Sava¸s∗1 1Usak University, Sciences and Arts Faculty, Department of Mathematics, Turkey

Keywords Abstract: In this paper, we investigate some topological results for the spaces [C(λ,σ), p]0, Multiple sequences and series, [C(λ,σ), p] and [C(λ,σ), p]∞, which will be respectively called the spaces of strongly λ- General summability methods, invariant summable to zero, strongly λ- invariant summable and strongly λ- invariant Matrix methods. bounded sequences.

1. Introduction We say that a bounded sequence x = (xk) is σ-convergent k if and only if x ∈ Vσ such that σ (n) 6= n for all n ≥ 0, Let s be the set of all sequences real or complex and `∞ de- ∞ k ≥ 1. note the Banach space of bounded sequences x = {xk}k=0 normed by ||x|| = supk≥0 |xk|. Let D be the shift operator Just as the concept of almost convergence lead naturally to ∞ 2 ∞ on w, that is, Dx = {xk}k=1, D x = {xk}k=2 and so on. It the concept of strong almost convergence, σ- convergence may be recalled that [see Banach [1]] Banach limit L is a leads naturally to the concept of strong σ-convergence. A nonnegative linear functional on ` such that L is invariant ∞ sequence x = (xk) is said to be strongly σ-convergent (see under the shift operator (that is, L(Dx) = L(x)∀x ∈ `∞) and Mursaleen [6]) if there exists a number L such that that L(e) = 1 where e = {1,1,...}. A sequence x ∈ `∞ is called almost convergent if all Banach limits of x coincide. 1 k x i − L → Let cˆ denote the set of all almost convergent sequences. ∑ σ (m) 0 (2) k i=1 Lorentz [3] proved that as k → ∞ uniformly in m. We write [Vσ ] as the set of all ( m ) 1 strong σ- convergent sequences. When (1.2) holds we cˆ = x : lim xn+i exists uniformly in n . m→∞ ∑ m + 1 i=0 write [Vσ ] − limx = `. Taking σ(m) = m + 1, we obtain [Vσ ] = [cˆ] so strong σ- convergence generalizes the con- Almost convergent sequences were studied by Lorentz [4], cept of strong almost convergence. Note that King [3], Duran[2], Nanda[8], Savas([11], [12],[13], [14]) and others. [Vσ ] ⊂ Vσ ⊂ l∞. Let σ be a one-to-one mapping of the set of positive inte- We suggest the readers for further motivation by including gers into itself. A continuous linear functional ϕ on l∞ is said to be an invariant mean or a σ- mean if and only if citations of some more papers such as Mursaleen[6], Saraswat and Gupta [9] and Savas [10] and others. 1. ϕ ≥ 0 when the sequence x = (xn) has xn ≥ 0 for all n. Let λ = (λn) be a non-decreasing sequence of positive 2. ϕ(e) = 1, where e = (1,1,...) and numbers tending to ∞ such that
λ ≤ λ + 1,λ = 1. 3. ϕ xσ(n) = ϕ(x) for all x ∈ l∞. n+1 n 1 For a certain kinds of mapping σ every invariant mean ϕ The generalized de la Valèe-Poussin mean is defined by extends the limit functional on space c, in the sense that 1 ϕ(x) = limx for all x ∈ c. Consequently, c ⊂ Vσ where Vσ tn(x) = ∑ xk λn is the bounded sequences all of whose σ-means are equal, k∈In ( see, [15]). where In = [n − λn + 1,n]. A sequence x = (xk) is said to be (V,λ)-summable to a number L, if t (x) → L as n → ∞. If x = (x ), set Tx = (Tx ) = x
it can be shown that n k k σ(k) Let C = (c ) be an infinite matrix of nonnegative real (see, Schaefer [15]) that nk numbers and p = (pk) be a sequence such that pk > 0. We   pk write Cx = {Cn(x)} if Cn(x) = ∑k cnk |xk| converges for Vσ = x ∈ l∞ : limtkm (x) = Le, L = σ − limx (1) k each n. The following sequence spaces were defined in [10]. where k 1 x + Tx + ... + T x pk m m m dmn(x) = ∑ Cσ n(i)(x) = ∑c(n,k,m)|xk| tkm(x) = and t−1,m = 0 λm k + 1 i∈Im k

∗ Corresponding author: [email protected] 60 / where 2. Main Results 1 c(n,k,m) = c n . λ ∑ σ (i),k We now study locally boundedness and r−convexity for m i∈Im the spaces of strongly almost summable sequences. We If λm = m,m = 1,2,3,.... start with some definitions. For 0 < r 5 1 a non-void 1 pk dmn(x) = C n (x) = c(n,k,m)|x | subset U of a linear space is said to be absolutely λ ∑ σ (i) ∑ k r r m i∈Im k r−convex if x,y ∈ U and |γ| + |µ| ≤ 1 together imply and that γx + µy ∈ U. It is clear that if U is absolutely 1 r−convex, then it is absolutely t− convex for t < r.A c(n,k,m) = c n . λ ∑ σ (i) linear topological space X is said to be r− convex if every m i∈Im neighbourhood of 0 ∈ X contains an absolutely r−convex reduces to neighbourhood of 0 ∈ X. The r−convexity for r > 1 is of 1 m pk little interest, since X is r−convex for r > 1 if and only if tmn(x) = ∑Cσ n(i)(x) = ∑c(n,k,m)|xk| m + 1 i=0 k X is the only neighbourhood of 0 ∈ X, [see Maddox and where Roles (1969)]. A subset B of X is said to be bounded if 1 m for each neighbourhood U of 0 ∈ X there exists an integer c(n,k,m) = ∑cσ n(i),k. N > 1 such that B ⊆ NU. X is called locally bounded if m + 1 i= 0 there is a bounded neighbourhood of zero. We now define, We first prove: C , p = {x : d (x) → 0 uniformly in n}; < p [C , p] [C , p] (λ,σ) 0 mn Let 0 k 5 1. Then (λ,σ) 0 and (λ,σ) ∞ are lo-   cally bounded if inf pk > 0. If (1.3) holds, then [C(λ,σ), p] C , p = {x : dmn(x − le) → 0 for some l (λ,σ) has the same property. uniformly in n Proof. We shall only prove for [C(λ,σ), p]∞. Let inf pk = θ > 0. If x ∈ [C(λ,σ), p]∞, then there exists a constant and K0 > 0 such that     C(λ,σ), p = x : suptmn(x) < ∞ . pk 0 ∞ n ∑c(n,k,m)|xk| 5 K (∀m,n). k The sets [C(λ,σ), p]0, [C(λ,σ), p] and [C(λ,σ), p]∞ will be re- spectively called the spaces of strongly (λ,σ) -summable For this K0 and given δ > 0 choose an integer N > 1 such to zero, strongly (λ,σ) -summable and strongly (λ,σ)- that bounded sequences. If = m,m = 1,2,3,...., the above K0 λm Nθ spaces reduces to the following sequence spaces. = δ 1 < p [Cσ , p]0 = {x : tmn(x) → 0 uniformly in n}; Since N 1 and k 5 θ we have [Cσ , p] = {x : tmn(x − le) → 0 for some l uniformly in n} 1 1 (∀k) and pk 5 θ   N N [Cσ , p]∞ = x : suptmn(x) < ∞ . Then for all m and n, we get n

If x is strongly (λ,σ)- summable to l we write xk → pk xk 1 pk c(n,k,m) c(n,k,m)|xk| l[A(λ,σ), p]. A pair (A, p) will be called strongly λ- in- ∑ N 5 Nθ ∑ variant regular if k k K0 δ. xk → l ⇒ xk → l[A , p]. 5 5 (λ,σ) Nθ In this paper we study r-convexity and locally boundedness Therefore by taking supremum over m and n we get, for these spaces which were defined above. The following theorem was proved in [10].  0 x : g(x) 5 K ⊆ N {x : g(x) 5 δ}. Let p ∈ ` . Then [B and [C (inf p > 0) are ∞ (λ,σ),p]0 (λ,σ),p]∞ k complete linear topological spaces paranormed by g. If For every δ > 0 ∃N > 1 for which the above inclusion holds and so ||C|| = sup c(n,k,r) < ∞. (3) ∑ x g(x) K0 r k : 5 and is bounded. This completes the proof.

∑c(n,k,r) → 0 uniformly in n. (4) It is known that every locally bounded linear topological k space is r− convex for some r such that 0 < r 5 1. But the hold then [C(λ,σ), p] has the same property. If further pk = following theorem gives exact conditions for r−convexity. p∀k, they are Banach spaces for 1 p < ∞ and p−normed Let 0 < p 1.Then C , p and C , p are 5 k 5 (λ,σ) 0 (λ,σ) ∞ spaces for 0 < p < 1. r−convex for all r where 0 < r < liminf pk. Moreover,   In this paper we study r-convexity and locally boundedness if pk = p 5 1 ∀k, then they are p−convex. C(λ,σ), p has for these spaces which were defined above. the same properties if (1.3) holds. 61 /

Proof. We shall prove the theorem only for C , p . [2] J. P. Duran (1972), Infinite matrices and almost con- (λ,σ) ∞ Let A , p and r ∈ (0,liminf p ). Then ∃k such that vergence, Math. Z. 128, 75-83. (λ,σ) ∞ k 0 r 5 pk (∀k > k0). Now define [3] J. P. King (1966), Almost summable sequences, Proc. Amer. Math. Soc. 17, 1219-25. " k0 ∞ k0 # r pk gˆ(x) = sup ∑c(n,k,m)|xk| + ∑ ∑a(n,k,m)|xk| . [4] G. G. Lorentz (1948), A contribution to the theory of m,n k=1 k=k0+1k=1 divergent sequences, Acta Math. 80, 167-190.

Since r 5 pk 5 1 (∀k > k0), gˆ is subadditive. Further for [5] I. J. Maddox and J. W. Roles, Absolute convexity in 0 < |γ| 5 1, certain topological linear spaces, Proc. Camb. Philos. pk r |γ| 5 |γ| (∀k > k0). Soc. 66,(1969), 541-45. Therefore for such γ we have [6] Mursaleen, Matrix transformation between some new sequence spaces, Houston J. Math. 9(1993), 505– r gˆ(γx) 5 |γ| gˆ(x). 509. Now for 0 < δ < 1, [7] Mursaleen, On some new invariant matrix methods of summability, Q.J. Math. 34(1983), 77-86. U = {x :g ˆ(x) 5 δ} [8] S. Nanda, Some sequence spaces and almost conver- gence, J. Austral. Math. Soc. 22(Series A), (1976), is an absolutely r−convex set, for |γ|r +|µ|r 1 and x,y ∈ 5 446-455. U imply that [9] S. K. Saraswat and S. K. Gupta, Spaces of strongly r r gˆ(γx + µy) 5 gˆ(γx) + gˆ(µy) 5 |γ| gˆ(x) + |µ| gˆ(y) σ-summable sequences, Bull. Cal. Math. Soc. r r 5 (|γ| + |µ| )δ 5 δ. 75,(1983), 179-184. [10] E. Sava¸s, On some new sequence spaces, J. BAUN If p = p (∀k), then for 0 < δ < 1, k Inst. Sci. Technol., 20(3) Special Issue, 154-162, (2018). V = {x : g(x) 5 δ} [11] E. Sava¸s, Strongly almost convergence and Almos is an absolutely p−convex set. This can be obtained by a λ -statistical convergence, Hokkaido Journal Math. similar analysis and therefore we omit the details. This Vol.29,(2000),63-68. completes the proof. [12] E. Sava¸s, Matrix Transformations and Absolute Al- The conclusions of Theorems 2.1 and 2.2 also hold for the most Convergence, Bull. Ins. Math. Academia Sinica, spaces of strongly summable sequences. The proofs are 15 (3),(1987). similar and therefore omitted. These results do not appear [13] E. Sava¸s, Almost Convergence and Almost Summa- anywhere, although Maddox and Roles [5] have obtained bility, Tamkang J. Math., 21 (4), (1990). some results in some special cases. [14] E. Sava¸s, Some Sequence Spaces and Almost Conver- References gence, Ann. Univ. Timiora, 30 (2-3), (1992). [15] P. Schaefer, Infinite matrices and invariant means, [1] S. Banach (1932), Theorie des Operations Lineaires Proc. Amer. Math. Soc. 36(1972), 104–110. (Warszawa).

62 5Th INTERNATIONAL CONFERENCE ON RECENT ADVANCES IN PURE AND APPLIED MATHEMATICS (ICRAPAM 2018) JULY 23-27 2018 at Karadeniz Technical University Prof. Dr. Osman Turan Convention Center in Trabzon, TURKEY ICRAPAM CONFERENCE PROCEEDING

Fixed Point Theorems for α-F-Suzuki Contraction Mapping with Rational Expressions in Branciari b-Metric Spaces

Neslihan Kaplan Kuru∗1, Mahpeyker Öztürk2 1Sakarya University, Department of Mathematics, Turkey 2Sakarya University, Department of Mathematics, Turkey

(Received: **.**.2018, Accepted: **.**.2018, Published Online: **.**.2018)

Keywords Abstract: In this paper, we introduce the notion of α-F-Suzuki contraction in Branciari Branciari b-metric space, b-metric space and prove some new fixed point theorems for α-F-Suzuki contraction Fixed point, mappings satisfying rational expressions in Branciari b-metric space. As an application Suzuki contraction, of the presented main theorem, we establish fixed point result endowed with a graph in Graph theory Branciari b-metric space.

1. Introduction with this new metric space have been published recently [9, 11, 20, 24]. In 2012, Samet et al. [26] introduced A fundamental result in the fixed point theory is the Banach the concept of α-admissible self-mappings and proved the contraction principle, which state that if X is complete met- existence of fixed point results using contractive condi- ric space, then every contraction has a unique fixed point. tions involving α-admissible mappings in complete metric [3] Due to its importance and simplicity, several authors spaces. They also gave some examples and applications of have obtained many interesting extensions and generaliza- the obtained results to ordinary differential equations. tions of the Banach contraction principle by using different After this, Salimi et al. [25] modified and generalized the forms of contractive conditions in various spaces. Some notions of α − ψ-contractive mappings and α-admissible of such generalizations are obtained by contraction condi- mappings. Also, they established new fixed point theorems tions described by rational expressions. for such mappings in complete metric spaces. Recently Some problems, particularly the problem of the conver- Karapinar et al. [14] introduced the notion of triangular gence of measurable functions with respect to a measure, α-admissible mapping in the setting of complete metric lead to a generalization of notation of metric. Using spaces and proved the existence and uniqueness of a fixed this idea, in 1989, Bakhtin [4] introduced the notion of point of such a mapping. b−metric space, while Czerwik [7] extensively used the In 2008, in order to characterize the completeness of un- concept of b−metric space for proving fixed point theo- derlying metric spaces, Suzuki [29] introduced a weaker rems for single-valued and multivalued mappings. Pacurar notion of contraction. Recently, Wardowski [31] intro- [21] obtained results on fixed point of sequences of al- duced a new contraction mapping, called F−contraction, most contractions in b-metric spaces. Successively, this and proved a fixed point result as a generalization of the notion has been reintroduced by Khamsi and Hussain [15], Banach contraction principle. with the name of metric-type space. In 2000, Branciari After this, Abbas et al. [1] generalized the idea of [6] introduced the notion of a generalized metric space F−contraction and proved fixed and common fixed point where the triangle inequality was replaced with a more results. Recently, Secelean [27] defined a large class of general inequality, namely, quadrilateral inequality. He functions using the condition (F20) instead of the condi- also proved the Banach contraction principle in this met- tion (F2) in the definition of F−contraction submited by ric space. The new metric has been mentioned different Wardowski [31]. Very recently, Piri and Kumam [22], de- names such as generalized metric, rectangular metric and veloped the result of Secelean [27], by using the condition Brianciari metric. Erhan et al. [8] discussed about exis- (F30) instead of the condition (F3). With these conditions, tence and uniqueness of fixed points of general class of they [22] proved the results by combining the concepts of (ψ,φ)-contractive mappings on complete rectangular met- Suzuki and F−contractions. Then, in 2016, Piri and Ku- ric spaces and generalized a fixed point theorem which was mam [23] extended this concept and obtained generalized introduced by Lakzian and Samet [17]. Moreover, they F−Suzuki contraction on b−metric spaces. also investigated fixed points of (ψ,φ)-contractions under In 2008, Jachymski [12] investigated a new approach in conditions involving rational expressions. metric fixed point theory by replacing order structure with Combining these two concepts(b-metric space and Bran- graph structure on a metric space. In this way, the re- ciari metric space), George et.al. [10] defined Branciari sults proved in ordered metric spaces are generalized. Ab- b-metric spaces. This new metric space is also referred to bas and Nazir [2] obtained some fixed point results for as rectangular b−metric spaces. Several articles related

∗ Corresponding author: [email protected] 63 / power graphic contraction pair endowed with a graph. The map d is called a Branciari b−metric and the pair Tiammee and Suantai [30] introduced graph-preserving (X,d) is called a Branciari b−metric space with a constant multi-valued mapping and a new type of multi-valued s ≥ 1. weak G-contraction on a metric space endowed with a The following example illustrate that the function d(x,y) directed graph. In this direction, we refer to [16, 19, 28] is not a metric, not a b−metric, not a Branciari metric but and references mentioned therein. only a Branciari b−metric with s = 2 : 2. Preliminaries  1 Example 2.5. [10] Let A = n ,n ∈ N , B = {0,3} and This section is devoted to supply the proof of the main X = A ∪ B. Define the function d(x,y) : X × X → [0,∞) theorems. The following well known definition of b-metric such that d(x,y) = d(y,x) in the following way. space:   0 if x = y,  4 if x,y ∈ A, Definition 2.1. [7] Let X be nonempty set and let s ≥ 1 be d (x,y) = 1 if x ∈ A, y ∈ B, a given real number. A function d : X ×X → [0,∞) is called  n a b−metric if for all x,y,z ∈ X the following conditions  2 if x,y ∈ B. are satisfied: The concepts of convergence, Cauchy sequence and com-

d1. d(x,y) = 0 if and only if x = y; pleteness and continuity on a Branciari b−metric space are defined below: d2. d(x,y) = d(y,x); Definition 2.6. [10] Let (X,d) be a Branciari b−metric d3. d(x,y) ≤ s[d(x,z) + d(z,y)]. space, {xn} be a sequence in X and x ∈ X. Then the se- quence {x } ⊂ X is said to converge to a point x ∈ X if, for The pair (X,d) is called b−metric space. The number n every ε > 0 there exists n ∈ N such that d (x ,x) < ε for s ≥ 1 is called the coefficient of (X,d). 0 n all n > n0. The convergence is also represented as follows. It is an obvious fact that a metric space is also a b−metric lim xn = x or xn → x as n → ∞. space with s = 1, but the converse is not generally true. n→∞ The following example support this fact: Definition 2.7. [10] Let (X,d) be a Branciari b−metric Example 2.2. [13] Consider the set X = [0,1] endowed space, {xn} be a sequence in X and x ∈ X. Then the se- + with the function d : X × X → R defined by d (x,y) = quence {xn} ⊂ X is said to be a Cauchy sequence if, for 2 |x − y| for all x,y ∈ X. Clearly (X,d,2) is a b−metric every ε > 0 there exists n0 ∈ N such that d (xn,xn+p) < ε space but it is not a metric space. for all n > n0, p > 0 or equivalently, if lim d (xn,xn+p) = 0 n→∞ Let us recall the definitions of the Branciari metric space: for all p > 0. Definition 2.3. [6] Let X be nonempty set and let d : X × Definition 2.8. [10] (X,d) is said to be a complete Bran- X → [0,∞) be function such that for all x,y ∈ X and all ciari b−metric space if for every Cauchy sequence in X distinct u,v ∈ X each of which is different from x and y, converges to some x ∈ X. the following conditions are satisfied: Definition 2.9. [10] Let (X,d) be a Branciari b−metric {x } X x X i. d(x,y) = 0 if and only if x = y; space, n be a sequence in and ∈ . Then a mapping T : X → X on is said to be continuous with respect to the ii. d(x,y) = d(y,x); Branciari b−metric d if, for any sequence {xn} ⊂ X which converges to some x ∈ X, that is lim d (xn,x) = 0 we have iii. d(x,y) ≤ d(x,u) + d(u,v) + d(v,y). n→∞ lim d (Txn,Tx) = 0. n→ The map d is called a Branciari metric and the pair (X,d) is ∞ called a Branciari metric space and abbreviated as (BMS). Because of properties listed below, When working with the Branciari and Branciari b−metric space one should be In some sources, (BMS) is known also as generalized careful. metric space and rectangular metric space. Combining the definitions of b−metric and Branciari metric, the so-called Lemma 2.10. [8] Let (X,d) be a Branciari and Branciari Branciari b−metric is defined as follows: b−metric space. Definition 2.4. [10] Let X be nonempty set and let d : 1. If we denote an open ball of radius r centered at x ∈ X X × X → [0,∞) be function such that for all x,y ∈ X and as all distinct u,v ∈ X each of which is different from x and y, Br (x,r) = { y ∈ X :|d (x,y) < r}, the following conditions are satisfied: such an open ball in (X,d) is not always an open set. i. d(x,y) = 0 if and only if x = y; ii. d(x,y) = d(y,x); 2. If τ is the collection of all subsets γ of X such that for each y ∈ γ there exist r > 0 with Br(y) ⊆ γ, then τ iii. d(x,y) ≤ s[d(x,u) + d(u,v) + d(v,y)] for some real defines a topology for (X,d) is not necessarily Haus- number s ≥ 1. dorff. 64 /

3. The limit of a convergent sequence {xn} ∈ X is not Denote the set of all functions satisfying (F1) − (F3) by necessarily unique. F . In [27] , Secelean changed the condition (F2) by an equivalent but a more simple condition (F20). 4. A convergent sequence in X is not necessarily a Cauchy sequence. (F20) infF = −∞,

5. Branciari and Branciari b−metric is not necessarily or also by continuous. 00 ∞ (F2 ) there exists a sequence {αn} of positive real num- Samet et al. [26] suggested a very interesting class of map- n=1 bers with lim F (αn) = −∞. pings, known as α − ψ−contractive mappings as follows: n→∞ Definition 2.11. Let T be a self-mapping on a metric space Recently, Piri and Kumam [22] used the following condi- 0 (X,d) and let α : X × X → [0,+∞) be a function. We say tion (F3 ) instead of (F3). that T is an α−admissible mapping if (F30) F is continuous on (0,∞). x,y ∈ X, α (x,y) ≥ 1 ⇒ α (Tx,Ty) ≥ 1. Then, they denote by ℑ the set of all functions satis- fying the conditions (F1), (F20) and F30. In Karapinar et al. [14] defined the concept of triangular Also, they defined the F−Suzuki contraction as follows: α−admissible mapping as the following: Definition 2.17. Let (X,d) be a metric space. A mapping Definition 2.12. Let T be a self-mapping on a metric space T : X → X is said to be an F−Suzuki contraction if there (X,d) and let α : X × X → [0,∞). then T is called a trian- exists τ > 0 such that for all x,y ∈ X with Tx 6= Ty gular α−admissible mapping if 1 i. T is α−admissible; d (x,Tx) < d (x,y) ⇒ τ + F (d (Tx,Ty)) ≤ F (d (x,y)), 2 ii. α (x,z) ≥ 1 and α (z,y) ≥ 1 imply α (x,y) ≥ 1. where F ∈ ℑ. Lemma 2.13. [14] Let T be a triangular α−admissible mapping. Assume that there exists x0 ∈ X such that 3. Fixed Points of α-F-Suzuki Contraction in n α (x0, f x0) ≥ 1. Define sequence {xn} by xn = f x0. Then Branciari b-Metric Space

α (xm,xn) ≥ 1 f or all m,n ∈ N with m < n. In this section, we introduce the notion of α-F-Suzuki con- traction in Branciari b-metric space and prove some new In 2008, Suzuki [29] proved generalized versions of Edel- fixed point theorems for α-F-Suzuki contraction mappings stein’s results in compact metric space as follows: satisfying rational expressions in Branciari b-metric space.

Theorem 2.14. Let (X,d) be a compact metric space and Definition 3.1. Let (X,d) be a Branciari b-metric space let T : X → X be a self-mapping. Assume that for all with a constant s ≥ 1 and a self mapping f : X → X is said x,y ∈ X with x 6= y, to be a α − F−Suzuki contraction if there exists F ∈ ℑ, α : X ×X → [0,∞) and τ > 0 such that for all x,y ∈ X with 1 d (x,Tx) < d (x,y) ⇒ d (Tx,Ty) < d (x,y). x 6= y, 2 1 d (x, f x) < d (x,y) Then T has a unique fixed point in X. 2s 2
Definition 2.15. [31] Let (X,d) be a metric space. A ⇒ τ + F s α (x,y)d ( f x, f y) ≤ F (M (x,y)) (2) mapping T : X → X is said to be an F−contraction if there where exists τ > 0 such that ∀ x,y ∈ X M (x,y) = max{d (x,y),d (x, f x),d ( f y,y) , d (Tx,Ty) > 0 ⇒ τ + F (d (Tx,Ty)) ≤ F (d (x,y)), (1) d ( f y,y)(1 + d (x, f x)) . where F : R+ → R is a mapping satisfying the following 1 + d (x,y) conditions: Theorem 3.2. Let be a complete Branciari b-metric space (F1) F is strictly increasing, that is, for α,β ∈ R+ such with a constant s ≥ 1 and a self mapping f : X → X is said that α < β implies F(α) < F(β); to be a α − F−Suzuki contraction satisfying the following conditions: (F2) for each sequence {αn} of positive numbers lim αn = n→∞ 0 if and only if lim F (αn) = −∞; i. f is triangular α−admissible mapping; n→∞ (F3) there exists k ∈ (0,1) such that lim αkF (α) = 0. ii. there exists x0 ∈ X such that α (xo, f xo) ≥ 1; α→0+ iii. f is continuous. Lemma 2.16. [31] From (F1) and (1) it is easy to con- clude that every F−contractions is necessarily continuous. Then f has a fixed point u ∈ X. 65 /

Proof. Regarding the condition (ii), we choose x0 ∈ X Continuing this process, we have such that α(x0, f x0) ≥ 1 and define the sequence {xn} as xn+1 = f xn for n ∈ N. First, we assume that any two F (d (xn,xn+1)) ≤ F (d (xn−1,xn)) − τ consecutive members of the sequence {xn} are distinct, ≤ F (d (xn−2,xn−1)) − 2τ that is, xn 6= xn+1 for all n ≥ 0. Otherwise, we would have xp = xp+1 = f xp for some p ∈ N, which means that xp is . a fixed point of f . Since f is α−admissible mapping and . α(x0,x1) = α(x0, f x0) ≥ 1, we deduce that α(x1,x2) = ≤ F (d (x0,x1)) − nτ. α( f x0, f x1) ≥ 1. Continuing this process, we get that By taking limit as n → ∞ in above inequality, we have, α(xn,xn+1) ≥ 1 for all n ∈ N. (3) lim F (d (xn,xn+1)) = −∞, and since F ∈ ℑ we obtain, n→∞ Now, we will prove that lim d (xn,xn+1) = 0. (7) n→∞ lim d (xn,xn+1) = 0. (4) n→∞ Now, we prove that {xn} is a Cauchy sequence, If not, For x = xn and y = xn+1 with the use of (3) there exists ε > 0 for which one can find subsequences 1 {xm } and {xn } of {xn} with ni > mi > i such that and 2s d (xn, f xn) < d (xn,xn+1), we can write i i d (x ,x ) ≥ , (8) d(xn,xn+1) = d( f xn−1, f xn) mi ni ε

2 where ni is the smallest integer satisfying (8), that is, ≤ s α(xn−1,xn)d( f xn−1, f xn).

Since F is strictly increasing, we have d (xmi ,xni−1) < ε. (9)

2 F(d(xn,xn+1)) ≤ F(s α(xn−1,xn)d( f xn−1, f xn)). From (8) and using the condition (iii) of the definition of Branciari b−metric space, we get From the contractive condition (2) and τ > 0, we have

ε ≤ d (xmi ,xni ) 2 τ + F(d(xn,xn+1)) ≤ τ + F(s α(xn−1,xn)d( f xn−1, f xn)). ≤ s[d (xmi ,xmi+1) + d (xmi+1,xni+1) + d (xni+1,xni )]. ≤ F(M(x ,x )), n−1 n Taking the upper limit as i → ∞ and using (7), we have for all n ≥ 1, where ε ≤ lim d (xm +1,xn +1) (10) s i→∞ i i M (xn−1,xn) = max{d (xn−1,xn) ,d (xn−1, f xn−1), also d ( f x ,x )(1 + d (x , f x )) d ( f x ,x ), n n n−1 n−1 n n + d (x ,x ) 1 n−1 n d (xmi ,xni )

= max{d (xn−1,xn),d (xn−1,xn),d (xn+1,xn) , ≤ s[d (xmi ,xni−1) + d (xni−1,xni+1) + d (xni+1,xni )]. d (x ,x )(1 + d (x ,x )) n+1 n n−1 n Then, taking into account (7) and (9), 1 + d (xn−1,xn) lim supd (xm ,xn ) ≤ sε. (11) = max{d (xn−1,xn),d (xn,xn+1)}, i→∞ i i we get In a similar way, by using b−rectangular inequality, we have τ + F (d (xn,xn+1))

d (xni ,xmi+1) ≤ F (max{d (xn−1,xn),d (xn,xn+1)}).

(5) ≤ s[d (xni ,xni−1) + d (xni−1,xmi ) + d (xmi ,xmi+1)]. If d (x ,x ) > d (x ,x ), then n n+1 n−1 n By letting as i → ∞ in the above inequality and regarding (7) and (9), we get max{d (xn,xn+1),d (xn−1,xn)} = d (xn,xn+1),

lim supd (xn ,xm +1) ≤ sε. (12) so (5) becomes i→∞ i i

τ + F (d (xn,xn+1)) ≤ F (d (xn,xn+1)), Since f is triangular α−admissible, we deduce from Lemma 2.13 that α (xmi ,xni ) ≥ 1. Also, because of (7) which is a contradiction from τ > 0. Thus, we conclude and (8), we can write 1 1 1 that 2s d (xmi , f xmi ) = 2s d (xmi ,xmi+1) < 2s ε < d (xmi ,xni ). From (2), we have F (d (xn,xn+1)) ≤ F (d (xn−1,xn)) − τ. 2
(6) τ + F s d ( f xmi , f xni ) 66 /

2
≤ τ + F s α (xmi ,xni )d ( f xmi , f xni ) ≤ F (M (u,w))

≤ F (M (xmi ,xni )) where (13) M (u,w) = max{d (u,w),d (u, f u),d ( f w,w), where d ( f w,w)(1 + d (u, f u)) M (x ,x ) = max{d (x ,x ), d (x , f x ), mi ni mi ni mi mi 1 + d (u, f u)  d ( f xn ,xn )(1 + d (xm , f xm )) = d (u,w). d ( f x ,x ), i i i i ni ni 1 + d (x ,x ) mi ni This implies τ + F (d (u,w)) ≤ F (d (u,w)), which is a

= max{d (xmi ,xni ),d (xmi ,xmi+1),d (xni+1,xni ), contradiction. Hence, u = w. Therefore, f has a unique fixed point. d (x ,x )(1 + d (x ,x )) ni+1 ni mi mi+1 . (14) 1 + d (xmi ,xni ) 4. Fixed Points Results with Graph Passing to the limit as i → ∞ in (14) and using (7), (12), The first result in this direction was given by Jachymski we obtain [12]: lim supM (xmi ,xni ) ≤ sε. (15) i→∞ Definition 4.1. Let (X,d) be a metric space endowed with Now, taking the upper limit as i → ∞ in (13), using (F1) a graph G. We say that a self-mapping f : X → X is a and from (10), (11), (12), (15), we have Banach G−contraction if f preserves the edge of G, that   is,  2 ε  τ + F s . ≤ F lim supM (xm ,xn ) (x,y) ∈ E (G) ⇒ ( f x, f y) ∈ E (G) s i→∞ i i for all x,y ∈ X and f decreases the weights of the edges of ≤ F (sε) G in the following way: hence τ + F (sε) ≤ F (sε), is a contradiction because of ∃ α ∈ (0,1) such that for all x,y ∈ X, τ > 0. So, {xn} is a Cauchy sequence in X. Since (X,d) is (x,y) ∈ E (G) ⇒ d ( f x, f y) ≤ αd (x,y). a complete Branciari b−metric space, there exists u ∈ X such that xn → u as n → ∞, that is We next review some basic notions in graph theory. Let (X,d) be a Branciari b−metric space and ∆ = {(x,x) : lim xn+1 = lim f xn = u. n→∞ n→∞ x ∈ X} denotes the diagonal of X × X. Let G be a directed graph such that the set V(G) of its vertices coincides with Since, the condition (iii) of the hypothesis, f is continuous, X and E(G) be the set of edges of the graph such that ∆ ⊆ we conclude that f u = u, which completes the proof. E(G). Also assume that G has no parallel edges and G is a weighted graph in the sense that each edge (x,y) is assigned In order to provide the uniqueness of the fixed point of the weight d (x,y). Whenever a zero weight is assigned triangular α-admissible mappings, extra condition is to some edge (x,y), it reduces to a (x,x) having weight 0. required. There are different versions of the uniqueness The graph G is identified with the pair (V(G),E(G)). condition, two of which are given below. If x and y are vertices of G, then a path in G from x to y of (U1) For every pair x and y of fixed points of f , length k ∈ N is a finite sequence {xn} of vertices such that α(x,y) ≥ 1. x = x0,...,xk = y and (xi−1,xi) ∈ E(G{)1for, i ∈ 2,...,k} (U2) For every pair x and y of fixed points of f , there Recall that a graph G is connected if there is a path be- z ∈ X (x,z) ≥ (y,z) ≥ exists such that α 1 and α 1. tween any two vertices and it is weakly connected if Ge is connected, where Ge denotes the undirected graph obtained We give uniqueness theorem employing the condition from G by ignoring the direction of edges. Denote by G−1 (U1). the graph obtained from G by reversing the direction of Theorem 3.3. If the condition (U1) is added to the con- edges. Thus, ditions of Theorem 3.2, then the mapping f has a unique E G−1
= {(x,y) ∈ X × X : (y,x) ∈ E (G)}. fixed point. It is more convenient to treat G as a directed graph for Proof. Let u,w ∈ Fix( f ) where u 6= w and α(u,w) ≥ 1, e which the set of its edges is symmetric, under this conven- 1 d (u, f u) < d (u,w). Then we can write that 2s tion; we have that 2 d (u,w) ≤ s (u,w)d ( f u, f w). −1 α E(Ge) = E(G) ∪ E(G ).

Since F ∈ ℑ, we get Let Gx be the component of G consisting of all the edges and vertices which are contained in some path in G begin- F (d(u,w)) ≤ F s2α (u,w)d ( f u, f w)
. ning at x. In V(G), we define the relation R in the following Then from τ > 0 and the condition (2) in the statement of way: For x,y ∈ V(G), we have xRy if and only if, there is the theorem implies that we have a path in G from x to y. If G is such that E(G) is symmet- ric, then for x ∈ V(G), the equivalence class [x]G in V(G) 2
τ + F (d(u,w)) ≤ τ + F s α (u,w)d ( f u, f w) . defined by the relation R is V(Gx). 67 /

Definition 4.2. Let be a Branciari b-metric space (with a References constant s ≥ 1) endowed with a graph G. We say that a self mapping f : X → X is said to be a α − FG−Suzuki [1] M. Abbas, T. Nazir, S. Romaguera, Fixed Point Re- contraction if f preserves the edges of G, that is, sults for Generalized Cyclic Contraction Mappings in Partial Metric Spaces. Revista de la Real Academia de (x,y) ∈ E (G) ⇒ ( f x, f y) ∈ E (G), for all x,y ∈ X; Ciencias Exactas, Fis. Nat., Ser. A Mat, 106 (2)(2012), 287-297. and decreases the weights of the edges of G in the follow- ing way: [2] M. Abbas, T. Nazir, Common Fixed Point of a Power 1 Graphic Contraction Pair in Partial Metric Spaces En- d (x, f x) < d (x,y) 2s dowed with a Graph. Fixed Point Theory and Appl., 2013(1)(2013), 8 pages. ⇒ τ + F s2d ( f x, f y)
≤ F (M (x,y)) [3] S. Banach, Sur Les Oprations Dans Les Ensembles where Abstraits et Leurs Applications Aux Quations Intgrales. Fund. Math., 3(1922), 133-181. M (x,y) = max{d (x,y),d (x, f x),d ( f y,y), [4] I. A. Bakhtin, The Contraction Principle in Quasi- d ( f y,y)(1 + d (x, f x)) metric Spaces. Func. An., Unianowsk, Gos. Ped. Ins., . 1 + d (x,y) 30(1989), 26-37, in Russian. for all (x,y) ∈ E(G). [5] M. Boriceanu, A. Petru¸sel,I. A. Rus, Fixed Point The- orems for Some Multivalued Generalized Contractions Theorem 4.3. Let (X,d) be a complete Branciari b-metric in b-Metric Spaces. Int. J. Math. Stat., 6(2010), 65-76. space endowed with a graph G and f : X → X. If the following conditions hold: [6] A. Branciari, A Fixed Point Theorem of Banach- Caccioppoli Type on a Class of Generalized Metric i. for all x,y ∈ X, (x,y) ∈ E(G), implies ( f x, f y) ∈ Spaces. Publ. Math. Debrecen, 57(2000), 31-37. E(G); [7] S. Czerwik, Contraction Mappings in b−Metric Spaces. Acta Math. Inform. Univ. Ostraviensis, ii. {xn} is a sequence in X such that xn → x as n → ∞ 1(1993), 5-11. and (xn, f xn) ∈ E(G), then (x, f x) ∈ E(G); [8] I. M. Erhan, E. Karapinar, T. Sekulic, Fixed Points iii. there exists (x , f x ) ∈ E(G); 0 0 of (ψ,φ) Contractions on Rectangular Metric Spaces. Then f has a fixed point x ∈ X. Fixed Point Theory and Appl., 2012(1)(2012), 12 pages. Proof. Define mapping α : X × X → [0,∞) by [9] I. M. Erhan, Geraghty Type Contraction Mappings  1, if (x,y) ∈ E (G) α (x,y) = on Branciari b-Metric Spaces. Advances in the Theory 0, otherwise. of Nonlinear Analysis and its Appl., 1(2)(2017), 147- Now, we show that f is α− triangular admissible map- 160. ping. Suppose that α(x,y) ≥ 1. Therefore, we have (x,y) ∈ E(G). From (i), we get ( f x, f y) ∈ E(G). So α( f x, f y) ≥ 1 [10] R. George, S. Radenovic, K. P. Reshma, S. Shukla, and f is α− admissible mapping. Also, if α(x,z) ≥ 1 Rectangular b-Metric Spaces and Contraction Princi- and α(z,y) ≥ 1, then because of the definition of α, we ple. J. Nonlinear Sci. Appl., 8(2015), 1005-1013. can write α(x,y) ≥ 1. So, f is α− triangular admissible [11] S. Gulyaz, E. Karapinar, I. Erhan, Generalized α- mapping. Hence, from the definition and inequality, we α Meir-Keeler Contraction Mappings on Branciari b- have Metric Spaces. Filomat, 31(2017), 5445-5456. ⇒ τ + F s2d ( f x, f y)
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69 5Th INTERNATIONAL CONFERENCE ON RECENT ADVANCES IN PURE AND APPLIED MATHEMATICS (ICRAPAM 2018) JULY 23-27 2018 at Karadeniz Technical University Prof. Dr. Osman Turan Convention Center in Trabzon, TURKEY ICRAPAM CONFERENCE PROCEEDING

Fixed Point Theorems for Mappings Satisfying Bϕ −Contraction

Abdurrahman Büyükkaya ∗1, Mahpeyker Öztürk2 1Karadeniz Technical University, Department of Mathematics, Turkey 2Sakarya University, Science Art Faculty, Department of Mathematics, Turkey

Keywords Abstract: In this paper, we introduced the class of Bϕ which includes nine-variable Bϕ −Contraction, functions and strong comparison function, which is a generalization of the class A that Fixed Point. introduced by Akram et al., in metric space and 2-metric space and and proved some fixed . point theorems.

1. Introduction Definition 1.1. [2] Let X be a non−empty set and let σ : X × X × X → R be a map satisfying the following Banach contraction mapping (or Banach contraction prin- conditions: ciple) implies briefly that every contraction mapping T on complete metric space (X,d) satisfying for all x,y ∈ X i. For every pair of distinct point x,y ∈ X, there exists a point z ∈ X such that σ (x,y,z) 6= 0. d (Tx,Ty) ≤ kd (x,y), where k ∈ (0,1) (1) ii. If at least two of three point x,y,z are the same, then σ (x,y,z) = 0. has a unique fixed point and for every x0 ∈ X, the sequence n {T x0} is convergent to fixed point. This theorem has iii. The symmetry: σ (x,y,z) = σ (x,z,y) = σ (y,x,z) = been a cornerstone for metric fixed point theory and any σ (y,z,x) = σ (z,x,y) = σ (z,y,x) for all x,y,z ∈ X. extensions of this principle have been done up to now. iv. The rectangle inequality: σ (x,y,z) ≤ σ (x,y,t) + On the other hand, in 2002, Branciari[26] obtained some σ (y,z,t) + σ (x,z,t) for all x,y,z,t ∈ X. fixed point results in complete metric space, which Then σ is a 2−metric on X and (X,σ) is called a 2−metric includes a general contractive conditions of integral type space which will be some times denoted by X if there is in complete metric space. Later many study have been no confusion. carry out on contractive conditions of integral type. Recently, Akram et al.[3] introduced a new class of Definition 1.2. [2] Let (X,σ) be a 2−metric space and contractive mapping, called A−contraction, which is a a,b ∈ X, r ≥ 0. The set proper superclass of Kannan’s[4], Bianchini’s[5] and Reich’s[6] type contractions. Later Saha et al.[28] proved B(a,b,r) = {x ∈ X : σ (a,b,x) < r} some fixed point results for A−contraction mapping is called a 2 − ball centered at a and b with radius r. The endow with a general contractive conditions of integral topology generated by the collection of all 2 − balls as a type. subbasis is called a 2−metric topology on X. Also, the notion 2−metric was defined by Gahler[2] in 1963 and as topologically, 2−metric space has been not Definition 1.3. [12] Let {xn} be a sequence in a 2−metric equivalent to ordinary metric space. Thus, there are not space (X,σ). easy relationship between results obtained in 2−metric 1. {x } is said convergent to x in (X,σ), written space and metric space. In fact, geometrically the value n lim xn = x, if for all a ∈ X, lim σ (xn,x,a) = 0, that of a 2-metric σ (x,y,z) represents the area of a triangle n→∞ n→∞ with vertices x,y and c, whereas, ordinary metric is not is, for each ε > 0 and a ∈ X, there exists n0 ∈ N such that lim d (xn,x,a) < ε for all n ≥ n0. so. Also, in this space, some basic fixed point results n→∞ have been established. For the fixed point theorems on 2−metric space, the readers may refer to [10 − 19]. 2. {xn} is said Cauchy in X, if for all a ∈ X, lim σ (xn,xm,a) = 0, that is, for each ε > 0, there In the sequel the letters N and R+ will denote the set of n,m→∞ all positive integer numbers and the set of all positive real exists n0 ∈ N such that lim d (xn,xm,a) < ε for all numbers, respectively. n,m→∞ m,n ≥ n0. First, we give some basic fact and definitions relation to 2−metric space. 3. (X,σ) is said to be complete if every Cauchy se- quence is a convergent sequence in X.

∗ Corresponding author: [email protected] 70 /

Remark 1.4. [22] y ≤ β (x,x,y,z,0,z,y,y,0), 1. This is straightforward from definition 1 that every or 2−metric is non−negative and every 2−metric space contains at least three distinct points. y ≤ β (x,y,x,z,0,z,y,y,0), or 2. A 2−metric σ (x,y,z) is sequentially continuous in one argument. Moreover, if a 2−metric σ (x,y,z) is y ≤ (y,x,x,z, ,z,y,y, ). sequentially continuous in two argument, then it is β 0 0 sequentially continuous in all three argument. In next definition, we give a definitions which will extend 3. A convergent sequence in a 2−metric space need not of the class of A−contraction by using following the class be a Cauchy sequence. of functions.

4. In a 2−metric space (X,σ), every convergent se- Definition 1.10. [21] Let Φ = {ϕ |ϕ : R+ → R+ } be class quence is a Cauchy sequence if σ is continuous. of function, which satisfies the following conditions. i. t ≤ t implies (t ) ≤ (t ); 5. There exists a 2−metric space (X,σ) such that every 1 2 ϕ 1 ϕ 2 convergent sequence is a Cauchy sequence but σ is n ii. (ϕ (t))n∈N converges to 0 for all t > 0; not continuous. ∞ iii. ∑ ϕn (t) converges for all t > 0; Lemma 1.5. [2] If lim xn = x in a 2−metric space (X,σ) n→∞ n=0 then lim σ (xn,y,z) = σ (x,y,z) for all x,y,z ∈ X. n→∞ If condition (i − ii) is hold then ϕ is called a comparison functions, and, the comparison function satisfies (iii), then Lemma 1.6. ([23], Lemma 5) If T : X → X is a continuous ϕ is called a strong comparison function. mapping from a 2−metric space X to a 2−metric space Y, then lim xn = x in X implies lim Txn = Tx in Y. Remark 1.11. [21] Any strong comparison function is a n→∞ n→∞ comparison function. Lemma 1.7. [8] Let {y } be a sequence in complete n Remark 1.12. [21] If ϕ : R → R is a comparison func- 2−metric space (X,σ). If there exists h ∈ (0,1) such that + + tion, then ϕ (t) < t, for all t > 0, ϕ (0) = 0 and ϕ is right σ (y ,y ,a) ≤ hσ (y ,y ,a) for all n ≥ 1 and a ∈ X, n n+1 n−1 n continuous at 0. then {yn} converges to a point in X. Recently, Ozturk et al.[29] extended the notion of Now, we recall the class of A which will useful in what A−contraction by using the function ϕ ∈ Φ and proved follows. some fixed point theorems and common fixed point theo- rems. Ozturk defined the class of Aϕ as follows. Definition 1.8. [3] Let R+ denotes the set of all non- negative real numbers and A be the set of all functions Definition 1.13. [29] Let R+ denotes the set of all non- 3 α : R+ → R+ satisfying: negative real numbers and Aϕ be the set of all functions 3 3 α : R+ → R+ satisfying: i. α is continuous on the set R+ (with respect to the 3 3 Euclidean metric on R+). i. α is continuous on the set R+ (with respect to the Euclidean metric on R3 ). ii. a ≤ kb for some k ∈ [0,1) whenever a ≤ + α (a,b,b) or a ≤ α (b,a,b) or a ≤ α (b,b,a) for all a,b. ii. for all u,v ∈ R+, u ≤ α (u,u,v) or u ≤ α (v,u,v) or u ≤ α (v,v,u), then u ≤ ϕ (v), Definition 1.9. [3] A self map T on a metric space X is called an A−contraction if for any x,y ∈ X and for α ∈ A, where ϕ is a strong comparison function. the following condition holds: In this definition, if we take ϕ (t) = kt as k ∈ (0,1) for all d (Tx,Ty) ≤ α (d (x,y),d (x,Tx),d (y,Ty)) t > 0, then we obtain α ∈ A.

On the other hand, Tran Van An et al.[24] extend the class 2. Main Results A in [3] to the class of nine-variable functions β : R9 → B + In this section, we established some fixed point theo- R satisfying + rems in metric space and 2−metric space by defining 9 −contraction. i. β is continuous on the set R+ (with respect to the Bϕ 9 Euclidean metric on R+) Definition 2.1. Let R+ denotes the set of all non-negative real numbers and be a class of nine variable functions ii. For all x,y,z ∈ R+, Bϕ ∗ 9 β : R+ → R+ satisfying: (a) If x ≤ β (0,0,x,x,0,0,0,0,x) or x ≤ ∗ 9 β (x,0,0,x,x,0,0,x,x), then x = 0. i. β is continuous on the set R+ (with respect to the Euclidean metric on R9 ) (b) There exists k ∈ [0,1) such that y ≤ kx provided + z ≤ x + y and ii. For all x,y,z ∈ R+, 71 /

(a) If Theorem 2.4. Let (X,σ) be a complete 2−metric space x ≤ β ∗ (0,0,x,x,0,0,0,0,x) and let T : X → X be a self mapping such that

σ (Tx,Ty,a) ≤ β ∗ (σ (x,y,a),σ (x,Tx,a),σ (y,Ty,a), or x ≤ β ∗ (x,0,0,x,x,0,0,x,x), σ (x,Ty,a),σ (y,Tx,a),σ T 2x,x,a
,σ T 2x,Tx,a

σ T 2x,y,a
,σ T 2x,Ty,a

then (4) x ≤ ϕ (x). ∗ for all x,y,a ∈ X and some β ∈ Bϕ . Then T has a unique fixed point x∗ and lim T nx = x∗ for all x ∈ X. n→∞ (b) There exists ϕ ∈ Φ such that y ≤ ϕ (x) provided 2.1. Consequences for Fixed Point Theorems z ≤ x + y and In this section, we give some results and our results apply ∗ y ≤ β (x,x,y,z,0,z,y,y,0), to contractive conditions of integral type in metric space or and 2−metric space. Also, our results are generalizations of some results on metric space and 2−metric space. The following Lemma is our first result. y ≤ β ∗ (x,y,x,z,0,z,y,y,0), Lemma 2.5. 1. If α ∈ A and or ϕ ∗ β (x1,x2,x3,x4,x5,x6,x7,x8,x9) = α (x1,x2,x3) ∗ y ≤ β (y,x,x,z,0,z,y,y,0), ∗ for all x1,x2,x3,x4,x5,x6,x7,x8,x9 ∈ R+, then β ∈ where ϕ is a strong comparison function. Bϕ . ∗ ∗ In this definition, we extended the class B in [24] to the 2. If β ∈ Bϕ and α (x,y,z) = β (x,y,z,0,0,0,0,0,0) for all x,y,z ∈ R , then α ∈ A . class of Bϕ and if we take ϕ (t) = kt as k ∈ (0,1) for all + ϕ ∗ t > 0, then we obtain β ∈ B. By using Lemma 2.5(1), Theorem 2.2 and Theorem 2.4, we get following corollaries. Theorem 2.2. Let (X,d) be a complete metric space and let T : X → X be a map such that Corollary 2.6. i. Let T be an A−contraction on a com- plete metric space X. Then T has a unique fixed point d (Tx,Ty) ≤ β ∗ (d (x,y),d (x,Tx,),d (y,Ty),d (x,Ty), in X such that the sequence {T nx} converges to the fixed point for any x ∈ X. d T 2x,x
,d (y,Tx),d T 2x,Tx
,d T 2x,y
,d T 2x,Ty

, (2) ii. Let T be an Aϕ −contraction on a complete metric ∗ space X. Then T has a unique fixed point in X such for all x,y ∈ X and some β ∈ Bϕ . Then T has a unique n fixed point x∗ and lim T nx = x∗ for all x ∈ X. that the sequence {T x} converges to the fixed point n→∞ for any x ∈ X. Next, we prove that a lemma which will be useful in what Corollary 2.7. i. Let T be an A−contraction on a com- follows in 2−metric space. plete 2−metric space X. Then T has a unique fixed point in X. Lemma 2.3. Let (X,σ) be a complete 2−metric space and let T : X → X be a self mapping such that ii. Let T be an Aϕ −contraction on a complete 2−metric space X. Then T has a unique fixed point in X. σ (Tx,Ty,a) ≤ β ∗ (σ (x,y,a),σ (x,Tx,a),σ (y,Ty,a), In 2002, Baranciari[26] proved a fixed point theorem, which include a general contractive condition of integral σ (x,Ty,a),σ (y,Tx,a),σ T 2x,x,a
,σ T 2x,Tx,a
, type in complete metric space as follows. σ T 2x,y,a
,σ T 2x,Ty,a

Theorem 2.8. (Branciari[26]) (3) Let (X,d) be a complete metric space, k ∈ (0,1), and let T be a mapping from X into itself such that for each x,y ∈ X, for all x,y,a ∈ X and some ∗ ∈ . Then β Bϕ d(Tx,Ty) d(x,y) 2
σ T x,Tx,x = 0. Also, R φ (t)dt ≤ k R φ (t)dt (5) 0 0 2
2
σ x,T x,a ≤ σ (x,Tx,a) + σ Tx,T x,a where φ is a locally integrable functions from [0, +∞) into itself and such that for all ε > 0, for all x,y,a ∈ X. ε Z In the following theorem, we set up a fixed point theorem φ (t)dt > 0 by using the class Bϕ in 2−metric space. 0 72 /

σ(Tx,Ty,a) σ(x,y,a) σ(x,Tx,a) Then T has a unique fixed point z ∈ X such that for each R ∗ R R x ∈ X, the sequence {T nx} converges to z. φ (t)dt ≤ β φ (t)dt, φ (t)dt, 0 0 0 Subsequently, many study have been done on contractive σ(y,Ty,a) σ(x,Ty,a) σ(y,Tx,a) conditions of integral type for different contractive R φ (t)dt, R φ (t)dt, R φ (t)dt, mappings, which is extended the result of Baranciari[26] 0 0 0 with various known properties. A fine study have been done by Rhoades[27] and Saha et al.[28] introduced the σ(T 2x,x,a) σ(T 2x,Tx,a) σ(T 2x,y,a) R R R analogues of some fixed point results for A−contraction φ (t)dt, φ (t)dt, φ (t)dt, mapping by setting integral as follows. 0 0 0  σ(T 2x,Ty,a) R Theorem 2.9. [28] Let T be a self mapping of a complete φ (t)dt 0 metric space (X,d) satisfying the following conditions: ∗ for each x,y,a ∈ X with some β ∈ Bϕ , where φ is a d(Tx,Ty)  d(x,y) d(x,Tx) d(y,Ty) locally integrable functions from [0, +∞) into itself and Z Z Z Z such that φ (t)dt ≤ α  φ (t)dt, φ (t)dt, φ (t)dt ε 0 0 0 0 Z f or all ε > 0, φ (t)dt > 0. for each x,y ∈ X with some α ∈ A, where φ : [0,+∞) → [0,+∞) is Lebesgue-integrable mapping which 0 is summable (i.e. with finite integral) on compact subset of Then T has a unique fixed point z ∈ X such that for each [0, +∞), non-negative, and such that x ∈ X, the sequence {T nx} converges to z.

ε Z As a results, if we take ϕ (t) = kt as k ∈ (0,1) for all f or each ε > 0, φ (t)dt > 0. t > 0, then we obtain β ∗ ∈ B, that is, our results ex- 0 tend, improve, and generalize some recent results in the literature, including the results of Tran Van An et Then T has a unique fixed point z ∈ X and for each x ∈ X, al. [Filomat 28(10), 2037-2045, 2014.], Akram et al. limT nx = z. n [Novi.Sad.J.Math., 38(1),(2008),25-33] and Ozturk et al. [Bangmod Int. J. Math. Comp. Sci., 1(1), 172-182, 2015]. Our result which inspired by above study is given as fol- lows. References Corollary 2.10. Let (X,d) be a complete metric space and let T : X → X be a map satisfying the following conditions: [1] S. Banach, Sur les operations dans les emsembles ab- d(Tx,Ty) d(x,y) d(x,Tx) straits et leurs applications aux equations integrales, R φ (t)dt ≤ β ∗ R φ (t)dt, R φ (t)dt, Fund. Math., 1(2012), 133-181. 0 0 0 [2] S. Gähler, 2-metrische raume und ihre topologiche struukture,Math.Nachr., 26(1963), 953-956. d(y,Ty) d(x,Ty) d(y,Tx) d(T 2x,x) R R R R φ (t)dt, φ (t)dt, φ (t)dt, φ (t)dt, [3] M. Akram, A. A. Zafar, A. A. Siddiqui, A general class 0 0 0 0 of contractions: A-contractions, Novi Sad J. Math.,  38:1(2002),25-33. d(T 2x,Tx) d(T 2x,y) d(T 2x,Ty) R R R φ (t)dt, φ (t)dt, φ (t)dt [4] R. Kannan, Some results on fixed points, Bull. Cal- 0 0 0 cutta. Math.Soc., 60(1968), 71-76. for each x,y ∈ X with some ∗ ∈ , where is a locally β Bϕ φ [5] R. Bianchini, Su un problema di S. Reich riguardante integrable functions from [ , + ) into itself and such that 0 ∞ la teori dei punti fissi, Boll. Un. Math. Ital., 5(1972), ε 103-108. Z f or all ε > 0, φ (t)dt > 0. [6] S. Reich, Kannan’s fixed point theorem, Boll. Un. 0 Math. Ital., 5(1971), 1-11. Then T has a unique fixed point z ∈ X such that for each [7] Lj. B. Ciric, A new fixed point theorem for contractive x ∈ X, the sequence {T nx} converges to z. mappings, Publ. Inst. Math. (Beograd), 30:44(1981), 25-27. Similarly, we have the following corollary for 2−metric [8] S. L. Singh, Some contractive type principles on 2- space. metric spaces and applications, Math. Sem. Notes Kobe Univ., 7:1(1979), 1-11. Corollary 2.11. Let (X,σ) be a com- [9] J. Matkowski, Fixed point theorems for contractive plete metric space and let T : X → X be mappings in metric spaces, Casopis Pro Pestovani a map satisfying the following conditions: Matematiky., 105:4(1980), 341-344. 73 /

[10] S. Gähler, Uber die uniformisierbakait 2-metrische [21] I. A. Rus, A. Petru¸sel,G. Petru¸sel,Fixed point theory, Räume, Math. Nachr., 28(1965), 235-244. Cluj Univ. Press, (2008). [11] S. Gähler, Zur geometric 2-metrische Räume, Rev. [22] SVR. Naidu, J. R. Prasad, Fixed point theorems in 2- Roumaine Math. Pures Appl., 11(1966), 655-664. metric spaces, Indian J. Pure Appl. Math., 17:8(1986), [12] K. Iseki, Fixed point theorems in 2-metric spaces. 974-993. Math. Semin. Notes, 3(1975), 133-136. [23] B. K. Lahiri, P. Das, L. K. Dey, Cantor’s theorem in 2-metric spaces and its applications to fixed point [13] K. Iseki, P.L. Sharma, B.K. Sharma, Contraction type problems, Taiwan J. Math., 15(2011), 337-352. mappings on 2-metric space, Math.Jap., 21(1976), 67- 70. [24] T. V. An, N. V. Dung, V. T. Hang, General fixed point [14] B.E. Rhoades, Contraction type mappings on a 2- theorems on metric spaces and 2-metric space, Filomat 28:10(2014), 2037-2045. metric space, Math. Nachr., 91(1979), 151-155. [15] M. Saha, D. Dey, On the theory of fixed points of [25] M. Saha, D. Dey, Fixed point theorems for a class of contractive type mappings in a 2-metric space, Int. A-contractions on a 2-metric space, Novi Sad J. Math., Journal of Math. Analysis, 3:6(2009), 283-293. 40:1(2010), 3-8. [16] A. K. Sharma, On fixed point in 2-metric space, [26] A. Branciari, A fixed point theorem for mappings Math.sem Notes, Kobe Univ., 6(1978), 467-473. satisfying a general contractive condition of integral type, Int. J. Math. Math. Sci., 29(2002), 531-536. [17] A. K. Sharma, A generalization of Banach Contrac- [27] B. E. Rhoades, Two Fixed point theorems for map- tion Principle to 2-metric space, Math. Sem. Notes, Kobe Univ., 7(1979), 291-295. pings satisfying a general contractive condition of in- tegral type, Int. J. of Math. and Mathematical Sci., [18] A. K. Sharma, On generalized contraction in 2-metric 63(2003), 4007-4013. space, Math.Sem Notes, Kobe Univ., 10(1982), 491- [28] M. Saha, D. Dey, Fixed point theorems for A- 506. contractions mappings of integral type, Journal of Non- [19] M. S. Khan, On fixed point theorems in 2- linear Sci. and Appl., 5(2012), 84-92 metric spaces, Publ. Inst. Math(Beogrand) (N.S), [29] M. Ozturk, E. Girgin, Some fixed point theorems 27:41(1980), 107-112. and common fixed point theorems in metric space [20] H. Chatterjee, On generalization of Banach contrac- involving a graph, Bangmod Int. J. Math. Comp. Sci., tion principle, Indian J. Pure Appl. Math., 10:4(1979), 1:1(2015), 172-182. 400-403.

74 5Th INTERNATIONAL CONFERENCE ON RECENT ADVANCES IN PURE AND APPLIED MATHEMATICS (ICRAPAM 2018) JULY 23-27 2018 at Karadeniz Technical University Prof. Dr. Osman Turan Convention Center in Trabzon, TURKEY ICRAPAM CONFERENCE PROCEEDING

Analysis of Neutron Capture Cross Section

Veli Capali∗1 1Usak University, Engineering Faculty, Department of Material Science and Nanotechnology, TURKEY

Keywords Abstract: The neutron capture cross–section reactions for some Cd isotopes have been Cross section, calculated with different level density models by using TALYS 1.6 and EMPIRE 3.2 codes Reactor materials, for incident neutron energies. The obtained results have been analyzed with available Level density models, experimental data. As an outcome, to obtain more precise and coherent calculations with the experimental values, level density model and its parameters have been determined.

1. Introduction choice of different parameters such as nuclear level densi- ties and nuclear models. In EMPIRE, the level densities The measurement of cadmium neutron capture cross sec- are described by several models with the corresponding tions for keV bombarding energies without using exist set parametrizations. Three of them are phenomenological of level density parameters could only be anticipated to be (Gilbert-Cameron Model, Generalized Superfluid Model, no more than a formality [1]. With the increasing excita- Enhanced Generalized Superfluid Model) and one is based tion energy, the spacing between levels decreases rapidly on Hartree-Fock-Bogoliubov microscopic model. TALYS and they cannot be resolved experimentally which caused is possible to choose 3 phenomenological and 3 micro- due to the uncertainty of the level densities between the scopic level densities models per nuclide, e.g. the Constant ground state and the neutron binding energy. The theoret- Temperature model for the target nucleus and the Back- ical models of nuclear reaction are generally required to shifted Fermi Gas model for the compound nucleus. The get the prediction of the reaction cross-sections, especially details of model parameters and options of TALYS 1.6 and if the no experimental data obtained or in cases where it EMPIRE 3.2 can be found in [2, 3]. is difficult to carry out the experimental measurements. The specific requests of scientists make them to find spe- 3. Results and Discussion cial characterized materials suitable to be used in neutron physics. This situation steers them to the cadmium due In the study, (n,) reaction cross–sections for some Cd to its unique cross section and it became a significantly isotopes have been calculated with different level density used metal in neutron physics research. As a benefit of cad- models. Relationship between the calculated results and miums characteristic which could be defined as its strong experimental values have been investigated statistically resonance around low energetic neutrons, capturing or fil- where the outcomes have been graphed as shown in Fig. 1. tering those low energy neutrons even with a relatively thin sheet manufactured from itself is possible.The theoretical cross–sections with respect to neutron energy have been 4. Summary and Conclusions calculated using different level density models of TALYS 1.8 [2] and EMPIRE 3.2 [3] computer codes. Level density Statistical analysis shows us that by taking into account of models are usually adjusted to experimental data on neu- all the studied reactions exist in this paper, TALYS 1.6 cal- tron resonance spacing at the neutron separation energy culations have the best acceptable agreement with respect available for many nuclei. Previous experimental works to experimental data. When the cross section calculations (cross-sections) were found in the literature and available performed on the Cd using Geant4 code have been exam- experimental data existing in the EXFOR [4]. The obtained ined, it has been observed that the total cross section results results have been analyzed with each level density model have been closed to the experimental ones. parameters available experimental data. Finally, we hope all the outcomes of this study will help to 2. Theoretical Calculation Methods improve the exist knowledge of the literature and provide researchers more reliable, compatible and certain informa- TALYS and EMPIRE are a bash-shell UNIX scripts code tion for the further studies. can be used for theoretical investigations of nuclear re- actions for nuclear data evaluation studies. Each code References are prediction and analysis of nuclear reactions that con- tain projectiles particle, in the board energy region. In [1] A. R. de L. Musgrovet, B. J. Allent and R. L. Macklin. TALYS and EMPIRE, several options are included for the J. Phys. G: Nucl. Phys. 4(5), 771 (1978).

∗ Corresponding author: [email protected] 75 /

[2] A. Koning, S. Hilaire, S. Goriely, TALYS-1.6 A Nu- PIRE–3.2 Rivoli Modular System for Nuclear Reac- clear Reaction Program, User Manual (NRG, The tion Calculations and Nuclear Data Evaluation, User’s Netherlands), First Edition: December 23, 2013 Manual (2013). (2013). [4] EXFOR, Brookhaven National Laboratory, National [3] M. Herman, R. Capote, M. Sin, A. Trkov, B. V. Carl- Nuclear Data Center. Database Version of 2018 son, P. Obložinsk, C. M. Mattoon, H. Wienkey, S. (http://www.nndc.bnl.gov/exfor/exfor.htm). Hoblit, Young-Sik Cho, V. Plujko, V. Zerkin, EM-

76 5Th INTERNATIONAL CONFERENCE ON RECENT ADVANCES IN PURE AND APPLIED MATHEMATICS (ICRAPAM 2018) JULY 23-27 2018 at Karadeniz Technical University Prof. Dr. Osman Turan Convention Center in Trabzon, TURKEY ICRAPAM CONFERENCE PROCEEDING

Some Fixed Point Theorems in Generalized b-metric Spaces

Awais Asif∗1, Ekrem Savas2, Waqas Ahmad3, Muhammad Arshad4 1,3,4International Islamic University Islamabad, Department of Mathematics and Statistics, Pakistan 2U¸sakUniversity, Department of Mathematics, Turkey

Keywords Abstract: The aim of this paper is to establish fixed point results of generalized contrac- b-metric space, tive mappings for single-valued mappings. To provide a more general concept of b-metric Cauchy sequence, space, fixed point of generalized contraction is proved in extended b-metric space. We Fixed Point support our results with examples. Our results unify and extend the existing results in the literature.

1. Introduction

Czerwik [3] defined b-metric space and generalized the renowned theorem of Banach contraction mapping. Fagin et al. [4] relaxed the triangular inequality in it and named this new metric as nonlinear Elastic Matching (NEM). Another kind of relaxed triangular inequality was used for the purpose of trading measure [5] as well as for measuring ice floes [6]. Later, b-metric space was generalized with the idea of Extended b-metric space by Kamran.T et al. [16] and some theorems of fixed point were proved for single-valued mappings. In continuation of that, we established some fixed point results of generalized contractions in extended b-metric spaces.

2. Preliminaries

In this portion, we discuss the basic notions related to b-metric, extended b-metric space and their results. [2] Consider a set X which is non-empty set and s ≥ 1 is some real number. The function ϕ : X × X → [0,∞) is known as b-metric if the following axioms are fulfilled, for each a,b,c ∈ X; 1. ϕ(a,b) = 0 ⇔ a = b 2. ϕ(a,b) = ϕ(b,a) 3. ϕ(a,c) ≤ s[ϕ(a,b) + ϕ(b,c)] Here (X,ϕ) is know as b-metric space. [16] Consider a set X which is non-empty and θ : X × X → [1,∞). The function ϕθ : X × X → [0,∞) is known as extended b-metric if and only if for each a,b,c ∈ X, the following axioms holds; 1. ϕθ (a,b) = 0 ⇔ a = b 2. ϕθ (a,b) = ϕθ (b,a) 3. ϕθ (a,c) ≤ θ(a,c)[ϕθ (a,b) + ϕθ (b,c)] Here (X,ϕθ ) is know as extended b-metric space. We get the definition of b-metric space if θ(a,c) = s for s > 1. + + [16] Consider X = {1,2,3}. Take θ : X × X → R and ϕθ : X × X → R as: θ(a,b) = 1 + a + b

ϕθ (1,1) = ϕθ (2,2) = ϕθ (3,3) = 0

ϕθ (1,2) = ϕθ (2,1) = 80,ϕθ (1,3) = ϕθ (3,1) = 1000,ϕθ (2,3) = ϕθ (3,2) = 600

Proof. (1) and (2) trivially hold. For (3) we have;

ϕθ (1,2) = 80,θ(1,2)(ϕθ (1,3) + ϕθ (3,2)) = 4(1000 + 600) = 6400

ϕθ (1,3) = 1000,θ(1,3)(ϕθ (1,2) + ϕθ (2,3)) = 5(80 + 600) = 3400

We get same calculations for ϕθ (2,3). Thus for every a,b,c ∈ X;

ϕθ (a,c) ≤ θ(a,c)[ϕθ (a,b) + ϕθ (b,c)].

∗ Corresponding author: [email protected] 77 /

Hence (X,ϕθ ) is extended b-metric space. [16] Consider C([a,b],R) = X be the space of all real-valued continuous mappings defined on [a,b]. Here X is complete 2 extended b-metric space by taking ϕθ (a,b) = supt∈[a,b] |a(t)−b(t)| , with θ(a,b) = |a(t)|+|b(t)|+2, where θ : X ×X → [1,∞). [16] Consider (X,ϕθ ) is an extended b-metric space. / / 1. The sequence {an} in the set X converges to a in X only if for every ε > 0, there is some n = n (ε) in N such as ϕθ (an,a) < ε, for every n ≥ N. Here, we write limn→∞ an = a. / / 2. A sequence {an} in the set X is Cauchy only if for every ε > 0, there is some n = n (ε) in N such as ϕθ (an,a) < ε, for every n ≥ N. [16] An extended b-metric space will be complete only if every Cauchy sequence in the set X is convergent. [16] Suppose (X,ϕθ ) is extended b-metric space and ϕθ be continuous, then each convergent sequence in the set X will have a unique limit. [16] Consider (X,ϕθ ) is complete extended b-metric space such as ϕθ be continuous functional. Consider T : X → X satisfying; ϕθ (Ta,Tb) ≤ δ(ϕθ (a,b)) forall a,b ∈ X n Where δ ∈ [0,1) be such that a0 ∈ X, limm,n→∞ θ(an,am) < 1/η, here an = T a0, n = 1,2,...Then the mapping T has exactly one fixed point a∗. Furthermore, for every b ∈ X, T nb → a∗. 2 [16] Let T : X → X be a self-mapping and for some a0 ∈ X, ϑ(a0) = {a0, f a0, f a0,...} is the orbit of a0. The function G : X → R is known as T-orbitally lower semi-continuous at t ∈ X if for {an} ⊂ ϑ(a0) and an → t,G(t)≤ limn→∞ infG(an). [16] Let (X,ϕθ ) is complete extended b-metric space such as ϕθ be continuous functional. Suppose T is a self mapping on X and there is some a0 ∈ X such as:

2 ϕθ (Tb,T b) ≤ δ(ϕθ (b,Tb)) forall b ∈ ϑ(a0);

n n ∗ Where δ ∈ [0,1) be such that for a0 ∈ X, limm,n→∞ θ(an,am) < 1/η, here an = T a0, n = 1,2,....Then T b → a ∈ X (as n → ∞). Furthermore a∗ is a fixed point of T if and only if G(a) = ϕ(a,Ta) is T-orbitally lower semi continuous at a∗.

3. Main Results

Suppose (X,ϕθ ) is complete extended b-metric space such as ϕθ be a continuous functional. Suppose T is a self mapping on X satisfying; ϕθ (Ta,Tb) ≤ δ(ϕθ (a,Ta) + ϕθ (b,Tb)) (1) n Where, δ ∈ [0,1/2) be such that a0 ∈ X, limm,n→∞ θ(an,am) < 1/η, here an = T a0, n = 1,2,....Then T has a unique fixed point. Proof. We choose any a0 ∈ X be arbitrary, define the iterative sequence {an} by;

2 n a0,a1 = Ta0,a2 = Ta1 = T(Ta0) = T (a0)...,an = T a0 .....

Then by applying the inequality (1) we obtain,

ϕθ (an,an+1)) ≤ ηϕθ (an−1,an) 2 ≤ η ϕθ (an−2,an−1) . .

n ≤ η ϕθ (ao,a1) (2) Where, η = δ/((1 − δ)). Now for m > n,

ϕθ (an,am) ≤ θ(an,am)ϕθ (an,an+1) + θ(an,am)θ(an+1,am)ϕθ (an+1,an+2)

+θ(an,am)θ(an+1,am)θ(an+2,am)ϕθ (an+2,an+3) + ···

+θ(an,am)θ(an+1,am)θ(an+2,am)···θ(am−1,am)ϕθ (am−1,am)

By using the inequality using in equation (2), we obtain

n n+1 ϕθ (an,am) ≤ ϕθ (a0,a1)(θ(a1,am)θ(a2,am)...θ(an,am)η + θ(a1,am)θ(a2,am)...θ(an,am)η m−1 +··· + θ(a1,am)θ(a2,am)...θ(an,am)η ))

78 /

∞ n n Since, limm,n→∞ θ(an+1,am)η < 1, So that the series ∑n=1 η Πi=1θ(ai,am) converges by the ratio test for each n ∈ N. Let, ∞ ∞ n n j j S = ∑ η Πi=1θ(ai,am), Sn = ∑ η Πi=1θ(ai,am) n=1 j=1 ∗ Letting limn→∞ we conclude that {an} is Cauchy sequence. Since X is complete, let an → a ∈ X:

∗ ∗ ∗ ∗ ∗ ϕθ (Ta ,a ) ≤ θ(Ta ,a)(ϕθ (Ta ,an) + ϕθ (an,a )) ∗ ∗ ∗ ∗ ≤ θ(Ta ,a )(ϕθ (Ta ,Tan−1)) + ϕθ (an,a ))

∗ ∗ ∗ ∗ ∗ ∗ ∗ [1 − θ(Ta ,a )δ]ϕθ (Ta ,a ) ≤ θ(Ta ,a )(δϕθ (an−1,an) + ϕθ (an,a )) ∗ ∗ ◦ ◦ Now Letting limn→∞, a∗ = Ta . Hence a is the fixed point of T. Let a = Ta is another fixed point of T, then;

∗ ◦ ∗ ◦ ϕθ (a ,a ) = ϕθ (Ta ,Ta ) a∗ = a◦

Hence T has a unique fixed point. + For X = [0,∞), Define ϕθ (a,b) : X × X → R and θ : X × X → [1,∞) as

2 ϕθ (a,b) = (a − b) , θ(a,b) = a + b + 2

a+1 Then ϕθ is a complete extended b-metric on X. Define T : X → X by Ta = 3 . We have, a + 1 b + 1 1 2a − 1 2b − 1 ϕ (Ta,Tb) = ( − )2 ≤ (( )2 + ( )2) θ 3 3 4 3 3 1 a + 1 b + 1 ≤ ((a − )2 + (b − )2) ≤ δ(ϕ (a,Ta) + ϕ (b,Tb)) 4 3 3 θ θ

n a+1 Now for each a ∈ X, T a = 3n . Thus it implies that, a + 1 a + 1 lim θ(T ma,T na) = lim θ( + ) m,n→∞ m,n→∞ 3m 3n lim θ(T ma,T na) < 4 m,n→∞

Hence all conditions of theorem (2) are satisfied, therefore T has a unique fixed point. Let (X,ϕθ ) be a complete extended b-metric space such that ϕθ is a continuous functional. Let T be a self mapping from X to X satisfies, ϕθ (Ta,Tb) ≤ δ(ϕθ (a,Tb) + ϕθ (b,Ta)) Forall a,b ∈ X 1 1 n Where, δ ∈ [0, 2 ) be such that for each a0 ∈ X, limm,n→∞ θ(an,am) < η , here an = T a0, n = 1,2,.... Then T has a unique fixed point. Proof. We choose any a0 ∈ X be arbitrary, define the iterative sequence {an} by;

2 n a0,a1 = Ta0,a2 = Ta1 = T(Ta0) = T (a0)...,an = T a0 .....

Then by Appling the inequality (1) we obtain:

ϕθ (an,an+1) δθ(an−1,an+1) δθ(an−2,an) δθ(an−3,an−2) δθ(a0,a2) ≤ ...... ϕθ (a0,a1) 1 − δθ(an−1,an+1) 1 − δθ(an−2,an) 1 − δθ(an−3,an−1) 1 − δθ(a0,a2)

n θ(an−1,an+1) θ(an−2,an) θ(a0,a2) ≤ δ . .... ϕθ (a0,a1) 1 − δθ(an−1,an+1) 1 − δθ(an−2,an) 1 − δθ(a0,a2)

Let ψ = θ(an−1,an+1) . θ(an−2,an) .... θ(a0,a2) Then 1−δθ(an−1,an+1) 1−δθ(an−2,an) 1−δθ(a0,a2) n ϕθ (an,an+1) ≤ δ ψϕθ (a0,a1)

Now for m > n, we have

ϕθ (an,am) ≤ θ(an,am)ϕθ (an,an+1) + θ(an,am)θ(an+1,am)ϕθ (an+1,an+2)

+θ(an,am)θ(an+1,am)θ(an+2,am)ϕθ (an+2,an+3) + ...

θ(an,am)θ(an+1,am)θ(an+2,am)...θ(am−1,am)ϕθ (am−1,am) 79 /

By using the inequality in equation (2), we obtain

n n+1 ϕθ (an,am) ≤ θ(an,am)δ ψϕθ (a0,a1) + θ(an,am)θ(an+1,am)δ ψϕθ (a0,a1) n+2 +θ(an,am)θ(an+1,am)θ(an+2,am)δ ψϕθ (a0,a1) + ... m−1 +θ(an,am)θ(an+1,am)θ(an+2,am)...θ(am−1,am)δ ψϕθ (a0,a1) n n+1 ≤ ψϕθ (a0,a1)(θ(a1,am)θ(a2,am)...θ(an,am)δ + θ(a1,am)θ(a2,am)...θ(an,am)δ m−1 +... + θ(a1,am)θ(a2,am)...θ(an,am)δ )

∞ n n Since limm,n→∞ θ(an+1,am)η < 1, so that the series ∑n=1 δ Πi=1θ(ai,am) converges by the ratio test for each n ∈ N. Let, ∞ ∞ n n j j S = ∑ δ Πi=1θ(ai,am), Sn = ∑ δ Πi=1θ(ai,am) n=1 j=1 Thus for m > n, above inequality implies

ϕθ (an,am) ≤ ϕθ (a0,a1)(Sm−1,Sn)ψ

∗ Letting the limm,n→∞, we conclude that {an} is Cauchy sequence in X. Since X is complete, let an → a ∈ X.

∗ ∗ ∗ ∗ ∗ ∗ ϕθ (Ta ,a ) ≤ θ(Ta ,a )(ϕθ (Ta ,an) + ϕθ (an,a )) ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ≤ θ(Ta ,a )δϕθ (a ,Tan−1)) + θ(Ta ,a )δϕθ (an−1,Ta ) + θ(Ta ,a )ϕθ (an,a )

∗ ∗ Now, Letting limm,n→∞, a = T. Hence a is the fixed point of T. Moreover, uniqueness can easily be proved by using inequality (1). + For X = [0,∞), Define ϕθ (a,b) : X × X → R and θ : X × X → [1,∞) as:

2 ϕθ (a,b) = (a − b) , θ(a,b) = a + b + 2

a Then ϕθ is a complete extended b-metric on X. Define T : X → X by Ta = 2 . We have, a b 1 b a 1 b a ϕ (Ta,Tb) = ( − )2 = ((a − )2 − (b − )2) ≤ ((a − )2 + (b − )2) θ 2 2 9 2 2 9 2 2 ≤ δ(ϕθ (a,Tb) + ϕθ (b,Ta))

n a Now for each a ∈ X, T a = 2n . Thus it implies that, a a lim θ(T ma,T na) = lim θ( + + 2) m,n→∞ m,n→∞ 2m 2n lim θ(T ma,T na) < 9 m,n→∞

Hence all conditions of theorem (2) are satisfied, therefore T has a unique fixed point. Let (X,ϕθ ) be complete extended b-metric space such that ϕθ is a continuous mapping. Let T be self mapping from X to X satisfying: ϕθ (Ta,Tb) ≤ a1ϕθ (a,b) + a2(ϕθ (a,Ta) + ϕθ (b,Tb)) + a3(ϕθ (a,Tb) + ϕθ (b,Ta)) 1 n For all a,b ∈ X. Where a1 + 2a2 + 2a3θ(a,b) < 1 be such that for each a0 ∈ X, limm,n→∞ θ(an,am) < η , here an = T a0, n = 1,2,.... Then T has a unique fixed point. Proof. We choose any a0 ∈ X be arbitrary, define the iterative sequence {an} by;

2 n a0,a1 = Ta0,a2 = Ta1 = T(Ta0) = T (a0)...,an = T a0 ..... Then

ϕθ (an,an+1) = ϕθ (Tan−1,Tan) (a1 + a2 + a3θ(an−1,an+1) (a1 + a2 + a3θ(an−2,an) ≤ ϕθ (an−1,an) ≤ ϕθ (an−2,an−1) (1 − a2 − a3θ(an−1,an+1)) (1 − a2 − a3θ(an−2,an)) . . (a1 + a2 + a3θ(a0,a1) ϕθ (a0,a1) (1 − a2 − a3θ(a0,a1) By letting the above equation becomes,

ϕθ (an,an+1) ≤ (δ1(an−1,an + 1).δ2(an−2,an).δ3(an−3,an−1)...δn(a0,a2))ϕθ (a0,a2) 80 /

Now suppose δ = δ1(an−1,an + 1).δ2(an−2,an).δ3(an−3,an−1)...δn(a0,a2)Then

ϕθ (an,an+1) ≤ δ.δ.δ ....δϕθ (a0,a2)

n ϕθ (an,an+1) ≤ δ ϕθ (a0,a2) (3) By triangular inequality, for m > n, we have:

ϕθ (an,am) ≤ θ(an,am)ϕθ (an,an+1) + θ(an,am)θ(an+1,am)ϕθ (an+1,an+2)

+θ(an,am)θ(an+1,am)θ(an+2,am)ϕθ (an+2,an+3) + ...

θ(an,am)θ(an+1,am)θ(an+2,am)...θ(am−1,am)ϕθ (am−1,am)

n n+1 ϕθ (an,am) ≤ θ(an,am)δ ϕθ (a0,a1) + θ(an,am)θ(an+1,am)δ ϕθ (a0,a1) n+2 +θ(an,am)θ(an+1,am)θ(an+2,am)δ ϕθ (a0,a1) + ... m−1 +θ(an,am)θ(an+1,am)θ(an+2,am)...θ(am−1,am)δ ϕθ (a0,a1) n n+1 ≤ ϕθ (a0,a1)(θ(a1,am)θ(a2,am)...θ(an,am)δ + θ(a1,am)θ(a2,am)...θ(an,am)δ m−1 +... + θ(a1,am)θ(a2,am)...θ(an,am)δ )

∞ n n Since limm,n→∞ θ(an+1,am)η < 1, so that the series ∑n=1 δ Πi=1θ(ai,am) converges by the ratio test for each n ∈ N. Let,

∞ ∞ n n j j S = ∑ δ Πi=1θ(ai,am), Sn = ∑ δ Πi=1θ(ai,am) n=1 j=1

Thus for m > n, above inequality implies

ϕθ (an,am) ≤ ϕθ (a0,a1)(Sm−1,Sn)

∗ Letting the limm,n→∞, we conclude that {an} is Cauchy sequence in X. Since X is complete, let an → a ∈ X.

∗ ∗ ∗ ∗ ∗ ∗ ϕθ (Ta ,a ) ≤ θ(Ta ,a )(ϕθ (Ta ,an) + ϕθ (an,a )) ≤ 0

This implies that a∗=Ta∗. Hence a∗ is the fixed point of T. We can easily prove its uniqueness in a similar way to theorem (3). + For X = [0,∞), Define ϕθ (a,b) : X × X → R and θ : X × X → [1,∞) as

2 ϕθ (a,b) = (a − b) , θ(a,b) = a + b + 2

a Then ϕθ is a complete extended b-metric on X. Define T : X → X by Ta = 2 . We have, a b 1 1 a b 1 b a ϕ (Ta,Tb) = ( − )2 = (a − b)2 + ((a − )2 − (b − )2) + ((a − )2 − (b − )2) θ 2 2 12 3 2 2 27 2 2 1 1 a b 1 b a ≤ (a − b)2 + ((a − )2 − (b − )2) + ((a − )2 − (b − )2) 12 3 2 2 27 2 2 ≤ δ(ϕθ (a,Tb) + ϕθ (b,Ta))

ϕθ (Ta,Tb) ≤ a1ϕθ (a,b) + a2ϕθ (a,Tb) + a3ϕθ (a,Tb) + a4ϕθ (a,Tb) + a5ϕθ (b,Ta) n a Now for each a ∈ X, T a = 2n . Thus it implies that, a a lim θ(T ma,T na) = lim θ( + + 2) m,n→∞ m,n→∞ 2m 2n lim θ(T ma,T na) < 27 m,n→∞

Hence all conditions of theorem (2) are satisfied, therefore T has a unique fixed point.

References

[1] Bourbaki, N. Topologie Generale; Herman: Paris, France, 1974. [2] Bakhtin, I.A. The contraction mapping principle in almost metric spaces. Funct. Anal. 1989, 30, 26–37. [3] Czerwik, S. Contraction mappings in b-metric spaces. Acta Math. Inform. Univ. Ostra. 1993, 1, 5–11. [4] Fagin, R.; Stockmeyer, L. Relaxing the triangle inequality in pattern matching. Int. J. Comput. Vis. 1998, 30, 219–231. 81 /

[5] Cortelazzo, G.; Mian, G.; Vezzi, G.; Zamperoni, P. Trademark shapes description by string matching techniques. Pattern Recognit. 1994, 27, 1005–1018. [6] McConnell, R.; Kwok, R.; Curlander, J.; Kober, W.; Pang, S. Y-S correlation and dynamic time warping: Two methods for tracking ice floes. IEEE Trans. Geosci. Remote Sens. 1991, 29, 1004–1012. [7] Heinonen, J. Lectures on Analysis on Metric Spaces; Springer: Berlin, Germany, 2001. [8] Czerwik, S. Nonlinear set-valued contraction mappings in b-metric spaces. Atti Sem. Mat. Univ. Modena1998, 46, 263–276. [9] Berinde, V. Generalized contractions in quasimetric spaces. In Seminar on Fixed Point Theory; Babe¸sBolyai University: Cluj-Napoca, Romania, 1993; pp. 3–9. [10] Boriceanu, M.; Bota, M.; Petru¸sel, A. Multivalued fractals in b-metric spaces. Cent. Eur. J. Math. 2010, 8,367–377. [11] Samreen, M.; Kamran, T.; Shahzad, N. Some fixed point theorems in b-metric space endowed with graph.Abstr. Appl. Anal. 2013, 2013, 967132. [12] Kadak, U. On the Classical Sets of Sequences with Fuzzy b-metric. Gen. Math. Notes 2014, 23, 2219–7184. [13] M. Kir, N. Kiziltun, H. On Some well known fixed point theorems in b-Metric spaces, Turkish J. Anal.Number. Theory 1 (1) (2013) 13-16. [14] Hussain, N.; Doric, D.; Kadelburg, Z.; Radenovic, S. Suzuki-type fixed point results in metric type spaces.Fixed Point Theory Appl. 2012, 2012, 126. [15] Hicks, T.L.; Rhoades, B.E. A Banach type fixed point theorem. Math. Jpn. 1979, 24, 327–330. [16] Kamran. T, Samreen. M, Ain. Q. U., A Generalization of b-Metric Space and Some Fixed Point Theorems, Mathemat- ics, 2017,5,19. [17] Wasfi Shatanawi, Kamaleldin Abodayeh, Aiman Mukheimer, Some fixed point theorems in extended b-metric spaces, U.P.B. Sci.Bull., Series A, 80(4), 71-78, 2018. [18] Maria Samreen, Tayyab Kamran, Mihai Postolache, Extended b-metric space, extended b-comparison function and nonlinear contractions, U.P.B. Sci.Bull., Series A, 80(4), 21-28, 2018. [19] Badr Alqahtani, Andreea Fulga, Erdal Karapınar, Common fixed point resultson an extended b-metric space, Journal of Inequalities and Applications, 2018:158, 2018, 15 pages. [20] Badr Alqahtani, Andreea Fulga, Erdal Karapınar, Non-unique fixed point results in extended b-metric space, Mathe- matics, 6, 68, 2018, 11 pages.

82 5Th INTERNATIONAL CONFERENCE ON RECENT ADVANCES IN PURE AND APPLIED MATHEMATICS (ICRAPAM 2018) JULY 23-27 2018 at Karadeniz Technical University Prof. Dr. Osman Turan Convention Center in Trabzon, TURKEY ICRAPAM CONFERENCE PROCEEDING

Influence of thermo diffusion, heat absorption and chemical reaction effect on non-Newtonian fluid flow

Hina Khursheed1, Muhammad Imran Asjad∗1, Rabia Naz1 1Department of Mathematics, University of Management and Technology Lahore, Pakistan

Keywords Abstract: Influence of thermo diffusion, heat absorption of a non-Newtonian fluid has Fractional second grade fluid, been analyzed taking into consideration the chemical reaction effect by means of Caputo- Oscillating vertical plate, Fabrizio derivative technique. Mathematical modeling of momentum, energy and diffusion Caputo Fabrizio, equations have been developed in term of partial differential equations accompanying by Thermo diffusion, some physical conditions. Closed form solution of the governing equations assisted by Heat absorption, suitable dimensionless variables have been attained through Laplace transform approach. Chemical reaction. Additionally, three comparisons have been performed as a limiting cases which are inline with the existing studies. Graphical illustrations have been constructed through the use of Mathcad software in order to find the numerical solution. Conclusively, flow parameters have been observed to have a significant influence on the flow field.

1. Introduction

Research on combined approach of heat and mass transfer has a long tradition. This physical phenomena has been influential in the field because of its industrial, engineering and biological implications such as oil recovery, wire drawing, food processing, piping and casting systems, paper making, metal spinning, cooling of metallic, generation of electric- ity, astrophysical flows, solar panel technology, making of artificial heart and treatments of some diseases like cancer and hyperthermia through magnetic targeted drug. For several years great effort has been devoted to the study of fluid models especially non-Newtonian fluid models accompanying by heat and mass transfer phenomenon. Hayat et al. [1, 2] investigated heat and mass transfer on steady and unsteady flow of second grade fluid along with chemical reaction and fluid film respectively. Sajid et al. [3] addressed second grade fluid lying above a stretching sheet with radiation effect. Cortell [4] measured heat transfer of an electrically conducting second grade viscous fluid lying over a stretching sheet. He studied two cases with constant and prescribed surface temperature. Abel et al. [5] described heat transfer in a non-Newtonian fluid passing through porous medium along with non-uniform heat source. Nadeem et al. [6] measured heat transfer effect on the boundary layer of a second grade fluid passing through a horizontal cylinder by applying homotopy analysis. In 2015, peristaltic flow of the second grade fluid kept in a cylindrical tube has been studied by Hameed et al. [7]. He studied the properties of magnetic field accompanying heat transfer. Furthermore, mixed convection heat and mass transfer problems along with chemical reaction and thermal radiation impact have been considered by several au- thors such as Makinde [8], Ibrahim et al. [9] Pal and Chatterjee [10], Olajuwon [11], Pal and Mondal [12], Rashidi et al.[13].

Among commonly used models for non-Newtonian fluids, second grade fractional fluid model is the most popular. Second grade fluid model was first introduced by Rivlin and Ericksen [14]. This model has been driven by numerous researchers in order to classifying the rheological properties and characteristic features. Moreover, many authors have demonstrated the unsteady flow of second grade fluid. Some interesting results has been proposed by Nazar et al. [15]. In their studies, they examined second order fluid in order to find accurate result through Laplace transformation. In addition, they stated the results as aggregate of steady state and transitory solutions. Subsequently, Farhad et al.[16] examined the electrical conductivity of second order fluid passing through porous medium. In some special cases, it has been found that the results obtained are much better as compared to those given by [15]. Primarily, this fluid model is recognized as a sub-model of differential type fluids. However, some other common models are further classified into subclass models, such as Maxwell, Oldroyd B and Burgers fluids. Friedrich [17] developed the Fractional Maxwell model from the ordinary Maxwell fluid model by considering it as a function of retardation and relaxation times. For several years, great effort has been made to analyze viscoelastic fluid flow with the fractional Maxwell model. Ariel [18] examined second grade fluid with two parallel porous walls in order to find an exact analytical solution.

∗ Corresponding author: [email protected] 83 /

The technique of Fractional calculus has been used to formulate mathematical modeling in various technological de- velopment, engineering applications and industrial sciences. Nowadays, it has proved to be an effective method for generalizing the complex dynamics of fluid flow. Particularly, for fluid dynamics, signal processing, tracer in fluent currents, viscoelasticity, bio-engineering, biological science, electrochemistry, and finance. Riemann-Liouville and Caputo fractional derivatives are among commonly applied fractional derivatives. [19, 20] However, these operators display difficulties in real problems. The Riemann-Liouville derivative contains non-zero constant. Resultantly, Laplace when applied on Riemann-Liouville derivative, comprises of terms lacking physical significance. Likewise, the kernel of Caputo fractional derivative is singular, which creates trouble while applying Laplace transform. In order to eliminate these difficulties, Ca- puto and Fabrizio in (2015) introduced the concept of fractional derivatives having exponential kernel removing singularities.

Itrat et al. [21] obtained fundamental solutions by working on time-fractional Caputo-Fabrizio derivative by means of Laplace transform. They calculated mathematical results for fractional and normal advection diffusion phenomenon. In a recent paper by Saqib et al. [22], applied the Caputo-Fabrizio derivative to get exact solutions of Jeffrey fluid at constant wall temperature above an infinite plate. They explained the momentum and energy equation through Laplace transformation. Shah and Khan [23] used the Caputo-Fabrizio fractional derivatives to examine second-grade fluid flow placed above a vertical plate and found the exact solutions through Laplace transformation.

Recently, several authors named Imran et al. [24], Zafar et al. [25], Caputo, Fabrizio [26], Atangana, Baleanu [27], Hristov [28] have proposed a new research to get valuable results using Caputo-Fabrizio fractional derivatives. Moreover, Vieru et al.[29] applied the concept of a fractional derivative in order to measure free convection flow of a viscous fluid over a vertical infinite plate with Newtonian heating and constant mass diffusion conditions. However, only a few studies are available regarding fractional model of a non-Newtonian fluid along with convection heat transfer.

Soret (thermal diffusion) effect named after Swiss scientist Charles Soret is discovered in 1879 [30]. According to this phenomenon, in fluid system consist of an enclosed medium, a temperature gradient is formed because of dynamic force until the system attains steady sate condition in the absence of any external force [31]. Soret effect has a numerous applications. It plays a vital role in hydrocarbon and crude oil reservoirs, solar ponds operating system, microstructure of the oceans, mass transport through biological membrane.

Ibrahim et al. [32] measured the Soret effect and chemical reaction along with heat and mass transfer phenomenon of non-Newtonian fluid in a porous medium. In 2015, Sheri et al. [33] discussed the Soret effect on unsteady MHD flow over a semi-infinite vertical plate with viscous dissipation. Hayat et al. [34] determined the Soret effect on the mixed convection fluid flow of electrically conducting second grade fluid emphasis on Hall ion slip effects.

Recently, Khan [35] examines heat and mass transfer in second grade fluid using Caputo-Fabrizio derivative and obtained exact solution. Nevertheless, they did not address the Soret effect and thermal radiation on the fluid flow. Based on the approach presented in [35] the purpose of this work is to analyze the influence of thermo diffusion and heat absorption on the convective flow of the the non-Newtonian fluid in the existence of heat transfer analysis involving chemical reaction effect above an oscillating vertical plate using fractional derivative method. For this purpose, governing equations are considered as a fractional differential equations created with the new fractional derivative in the absence of singular kernel. By using Caputo-Fabrizio fractional derivatives, the results thus obtained are simpler and the solution is appropriate for numerical calculations.

2. Statement of the problem

Consider an unsteady non-Newtonian fluid for natural convection that is placed over an infinite vertical flat plate. The plate occupies the flow coordinate plane xy such that horizontal axis is in the vertically upward direction and vertical axis is in the perpendicular direction. In the beginning, at t = 0, both the plate and the fluid are in static position having + uniform temperature T∞ and concentration C∞. After some time at t = 0 , the plate starts oscillation in its own plane due to which motion is created in the fluid with velocity u0H(t)exp(iωt) against the gravitational field. Where H(t) is the unit step function, u0 is the constant having dimension of velocity and ω is the oscillation frequency of the plate. In the meantime, the temperature and the concentration of the plate are elevated to Tω and Cω . It is implied that the temperature, concentration and velocity are the functions of y and t only. By taking the Boussinesq’s approximation, governing equations for the flow are as follow: 2 3 ∂u(y,t) ∂ u(y,t) α1 ∂ u(y,t) = ν + + gβT (T(y,t) − T ) + gβc(C(y,t) −C ), (1) ∂t ∂y2 ρ ∂y2∂t ∞ ∞

2 3 2 ∂T ∂ T 16σT∞ ∂ T ρCp = K + 2 − Qo(T − T∞), (2) ∂t ∂y 3K1 ∂y 84 /

2 2 ∂C ∂ C DkT ∂ T = D 2 + 2 − kr(C −C∞), (3) ∂t ∂y Tm ∂y The suitable initial and boundary conditions for the problem:

u(y,0) = 0, T(y,0) = T∞, C(y,0) = C∞, y ≥ 0, (4)

u(0,t) = uoH(t)exp(iωt), T(0,t) = Tω , C(0,t) = Cω , t > 0, u0 > 0, (5)

u(y,t) → 0, T(y,t) → T∞, C(y,t) → C∞, as y → ∞, t > 0. (6) To make the governing equations non-dimensional, following parameters and variables are introduced 2 2 ∗ u ∗ u0t T − T∞ ∗ u0y α1u0 u = , t = , θ = , y = , α2 = , u0 ν Tω − T∞ ν µν 3 2 C −C∞ 16σT∞ν µCp Qoν φ = , R = 2 , Pr = , QH = 2 , Cω −C∞ 3k1uo k ρCpuo 2 1 + R 1 ν D kT Krν µ = νρ, = , Sc = , Sr = , γ = 2 (7) Pr W D ν Tm uo into Eqs.(2.1)-(2.6) and also removing the star notations and the corresponding fractional model is given by 2 CF α 1 ∂ θ D θ = − θ(y,t)QH , (8) t W ∂y2

1 ∂ 2φ ∂ 2θ CF Dα φ = + Sr − γφ, (9) t Sc ∂y2 ∂y2

∂ 2u ∂ 2u CF Dα u = + α CF Dα + Grθ + Gmφ. (10) t ∂y2 2 t ∂y2 The dimensionless initial and boundary conditions are: u(y,0) = 0, T(y,0) = 0, C(y,0) = 0 y ≥ 0, (11)

u(0,t) = H(t)exp(iωt), T(0,t) = 1, C(0,t) = 1, t > 0, (12)

u(y,t) → 0, T(y,t) → 0, C(y,t) → 0, as y → ∞. (13) 3. Calculation of temperature field

By applying Laplace transform to Eqs. (2.8), (2.12)2 and (2.13)2, we obtain ∂ 2θ¯(y,s)  s  − θ¯(y,s) W( + QH ) = 0, (14) ∂y2 (1 − α)s + α

1 T¯(y,s) → 0 as y → ∞, T¯(0,s) = . (15) s Thus, the solution of the Eq. (3.1) subject to the conditions (3.2) is given by 1  r s  θ¯(y,s) = exp −y W( + QH ) . (16) s (1 − α)s + α Simplifying Eq. (3.3), we get ! 1 √ rs + b θ¯(y,s) = exp −y a , (17) s 2 s + c where,

1 QH αa1 a1 = , a2 = W(a1 + QH ), b = , c = αa1. (18) 1 − α a1 + QH

Applying inverse Laplace transform on Eq.(3.4) and using formula (A3) from Appendix, we get the exact solution p ∞ t 2√ √ y a2(b − c) Z Z 1 y a2 θ(y,t) = exp(−y a2) − √ √ exp(−ct − − u) 2 π 0 0 π 4u p ×I1(2 (b − c)ut)dtdu. (19) 85 /

3.1. Temperature for ordinary case (α → 1)

By taking α → 1 into Eq. (3.3), we get 1  p  θ¯(y,s) = exp −y W(s + Q ) . (20) s H

Applying inverse Laplace by using formula (A9), we get " r !  # 1 QH y p θ(y,t) = exp −y er f c √ − QHt 2 W 2 t " r !  # 1 QH y p + exp y er f c √ + QHt . (21) 2 W 2 t

4. Calculation of concentration field

By applying Laplace transformation to Eqs.(2.9), (2.12)3 and (2.13)3, we obtain ∂ 2φ¯(y,s)   s  ∂ 2θ¯(y,s) − φ¯(y,s) Sc + γ + SrSc = 0, (22) ∂y2 (1 − α)s + α ∂y2

1 C¯(y,s) → 0 as y → ∞, C¯(0,s) = . (23) s Thus, the solution of the Eq. (4.1) subject to the condition (4.2) is given by s ! 1  s  φ¯(y,s) = exp −y Sc + γ s (1 − α)s + α  r  h  s i  s  SrSc W (1−α)s+α + QH exp −y Sc (1−α)s+α + γ + h  s   s i s W (1−α)s+α + QH − Sc (1−α)s+α + γ  r  h  s i  s  SrSc W (1−α)s+α + QH exp −y W (1−α)s+α + QH − . (24) h  s   s i s W (1−α)s+α + QH − Sc (1−α)s+α + γ In suitable form ! ! 1 √ rs + d A √ rs + d φ¯(y,s) = exp −y a + exp −y a s 3 s + c s 3 s + c ! ! B √ rs + d A √ rs + b + exp −y a3 − exp −y a2 s + a4 s + c s s + c ! B √ rs + b − exp −y a2 , (25) s + a4 s + c

αa1γ a2b−a3d 1 where, a3 = Sc(a1 + γ), d = , a4 = , a5 = a2SrSc,a1 = , a2 = W(a1 + QH ), a1+γ a2−a3 1−α QH αa1 a5b a5(b−a4) b = , c = αa1,A = , B = , a1+QH a4 a4

! 1 √ rs + d ψ (y,s;d,c) = exp −y a , (26) 1 s 3 s + c ! 1 √ rs + d ψ2(y,s;d,c) = exp −y a3 , (27) s + a4 s + c ! 1 √ rs + b ψ (y,s;b,c) = exp −y a , (28) 3 s 2 s + c ! 1 √ rs + b ψ4(y,s;b,c) = exp −y a2 . (29) s + a4 s + c 86 /

Applying inverse Laplace transform to Eq. (4.4) and using formula’s A1 and A2 from the Appendix, we obtain the exact solution for concentration field

φ(y,t) = ψ1(y,t;d,c) + Aψ1(y,t;d,c) + Bψ2(y,t;d,c) + Aψ3(y,t;b,c)

+Bψ4(y,t;b,c), (30) where,

p ∞ t √ y a3(d − c) Z Z 1 ψ1(y,t;d,c) = exp(−y a3) − √ √ 2 π 0 0 π 2√ y a3 p exp(−ct − − u) × I (2 (d − c)ut)dtdu, (31) 4u 1

p ∞ t √ y a3(d − c) Z Z 1 ψ2(y,t;d,c) = exp(−a4 − y a3) − √ √ 2 π 0 0 π 2√ y a3 p exp(−a t − ct − − u) × I (2 (d − c)ut)dtdu, (32) 4 4u 1

p ∞ t √ y a2(b − c) Z Z 1 ψ3(y,t;b,c) = exp(−y a2) − √ √ 2 π 0 0 π 2√ y a2 p exp(−ct − − u) × I (2 (b − c)ut)dtdu, (33) 4u 1

p ∞ t √ y a2(b − c) Z Z 1 ψ4(y,t;b,c) = exp(−a4 − y a2) − √ √ 2 π 0 0 π 2√ y a2 p exp(−a t − ct − − u) × I (2 (b − c)ut)dtdu. (34) 4 4u 1 4.1. Concentration for ordinary case (α → 1)

Taking α → 1 in equation (4.3), we get       1 −y p E F −y p φ¯(y,s) = exp √ (s + γ) + e0 + exp √ (s + γ) s Sc s s + e Sc     E F −y p −e0 + exp √ (s + QH ) (35) s s + e W

Applying Laplace transform to Eq. (4.14) and using the formula (A9) and (A10) from the Appendix,we get

φ(y,t) = ξ3(y,t,γ) + e0Eξ4(y,t,e,γ) + e0Fξ4(y,t,e,γ)

−e0Eξ4(y,s,QH ) − e0Fξ5(y,t,e,QH ) (36) where, Q e − Q WQ − Scγ E = H , F = H , e = WSrSc, e = H . (37) e e 0 W − Sc

5. Solution of velocity field

By applying Laplace transform to Eqs.(2.10), (2.12)1 and (2.13)1, we obtain

su¯(y,s)   s  ∂ 2u¯(y,s) = 1 + α + Grθ¯(y,s) + Gmφ¯(y,s) (38) (1 − α)s + α 2 (1 − α)s + α ∂y2

1 u¯(y,t) → 0 as y → ∞, u¯(0,t) = . (39) s − iω

87 /

Thus, the solution of the Eq. (5.1) subject to conditions Eq. (5.2) in a suitable form is given by s u¯(y,s) = .ψ (y,s;b ,a) s − iω 1 2  G H  +b A + + .ψ (y,s;b ,a) 4 s + n s + m 1 2  G H  +b A + + .ψ (y,s;b ,a) 5 s + n s + m 1 2  K L M  +b 1 + + .ψ (y,s;b ,a,a ) 6 s s + r s + p 1 2 4  G H  −b A + + .ψ (y,s;b ,a) 6 s + n s + m 1 2  G H  −b A + + .ϕ (y,s;b,c) 4 s + n s + m 1  G H  −b A + + .ϕ (y,s;b,c) 5 s + n s + m 2  K L M  −b 1 + + .ϑ (y,s;d,c,a ) 6 s s + r s + p 1 4  K L M  +b 1 + + .ϑ (y,s;d,c,a ), (40) 6 s s + r s + p 2 4 where, ! 1 r b s ψ (y,s;b ,a) = exp −y 2 , (41) 1 2 s s + a r ! 1 b2s ψ1(y,s;b2,a,a4) = exp −y , (42) s + a4 s + a ! 1 √ rs + b ϕ (y,s;b,c) = exp −y a , (43) 1 s 2 s + c ! 1 √ rs + d ϕ (y,s;d,c) = exp −y a , (44) 2 s 3 s + c ! 1 √ rs + d ϑ1(y,s;d,c,a4) = exp −y a3 , (45) s + a4 s + c ! 1 √ rs + b ϑ2(y,s;b,c,a4) = exp −y a2 , (46) s + a4 s + c

b3Gr = b4, b3Gm = b5, a5b3Gm = b6. (47)

Applying inverse Laplace transform to Eq. (5.3) termwise and using (A1) to (A8) from Appendix, for this we consider

u(y,t) = u1(y,t) + u2(y,t) + u3(y,t) + u4(y,t) − u5(y,t)

−u6(y,t) − u7(y,t) − u8(y,t) + u9(y,t), (48) where,

−iωt u1(y,t) = (δt + iωe ) ∗ ψ1(y,t;b2,a) (49)

 −nt  u2(y,t) = b4A.ψ1(y,t;b2,a) + b4G (e ) ∗ ψ1(y,t;b2,a)  −mt  +b4H.ψ1(y,t;b2,a) + b4G (e ) ∗ ψ1(y,t;b2,a) (50)

 −nt  u3(y,t) = b5A.ψ1(y,t;b2,a) + b5G (e ) ∗ ψ1(y,t;b2,a)  −mt  +b5H.ψ1(y,t;b2,a) + b5G (e ) ∗ ψ1(y,t;b2,a) (51)

88 /

 −a4   −a4t  u4(y,t) = b6 (δt + a4e ) ∗ ψ1(y,t;b2,a) + b6K (e ) ∗ ψ1(y,t;b2,a)  −rt   −pt  +b6L (e ) ∗ ψ1(y,t;b2,a) + b6M (e ) ∗ ψ1(y,t;b2,a) , (52)

 −nt  u5(y,t) = b6A.ψ1(y,t;b2,a) + b6G (e ) ∗ ψ1(y,t;b2,a)  −mt  +b6H.ψ1(y,t;b2,a) + b6G (e ) ∗ ψ1(y,t;b2,a) (53)

 −nt  u6(y,t) = b4A.ϕ1(y,t;b,c) + b4G (e ) ∗ ϕ1(y,t;b,c)  −mt  +b4H.ϕ1(y,t;b,c) + b4G (e ) ∗ ϕ1(y,t;b,c) (54)

 −nt  u7(y,t) = b5A.ϕ1(y,t;d,c) + b5G (e ) ∗ ϕ1(y,t;d,c)  −mt  +b5H.ϕ1(y,t;d,c) + b5G (e ) ∗ ϕ1(y,t;d,c) (55)

 −a4   −a4t  u8(y,t) = b6 (δt + a4e ) ∗ ϑ1(y,t;d,c,a4) + b6K (e ) ∗ ϑ1(y,t;d,c,a4)  −rt   −pt  +b6L (e ) ∗ ϑ1(y,t;d,c,a4) + b6M (e ) ∗ ϑ1(y,t;d,c,a4) (56)

 −a4   −a4t  u9(y,t) = b6 (δt + a4e ) ∗ ϑ1(y,t;b,c,a4) + b6K (e ) ∗ ϑ1(y,t;b,c,a4)  −rt   −pt  +b6L (e ) ∗ ϑ1(y,t;b,c,a4) + b6M (e ) ∗ ϑ1(y,t;b,c,a4) , (57) where (∗) denotes convolution product.

5.1. Limiting cases

1. Velocity field for the comparison of fractional Newtonian with fractional second grade when α2 → 0. 2. Velocity field in the absence of Gm = R = QH = 0 is identical to those obtained by [35]. 3. Concentration field for the comparison of Caputo-Fabrizio fractional derivative approach with Atangana-Baleanu [36].

6. Results

This study determines the exact solution of second grade non-Newtonian fluid over an oscillating vertical plate. These theoretical results have been carried out by using Laplace transformation and Caputo- Fabrizio derivative. To see the behavior of the fluid flow parameters, for instance, fractional parameter α, second grade parameter α2, Mass Grashof number Gm, Prandtl number Pr, Grashof number Gr, Soret number Sr, radiation parameter R, Schmidt number Sc, chemical reaction parameter γ and heat absorption parameter QH several numerical simulations were performed. The results computed have been discussed and presented in the graphs considering that all the parameters are dimensionless. In figures 1 to 6 temperature profile θ(y,t) has been plotted for different flow parameters. In figures 7 to 14 concentration profile φ(y,t) has been plotted. Likewise the velocity profile u(y,t) has been presented in the figures 15 to 28.

The figures 1,7 and 14 is outlining the variation of the fractional parameter α for the temperature, concentration and velocity field. It can be observed that the θ(y,t), φ(y,t) and u(y,t) related to fluid surges as we grow the value of parameter α. Further, the figures 5,12 and 13 is displaying the similar results for the variant values of time t against y. It can be depicted from the above results that the thickness of the boundary layer enhances as time (t) and fractional parameter (α) increases. The figures 2,8 and 16 is demonstrating the temperature, concentration and velocity field for various values of the Prandtl number Pr. By keeping the values of rest of parameters constant, it is noted that θ(y,t), φ(y,t) and u(y,t) of the fluid decreases as the value of Pr increases. Practically, it is because the thermal conductivity is greater for the smaller values of Prandtl number Pr. Thus, resulting a decrease in the thickness of the boundary layer. The figures 11 and 19 is representing the impact that chemical reaction parameter γ has on concentration and velocity field. From the figures, it is evident that as the values of γ increases the concentration and velocity of the fluid reduces. Figures 3 and 17, indicates the consequence of heat absorption parameter QH on the temperature and velocity field versus y. As expected, the temperature and velocity of the fluids are reduced by enhancing the amount of QH. Actually, it occurs due to the reason that boundary layers of the fluid absorbs some amount of heat. Resultantly, temperature and velocity of the fluid decrease along with boundary layer thickness. The impact of radiation parameter R on the temperature and velocity field can be seen in the figures 4 and 18. As shown in the figures 4 and 18, both temperature and velocity are the increasing functions of R . Furthermore, R enhances the boundary layer thickness. The influence of the second grade parameter α2 can be found in figure 15. It is noticed that the fluid velocity is the decreasing function of α2. Figure 9 and 20 is describing the outcome of Schmidt number Sc while the figure 10 and 21 is outlining the influence of Soret number Sr on the concentration and 89 / velocity field. It is detected that the influence of both the numbers is opposite on fluid concentration and velocity. By increasing the value of Sc decreases both φ(y,t) and u(y,t) whereas the increasing value of Sr increases both φ(y,t) and u(y,t). From the physical aspect, it is because of the fact that by enhancing the value of Sc molecular diffusivity reduces which lead to the reduction of thickness along the boundary layer. The figures 22 and 23 is illustrating the impact of mass Grashof number Gm and Grashof number Gr respectively. Clearly, it is revealed from the figures that both have the similar influence on the fluid velocity which turns out to be the increasing function of Gm and Gr parameters. In physical aspect, it is due to the thermal buoyancy and viscous forces followed by natural convection. The figures 6,13 and 25 are presenting the fractional parameter α when it approaches 1 for temperature θ, concentration (φ) and velocity (u) fields. On the basis of the graphical results it can be confirmed that both α and α → 1 are in complete agreement. At the end some comparisons has been shown as a limiting cases in figures 26 − 28. Firstly, the comparison of fractional second grade with fractional Newtonian is shown in the figure 26. It can be examined that for α2 → 0 both fluid models are in good agreement. Then, comparison of our results with Nehad and Khan [35] has been performed in figure 27. It can be depicted from the graphical result that by taking Gm = QH = R = 0 both the results are in good agreement. Likewise, Imran et al. [36] solved the concentration field by Atangana-Baleanu fractional derivative whereas we solved the same function with Caputo-Fabrizio fractional derivative. It has been observed that by varying the values of αi.e.(α = 0.2,α = 0.4,α = 0.6) fractional model with Atangana-Baleanu exhibited the more memory effect of the same function as compared to Caputo-Fabrizio. As soon as the value of α reaches 1 both the results coincide which can be seen in figure 28.

Nomenclature: cp Specific heat at constant pressure ρ Fluid density γ Chemical reaction parameter ν Kinematic viscosity g Gravitational acceleration k Thermal conductivity D Mass diffusivity γ Chemical reaction parameter QH Heat absorption parameter µ Dynamic viscosity Gm Mass Grashof number Cw Wall concentration Gr Grashof number C∞ Concentration far away from the plate Pr Prandtl number Tw Wall temperature Sc Schmidt number T∞ Temperature far away from the plate Sr Soret number βT Thermal expansion coefficient s Laplace transform parameter βc Concentration expansion coefficient α Fractional parameter α1 Second grade parameter H(t) Heaviside unit step function α2 Normal stress Moduli

90 /

Figure 1. Temperature profile projected for t = 4, Pr = 10, R = 4, QH = 0.5 with variant values of α against y.

Figure 2. Temperature profile projected for t = 4, α = 0.4, R = 4, QH = 0.6 with variant values of Pr against y.

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Figure 3. Temperature profile projected for t = 4, Pr = 10, R = 5, α = 0.4 with variant values of QH against y.

Figure 4. Temperature profile projected for t = 4, Pr = 10, α = 0.6, QH = 2.5 with variant values of R against y.

Figure 5. Temperature profile projected for α = 0.2, Pr = 10, R = 4, QH = 0.4 with variant values of time against y.

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Figure 6. Temperature projected comparison for ordinary case when α → 1.

Figure 7. Concentration profile projected for t = 4, Pr = 15, γ = 0.1, R = 5, QH = 3, Sc = 5, Sr = 10 with variant values of α against y.

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Figure 8. Concentration profile projected for t = 1, α = 0.1, γ = 8, R = 0.1, QH = 0.01, Sc = 1.5, Sr = 1.5 with variant values of Pr against y.

Figure 9. Concentration profile projected for t = 4, α = 0.4, γ = 0.1, R = 5, QH = 2.5, Pr = 15, Sr = 10 with variant values of Sc against y.

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Figure 10. Concentration profile projected for t = 4, α = 0.4, γ = 0.1, R = 5, QH = 3, Pr = 15, Sc = 5 with variant values of Sr against y.

Figure 11. Concentration profile projected for t = 4, α = 0.4, R = 5, QH = 3, Pr = 15, Sc = 5,Sr = 10 with variant values of γ against y.

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Figure 12. Concentration profile projected for α = 0.2, γ = 0.1, R = 5, QH = 3, Pr = 15, Sc = 5,Sr = 10 with variant values of time against y.

f (,)yt

Figure 13. Concentration projected comparison for ordinary case when α → 1.

Figure 14. Concentration projected comparison of Caputo-Fabrizio and Atangana-Baleanu derivative[36] when α → 1.

96 /

uyt(,)

Figure 15. Velocity profile projected for t = 4, γ = 0.5, R = 0.1, QH = 0.01, Pr = 10, Sc = 1.5, Sr = 1.5, Gr = 10, Gm = 10, α2 = 0.5 with variant values of α against y.

uyt(,)

Figure 16. Velocity profile projected for t = 4, γ = 0.1, R = 0.1, QH = 0.01, Pr = 5, Sc = 1.5, Sr = 1.5, Gr = 10, Gm = 10,α = 0.6 with variant values of α2 against y.

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uyt(,)

Figure 17. Velocity profile projected for t = 4, γ = 1, α2 = 1, R = 0.1, QH = 0.01, Sc = 0.5, Sr = 0.5, Gr = 10, Gm = 10,α = 0.1 with variant values of Pr against y.

uyt(,)

Figure 18. Velocity profile projected for t = 4, γ = 1, α2 = 0.5, Pr = 1, R = 0.1, Sc = 0.5, Sr = 0.5, Gr = 5, Gm = 5,α = 0.1 with variant values of QH against y.

98 /

uyt(,)

Figure 19. Velocity profile projected for t = 4, γ = 5, α2 = 0.5, Pr = 5, QH = 0.01, Sc = 0.5, Sr = 0.5, Gr = 10, Gm = 10,α = 0.3 with variant values of R against y.

Figure 20. Velocity profile projected for t = 1, R = 0.1 , α2 = 0.5, Pr = 5, QH = 0.01, Sc = 1.5, Sr = 5, Gr = 5, Gm = 5, α = 0.1 with variant values of γ against y.

99 /

uyt(,)

Figure 21. Velocity profile projected for t = 4, γ = 0.1, α2 = 0.5, Pr = 5, QH = 0.01, R = 0.5 , Sr = 1, Gr = 10, Gm = 10,α = 0.4 with variant values of Sc against y.

uyt(,)

Figure 22. Velocity profile projected for t = 6, γ = 0.5, α2 = 1, Pr = 5, QH = 0.01, R = 1 , Sc = 1.5, Gr = 5, Gm = 5,α = 0.3 with variant values of Sr against y.

100 /

uyt(,)

Figure 23. Velocity profile projected for t = 6, γ = 3, α2 = 0.5, Pr = 10, QH = 0.01, R = 0.1 , Sc = 1.5, Sr = 5, Gr = 7, α = 0.4 with variant values of Gm against y.

uyt(,)

Figure 24. Velocity profile projected for t = 4, γ = 0.5, α2 = 0.5, Pr = 10, QH = 0.01, R = 1 , Sc = 1.5, Sr = 7, Gm = 5,α = 0.4 with variant values of Gr against y.

101 /

uyt(,)

Figure 25. Velocity profile projected for γ = 0.1, α2 = 0.5, Pr = 5, QH = 0.01, R = 0.1 , Sc = 1.5, Sr = 5, Gm = 5, Gr = 5, α = 0.1 with variant values of time against y.

Figure 26. Velocity projected comparison of fractional second grade and fractional Newtonian.

7. Discussion and conclusion

The exact solution for non-Newtonian fluid along with the chemical reaction, heat absorption, Soret effect and thermal radiation have been analyzed. Fractional Caputo-Fabrizio derivative approach has been considered to find out solution of temperature, concentration and velocity field. Afterward, the results obtained are projected graphically to measure physical impact of flow parameters and further discussed in detail. The findings of this study can be concluded as follow:

• Temperature is the increasing function of time t, fractional parameter α and thermal radiation R whereas it reduces when the value of Prandtl number Pr as well as heat absorption parameter QH increase.

• Concentration is the increasing function of time t, fractional parameter α and Soret number Sr whereas, it decreases as the values of Schmidt number Sc, chemical reaction γ and Prandtl number Pr increase. • Velocity is the increasing function of time t, fractional parameter α , thermal radiation R, Soret number Sr, Grashof number Gr and the Mass Grashof number Gm whereas, it reduces as the values of Schmidt number Sc, chemical reaction γ, Prandtl number Pr, heat absorption parameter QH and second grade parameter α2 increase.

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Figure 27. Velocity projected comparison of present result when Gm=QH=R=0 and Nehad and Khan[35].

8. Appendix

 2  −1 2b2 R ∞ sin(yx) −atx •Ł [ψ1(y,s;b2,a)] = 1 − 2 exp 2 dx = ψ1(y,t;b2,a) π 0 x(b2+x ) b2+x ...... (A1)  q  −1  1  b2c 2b2c •Ł s ψ1(y,s;b2,a,a4) = exp ct − y a+c − 1 − π R ∞ sin(yx)  −atx2  0 2 exp 2 = ψ1(y,t;b2,a,a4) ...... (A2) x(b2c+(a+c)x ) b2+x √ √ √ y a (b−c) y2 a •Ł−1 [ (y,s;b,c)] = exp(−y) a − √2 R ∞ R t √1 exp(−ct − 2 − u) ϕ1 2 2 π 0 0 π 4u p × I1(2 (b − c)ut)dtdu = ϕ1(y,t;b,c) ...... (A3) √ √ y a (d−c) •Ł−1 [ (y,s;d,c)] = exp(−y) a − √3 R ∞ R t √1 ϕ2 3 2 π 0 0 π √ 2 a3y p exp(−ct − 4u − u) × I1(2 (b − c)ut)dtdu = ϕ2(y,t;d,c) ...... (A4) √ y a (d−c) •Ł−1 [ (y,s;d,c,a )] = exp(−a − y) − √3 R ∞ R t √1 ϑ1 4 4 2 π 0 0 π √ 2 a3y p exp(−a4t − ct − 4u − u) × I1(2 (b − c)ut)dtdu = ϑ1(y,t;d,c,a4) ...... (A5) √ y (b−c) •Ł−1 [ (y,s;b,c,a )] = exp(−a − y) − √ R ∞ R t √1 ϑ2 4 4 2 π 0 0 π y2 p exp(−a4t − ct − 4u − u) × I1(2 (b − c)ut)dtdu = ϑ2(y,t;b,c,a4) ...... (A6) −1  1  •Ł s−a = exp(at) ...... (A7) R t • f ∗ g(t) = 0 f (τ)g(t − τ)dτ ...... (A8) h  i h  √   √ i •Ł−1 1 exp −yp(s + b) = 1 exp −y b er f c √y − bt s 2 2 t h  √   √ i + 1 exp y b er f c √y + bt = (y,t,b)...... (A ) 2 2 t ξ1 9

h  i et h √  i •Ł−1 1 exp −yp(s + b) = e exp−y b + e
er f c √y − p(b + e)t s+e 2 2 t

et h √  i + e expy b + e
er f c √y + p(b + e)t = (y,t,b,e)...... (A ) 2 2 t ξ2 10

103 /

9. Acknowledgment The authors are highly thankful and grateful to the Department of Mathematics, University of Management and Technology Lahore, Pakistan for supporting and facilitating the research.

References [1] Hayat, T., Abbas, Z. Sajid, M. (2008), Heat and mass transfer analysis on the flow of a second grade fluid in the presence of chemical reaction, Physics letters A, 372(14), 2400-2408. [2] Hayat, T., Saif, S. Abbas, Z. (2008), The influence of heat transfer in an MHD second grade fluid film over an unsteady stretching sheet, Physics letters A, 372(30), 5037-5045. [3] Sajid, M., Ahmad, I. Hayat, T. Ayub, M. (2009), Unsteady flow and heat transfer of a second grade fluid over a stretching sheet, Communications in Nonlinear Science and Numerical Simulation, 14(1), 96-108. [4] Cortell, R. (2007), Viscous flow and heat transfer over a nonlinearly stretching sheet, Applied Mathematics and Computation, 184(2), 864-873. [5] Abel, M. S. Nandeppanavar, M. M. Malipatil, S. B. (2010), Heat transfer in a second grade fluid through a porous medium from a permeable stretching sheet with non-uniform heat source/sink, International Journal of Heat and Mass Transfer, 53(9-10), 1788-1795. [6] Nadeem, S., Rehman, A., Lee, C. Lee, J. (2012), Boundary layer flow of second grade fluid in a cylinder with heat transfer, Mathematical Problems in Engineering. [7] Hameed, M., Khan, A. A. Ellahi, R. Raza, M. (2015), Study of magnetic and heat transfer on the peristaltic transport of a fractional second grade fluid in a vertical tube, Engineering Science and Technology, an International Journal, 18(3), 496-502. [8] Makinde, O. D. (2005), Free convection flow with thermal radiation and mass transfer past a moving vertical porous plate, International Communications in Heat and Mass Transfer, 32(10), 1411-1419. [9] Ibrahim, F. S., Elaiw, A. M. Bakr, A. A. (2008), Influence of viscous dissipation and radiation on unsteady MHD mixed convection flow of micropolar fluids, Appl. Math. Inf. Sci, 2, 143-162. [10] Pal, D. Chatterjee, S. (2010), Heat and mass transfer in MHD non-Darcian flow of a micropolar fluid over a stretching sheet embedded in a porous media with non-uniform heat source and thermal radiation, Communications in Nonlinear Science and Numerical Simulation, 15(7), 1843-1857. [11] Olajuwon, B. I. (2011). Convection heat and mass transfer in a hydromagnetic flow of a second grade fluid in the presence of thermal radiation and thermal diffusion. International Communications in Heat and Mass Transfer, 38(3), 377-382. [12] Pal, D. Mondal, H. (2012), Influence of chemical reaction and thermal radiation on mixed convection heat and mass transfer over a stretching sheet in Darcian porous medium with Soret and Dufour effects. Energy conversion and management, 62, 102-108. [13] Rashidi, M. M., Rostami, B. Freidoonimehr, N. Abbasbandy, S. (2014), Free convective heat and mass transfer for MHD fluid flow over a permeable vertical stretching sheet in the presence of the radiation and buoyancy effects. Ain Shams Engineering Journal, 5(3), 901-912. [14] Rivlin, R. S. (1955), Further remarks on the stress-deformation relations for isotropic materials, Journal of Rational Mechanics and Analysis, 4, 681-702. [15] Nazar, M., Fetecau, C. Vieru, D.Fetecau, C. (2010), New exact solutions corresponding to the second problem of Stokes for second grade fluids, Nonlinear Analysis: Real World Applications, 11(1), 584-591. [16] Ali, F., Khan, I. Shafie, S. (2014), Closed form solutions for unsteady free convection flow of a second grade fluid over an oscillating vertical plate, PLoS One, 9(2), e85099. [17] Friedrich, C. H. R. (1991), Relaxation and retardation functions of the Maxwell model with fractional derivatives, Rheologica Acta, 30(2), 151-158. [18] Ariel, P. D. (2001), Axisymmetric flow of a second grade fluid past a stretching sheet, International Journal of Engineering Science, 39(5), 529-553. [19] Hilfer, R. (2008). Threefold introduction to fractional derivatives. Anomalous transport: Foundations and applications, 17-73. [20] Gorenflo, R., Mainardi, F. Moretti, D. Paradisi, P. (2002), Time fractional diffusion: a discrete random walk approach, Nonlinear Dynamics, 29(1-4), 129-143. 104 /

[21] Mirza, I. A. Vieru, D. (2017), Fundamental solutions to advection–diffusion equation with time-fractional Ca- puto–Fabrizio derivative, Computers and Mathematics with Applications, 73(1), 1-10. [22] Saqib, M., Ali, F., Khan, I., Sheikh, N. A. Jan, S. A. A. (2017), Exact solutions for free convection flow of generalized Jeffrey fluid: A Caputo-Fabrizio fractional model. Alexandria Engineering Journal. [23] Sheikh, N. A., Ali, F., Khan, I. Saqib, M. (2016), A modern approach of Caputo–Fabrizio time-fractional derivative to MHD free convection flow of generalized second-grade fluid in a porous medium. Neural Computing and Applications, 1-11. [24] Asjad, M. I., Shah, N. A., Aleem, M. Khan, I. (2017). Heat transfer analysis of fractional second-grade fluid subject to Newtonian heating with Caputo and Caputo-Fabrizio fractional derivatives: A comparison. The European Physical Journal Plus, 132(8), 340. [25] Zafar, A. A. Fetecau, C. (2016), Flow over an infinite plate of a viscous fluid with non-integer order derivative without singular kernel. Alexandria Engineering Journal, 55(3), 2789-2796. [26] Caputo, M. Fabrizio, M. (2016), Applications of new time and spatial fractional derivatives with exponential kernels. Progr. Fract. Differ. Appl, 2(2), 1-11. [27] Atangana, A. Baleanu, D. (2016), New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model. arXiv preprint arXiv:1602.03408. [28] Hristov, J. (2016), Steady-state heat conduction in a medium with spatial non-singular fading memory: derivation of Caputo-Fabrizio space-fractional derivative with Jeffrey’s kernel and analytical solutions. Thermal science, (00), 115-115. [29] Vieru, D., Fetecau, C. Fetecau, C. (2015), Time-fractional free convection flow near a vertical plate with Newtonian heating and mass diffusion. Thermal Science, 19(suppl. 1), 85-98. [30] De Groot, S. R. Mazur, P. (2013), Non-equilibrium thermodynamics. Courier Corporation. [31] Platten, J. K. Costesèque, P. (2004), Charles Soret. A short biography a. The European Physical Journal E, 15(3), 235-239. [32] Ibrahim, F. S. Hady, F. M. Abdel-Gaied, S. M. Eid, M. R. (2010), Influence of chemical reaction on heat and mass transfer of non-Newtonian fluid with yield stress by free convection from vertical surface in porous medium considering Soret effect. Applied Mathematics and Mechanics, 31(6), 675-684. [33] Sheri, S. R. Srinivasa Raju, R. (2015), Soret effect on unsteady MHD free convective flow past a semi-infinite vertical plate in the presence of viscous dissipation, International Journal for Computational Methods in Engineering Science and Mechanics, 16(2), 132-141. [34] Hayat, T. Nawaz, M. (2011). Soret and Dufour effects on the mixed convection flow of a second grade fluid subject to Hall and ion?slip currents. International Journal for Numerical Methods in Fluids, 67(9), 1073-1099. [35] Shah, N. A. Khan, I. (2016). Heat transfer analysis in a second grade fluid over and oscillating vertical plate using fractional Caputo-Fabrizio derivatives, The European Physical Journal C, 76(7), 362. [36] Asjad, M. I., Miraj, F., Khan, I. (2018). Soret effects on simultaneous heat and mass transfer in MHD viscous fluid through a porous medium with uniform heat flux and Atangana-Baleanu fractional derivative approach. The European Physical Journal Plus, 133(6), 224.

105 5Th INTERNATIONAL CONFERENCE ON RECENT ADVANCES IN PURE AND APPLIED MATHEMATICS (ICRAPAM 2018) JULY 23-27 2018 at Karadeniz Technical University Prof. Dr. Osman Turan Convention Center in Trabzon, TURKEY ICRAPAM CONFERENCE PROCEEDING

On The Fekete–Szegö Problem For a New Class of m-Fold Symmetric Bi-Univalent Functions Satisfying Subordination Condition Given by Symmetric Q-Derivative Operator

Arzu Akgül∗1 1Department of Mathematics, Faculty of Arts and Science, Kocaeli University, Kocaeli, Turkey

q, Keywords Abstract: In this paper, we investigate a new subclass S λ of Σ consisting of analytic Σm m Analytic functions, and m-fold symmetric bi-univalent functions given by symmetric q-derivative operator m-fold symmetric bi-univalent and satisfying subordination in the open unit disk U. We consider the Fekete-Szegö functions, inequalities for this class. Also, we establish estimates for the coefficients for this subclas Coefficient bounds. and several related classes are also considered and connections to earlier known results are made.

1. Introduction

Let A denote the class of functions of the form ∞ n f (z) = z + ∑ anz , (1) n=2 which are analytic in the open unit disk U = {z : |z| < 1}, and let S be the subclass of A consisting of the form (1) which are also univalent in U. 1 The Koebe one-quarter theorem [11] states that the image of U under every function f from S contains a disk of radius 4 . Thus every such univalent function has an inverse f −1 which satisfies

f −1 ( f (z)) = z (z ∈ U) and  1 f f −1 (w)
= w |w| < r ( f ) , r ( f ) ≥ , 0 0 4 where −1 2 2
3 3
4 f (w) = w − a2w + 2a2 − a3 w − 5a2 − 5a2a3 + a4 w + ··· . (2) A function f ∈ A is said to be bi-univalent in U if both f and f −1 are univalent in U. Let Σ denote the class of bi-univalent functions defined in the unit disk U. For a brief history and interesting examples in the class Σ, see [27]. Examples of functions in the class Σ are

z 1 1 + z , −log(1 − z), log 1 − z 2 1 − z and so on. However, the familier Koebe function is not a member of Σ. Other common examples of functions in S such as

z2 z z − and 2 1 − z2 are also not members of Σ (see [27]). An analytic function f is said to be subordinate to another analytic function g, written

f (z) ≺ g(z), (3) provided that there is an analytic function w defined on U with

w(0) = 0 and |w(z)| < 1

∗ Corresponding author: [email protected] 106 / satisfying the following condition:

f (z) = g(w(z)).

Lewin [16] studied the class of bi-univalent functions, obtaining the√ bound 1.51 for modulus of the second coefficient |a2|. Subsequently, Brannan and Clunie [8] conjectured that |a2| ≤ 2 for f ∈ Σ. Later, Netanyahu [21] showed that 4 max|a2| = 3 if f (z) ∈ Σ. Brannan and Taha [9] introduced certain subclasses of the bi-univalent function class Σ similar to the familiar subclasses. S? (β) and K (β) of starlike and convex function of order β (0 ≤ β < 1) respectively (see [21]). ? The classes SΣ (α) and KΣ (α) of bi-starlike functions of order α and bi-convex functions of order α, corresponding to the ? ? function classes S (α) and K (α), were also introduced analogously. For each of the function classes SΣ (α) and KΣ (α), they found non-sharp estimates on the initial coefficients. In fact, the aforecited work of Srivastava et al. [27] essentially revived the investigation of various subclasses of the bi-univalent function class Σ in recent years. Recently, many authors investigated bounds for various subclasses of bi-univalent functions ([1], [2], [3], [4],[6], [12], [19], [25], [26],[27], [28],[5][31]). Not much is known about the bounds on the general coefficient |an| for n ≥ 4. In the literature, the only a few works determining the general coefficient bounds |an| for the analytic bi-univalent functions , [13], [15]). The coefficient estimate problem for each of |an| ( n ∈ N\{1,2}; N = {1,2,3,...}) is still an open problem. For each function f ∈ S, the function

pm h(z) = f (zm)(z ∈ U, m ∈ N) (4) is univalent and maps the unit disk U into a region with m-fold symmetry. A function is said to be m-fold symmetric (see [17], [24]) if it has the following normalized form:

∞ mk+1 f (z) = z + ∑amk+1z (z ∈ U, m ∈ N). (5) k=1

We denote by Sm the class of m-fold symmetric univalent functions in U, which are normalized by the series expansion (4). In fact, the functions in the class S are one-fold symmetric. Analogous to the concept of m-fold symmetric univalent functions, we here introduced the concept of m-fold symmetric bi-univalent functions. Each function f ∈ Σ generates an m-fold symmetric bi-univalent function for each integer m ∈ N. The normalized form of f is given as in (4) and the series expansion for f −1, which has been recently proven by Srivastava et al. [29], is given as follows:

m+1  2
 2m+1 g(w) = w − am+1w + m + 1)am+1 − a2m+1 w (6) 1  − (m + 1)(3m + 2)a3 − (3m + 2)a a + a w3m+1 + ··· . 2 m+1 m+1 2m+1 3m+1

−1 where f = g. We denote by Σm the class of m-fold symmetric bi-univalent functions in U. For m = 1, the formula (5) coincides with the formula (2) of the class Σ. Some examples of m-fold symmetric bi-univalent functions are given as follows: m 1 " m 1 #  z  m 1 1 1 + z  m , [−log(1 − zm)] m , log . 1 − zm 2 1 − zm

In the present paper, we also denote by P the class of analytic functions of the form:

2 3 p(z) = 1 + p1z + p2z + p3z + ··· such that

R(p(z)) > 0 (z ∈ U). According the work of Pommerenke [24], the m-fold symmetric function p in the class P is of the form:

2m 3m p(z) = 1 + pmz + p2mz + p3mz + ··· (7)

From beginning to the end of this study, it will be observed that ϕ is an analytic function with positive real part in the unit disk U such that

ϕ(0) = 1 and ϕ(0) > 0. and ϕ(U) is symmetric with respect to the real axis. The function ϕ has a series expansion of the form:

2 3 ϕ (z) = 1 + B1z + B2z + B3z + ··· (B1 > 0). (8) 107 /

Let u(z) and v(z) be two analytic functions in the unit disk U with

u(0) = v(0) = 0 and max{|u(z)|,|v(z)|} < 1.

We assume that

m 2m 3m u(z) = bmz + b2mz + b3mz + ··· (9) and

m 2m 3m v(w) = cmw + c2mw + c3mw + ···. (10) Also we notice that 2 2 |bm| ≤ 1, |b2m| ≤ 1 − |bm| ,|cm| ≤ 1, |c2m| ≤ 1 − |cm| (11) By some simple calculations we can state that

m 2
2m ϕ (u(z)) = 1 + B1bmz + B1b2m + B2bm z + ···(|z| < 1) (12) and

m 2
2m ϕ (v(w)) = 1 + B1cmw + B1c2m + B2cm z + ···(|w| < 1) (13)

Quantum calculus, sometimes denominated calculus with no limit is equivalent to classic infinite estimal calculus without the notion of limits. It defines q-calculus and and h-calculus , where h apparently represents Plank’s contact while q represents quantum. The q-calculus , while dating in one respect to Leonhard Euler and Carl Custav Jakobi, is beginning to see more usefulness in quantum mechanics in recent years, having an intimate connection with commutativity relations and Lie algebra. Recently the researchers worry about the area of q-calculus . The reason of this great interest is its applications in diversified branch of mathematics, phisics and mechanics, as for example, in the areas of ordinary fractional mathematics, optimal control problems, q-difference and q-integral equations and in q-transform analysis. The application of q-calculus was started by Jackson [14], Arals et all [7]. We give the definitions of fractional q-calculus operators of complex valued function f (z). Let q ∈ (0,1). For k ∈ N the q-fractional [k]q! is defined by

 n   ∏ [k]q, n = 1,2,...  [k]q! = k=1 ,  1, n = 0  where 1 − qn [k] = . q 1 − q Definition 1.1. For a function f ∈ A given by (1.1) and 0 < q < 1, the q-derivate of function f is defined by Jackson (see[14])

f (qz) − f (z) D f (z) = (z 6= 0). q (q − 1)z and 0 Dq f (0) = ( f ) (0). Thus we have ∞ k−1 Dq f (z) = z + ∑[k]qakz , (z 6= 0), k=2 where [k]q given by the Definition 1. k Also as q → 1,[k]q −→ k. If we choose the function (z) = z , then we have

k k−1 0 Dqg(z) = Dq(z ) = [k]qz = (g) (z), where g0 is the ordinary derivative. The aim of the this paper is to introduce two new subclasses of the function class Σm and derive estimates on the initial coefficients |am+1| and |a2m+1| for functions in these new subclasses, Motivated essentially by the work of Ma and Minda [18]. 108 /

2. Coefficient bounds for the function class q,ϕ SΣm

A function f ∈ Σ is said to be in the class q,ϕ if the following conditions are satisfied: m SΣm

Dq f (z) ≺ ϕ(z)(0 < q < 1, z ∈ U) and Dqg(w) ≺ ϕ(w)(0 < q < 1, w ∈ U) where the function g = f −1. For m−fold symmetric functions, we have

q,ϕ ϕ ϕ lim S = S = H q→1 Σm Σm Σ,m introduced by Tang et al.(see Def.1.in [30]). In the case of q → 1, we obtain following well known classes:

1. For m = 1, we have a the class q,ϕ q,ϕ ϕ lim S = lim S = H q→1 Σ1 q→1 Σ Σ introduced by Tang et al [30].

1+z
γ 2. For m = 1 and ϕ = 1−z we have the class

1+z γ  γ q,ϕ q,( 1−z ) 1 + z lim S = lim S = HΣ,1 q→1 Σ1 q→1 Σ 1 − z introduced by Srivastava et al. [27].

 1+(1−2α)z  3. For m = 1 and ϕ = 1−z we have the class

 1+(1−2α)z    q,ϕ q, 1−z 1 + (1 − 2α)z lim S = lim S = HΣ,1 q→1 Σ1 q→1 Σ 1 − z introduced by Srivastava et al. [27].

We first state and prove the following theorem. Let f given by (5) be in the class q,ϕ . Then SΣm √ B 2B |a | ≤ 1 1 (14) m+1 r 2 2 2 2 B1 [1 + 2m]q (1 + m) − 2B2 [1 + m]q + 2B1 [1 + m]q and   2  [1+m] 2B3  m+1 q 1 B1  − 2 2 + 2[1+m]  2 2[1+2m]qB1 |B2[1+2m] (1+m)−2B2[1+m] |+2B [1+m] [1+2m]q q  1 q 2 q 1 q , B1 ≥ B [1+2m]q |a2m+1| ≤ + 1 (15) [1+2m]q  2[1+m]  B1 q  , B1 <  [1+2m]q [1+2m]q where 0 < q < 1, z ∈ U. Proof. Let f ∈ q,ϕ . Then there are analytic functions u : U → U and v : U → U, with SΣm u(0) = v(0) = 0, satisfying the following conditions:

f 0(z) = ϕ(u(z)) and g0(w) = ϕ(v(w)). (16)

Using the equalities (12), (13) in (16) and comparing the coefficient of (16) , we have

[1 + m]q am+1 = B1bm, (17) 2 [1 + 2m]q a2m+1 = B1b2m + B2bm, (18) 109 / and − [1 + m]q am+1 = B1cm, (19)  2  2 [1 + 2m]q (m + 1)am+1 − a2m+1 = B1c2m + B2cm. (20) From (8) and (10) we obtain bm = −cm. (21) and 2 2 2 2 2 [1 + m]q am+1 = B1(bm + cm). (22) Also from (9), (20) and (22) we have

B2 [ + m]2 2 2 2 1 q 2 [1 + 2m]q (1 + m)am+1 = B1 (b2m + c2m) + 2 am+1. B1 Therefore, we have h 2 2i 2 3 [1 + 2m]q (1 + m)B1 − 2B2 [1 + m]q am+1 = B1 (b2m + c2m) (23)

Appying Lemma 1 for the coefficients b2m and c2m, we obtain h i 2 2 2 3 2
[1 + 2m]q (1 + m)B1 − 2B2 [1 + m]q am+1 ≤ 2B1 1 − bm (24) and by using (8) in (24) we have

3 2 2B1 |am+1| ≤ (25) 2 2 2 [1 + 2m]q (1 + m)B1 − 2B2 [1 + m]q + 2B1 [1 + m]q which implies the assertion (14) . Next, in order to find the bound on |a2m+1|, by subtracting (20) from (18), we obtain

2 2[1 + 2m]q a2m+1 = [1 + 2m]q (m + 1)am+1 + B1 (b2m − c2m). (26)

Then, in view of (17) ,(21) and (26) , and appying Lemma 1 for the coefficients p2m, pm and q2m,qm we have

2
(m + 1) 2 B1 1 − bm |a2m+1| ≤ |am+1| + . (27) 2 [1 + 2m]q

2 Substituting |am+1| from the inequality (25) and putting in the (27) , we have desired result.

For the case of m-fold symmetric functions, for q → 1 ,Theorem 5 reduces to the corresponding results of Tang et al.(Thm1.,p.10067 in [30]), which we recall here as Corollary 7 below ( See [30]) Let f given by (5) be in the class ϕ = ϕ . Then SΣm HΣ,m √ B1 2B1 |am+1| ≤ q (28) 2 (1 + m) B1 (1 + 2m) − 2B2 (1 + m) + 2B1 (1 + m) and    3 2(1+m) 2B1 B1 2(1+m)  1 − + , B1 ≥ (1+2m)B1 |B2(1+2m)−2B (1+m)|+2B (1+m) 1+2m (1+2m) |a2m+1| ≤ 1 2 1 (29) B1 2(1+m)  1+2m , B1 < (1+2m)

For the case of one-fold symmetric functions, for q → 1 ,Theorem 5 reduces the corresponding results of Peng et al. ([23]) which we recall here as Corollary 9: ([23]) Let f given by (5) be in the class ϕ . Then SΣm √ B1 B1 |a2| ≤ q (30) 2 3B1 − 4B2 + 4B1 and

  3  4 B1 1 − 2 4  3B1 |3B −4B2|+4B1 1 , B1 ≥ 3 |a3| ≤ B1 (31) + 3  B1 4 3 , B1 < 3 140 /

From among the many choices of the function ϕ which would provide the followig corollaries: For the case of one-fold symmetric functions, for q → 1 , Theorem 5 reduces the corresponding results of Peng et al. ([23]) which we recall here as Corollaries:

From among the many choices of the function ϕ which would provide the followig corollaries:

β i) In the case of ϕ = 1+z
we have 1−z √ 2β |a2| ≤ p (32) 2 + β

( 8β 2 2 6+3β , 3 ≤ β ≤ 1 |a3| ≤ (33) 2β 2 3 , 0 ≤ β ≤ 3 and The estimate for |a3| asserted by Corollary 11 i) is more accurate than those given by Theorem 2 in Srivastava et al. [27].   ii) for ϕ = 1+(1−2α)z we have 1−z √ 2(1 − α) |a2| ≤ (34) p2 + |1 − 3α| and  8−12α 1 9 , 0 ≤ α ≤ 3 |a3| ≤ 2(1−α) 1 (35) 3 , 3 ≤ β ≤ 1

The estimates for |a2| and |a3| asserted by Corollary 11 ii) are more accurate than those given by Theorem 2 in Srivastava et al. [27]. In the next section, we will give the Fekete-Szegö Problem for the class ϕ (λ, µ). SΣm

3. The Fekete-Szegö Theorem for the Class q,ϕ SΣm

Let f given by (5) be in the class q,ϕ . Then SΣm

( B1 , 0 ≤ |h(δ)| < 1 2 [1+2m]q 2[1+2m]q a2m+1 − δam+1 ≤ 1 (36) 2B1 |h(δ)| , |h(δ)| ≥ 2[1+2m]q where

m + 1  B2 h(δ) = − δ 1 . (37) 2 2 2 2 B1 [1 + 2m]q (1 + m) − 2B2 [1 + m]q Proof. From the equations (23) and (25),

B3 (b + c ) a2 = 1 2m 2m (38) m+1 2 2 2 B1 [1 + 2m]q (1 + m) − 2B2 [1 + m]q and m + 1 2 B1 (b2m − c2m) a2m+1 = am+1 + (39) 2 [1 + 2m]q By using the equalities (38) and (39), we have " ! ! # 2 1 1 a2m+1 − δam+1 = B1 h(δ) + b2m + h(δ) − c2m , (40) 2[1 + 2m]q 2[1 + 2m]q

m + 1  B2 h(δ) = − δ 1 . (41) 2 2 2 2 B1 [1 + 2m]q (1 + m) − 2B2 [1 + m]q

141 /

Due to the fact that all Bi are real and B1 > 0, which holds the assertion (36), the proof of the theorem is copmleted For m-fold symmetric functions, if we choose for q → 1 the Theorem 14 reduces to the corresponding result of the Tang et al. ( Thm 2, page 1070 in [30]) which we recall here as Corollary 16: ( see [30]) Let f given by (5) be in the class ϕ . Then SΣm

( B1 1 2 1+2m , 0 ≤ |h(δ)| < 2(1+2m) a2m+1 − δam+1 ≤ 1 (42) 2B1 |h(δ)| , |h(δ)| ≥ 2(1+2m) where

  2 m + 1 B1 h(δ) = − δ  2 . (43) 2 (1 + m) (1 + 2m)B1 − 2B2 (1 + m) For one-fold symmetric functions, if we choose for q → 1 the Theorem 14 reduces to the following corollary: ϕ Let f given by (5) be in the class SΣ . Then    B1 , 1 ≤ 1 − 4  3 3B2 |a | ≤ 1 , (44) 3  2(µ+λ)2   2B1 |h(0)| , 1 > 1 − 2 (µ+1)(µ+2λ)B1

m + 1 B2 h(0) = 1 . (45) 2 2 2 (µ + 2mλ)(µ + m)B1 − 2B2 (µ + mλ) For the case of m-fold symmetric functions, for q → 1, taking δ = 1 and δ = 0 in Theprem 14, we have the following corollaries: Let f given by (5) be in the class q,ϕ . Then SΣm

( B1 1 2 (1+2m) , 0 ≤ |h(1)| < 2(1+2m) a2m+1 − am+1 ≤ 1 (46) 2B1 |h(δ)| , |h(1δ)| ≥ 2(1+2m) where

  2 m − 1 B1 h(1) = 2 2 . (47) 2 B1 (1 + 2m)(1 + m) − 2B2 (1 + m) For the case of one-fold symmetric functions, for q → 1, taking δ = 1 in Theprem 14, we have the following corollary: Let f given by (5) be in the class q,ϕ . Then SΣm 2 B1 a3 − a ≤ (48) 2 3

For the case m-fold symmetric functions, for q → 1, taking δ = 0 in Theprem 14, we have the following corollary:

 B1 B2  (1+2m) , 2 ∈ (−∞,0) ∪ ((1 + 2m),∞) B1 |a2m+1| ≤     (49) B2 (1+2m) (1+2m) (1+2m)  2B1 |h(δ)| , 2 ∈ 2(1+m) , 2 ∪ 2 ,0 B1 (1+m) 2(1+m)

For the case one-fold symmetric functions, for q → 1, taking δ = 0 in Theprem 14, we have the following corollary:

 B1 B2 3
3 , 2 ∈ (−∞,0) ∪ 2 ,∞  B1 |a3| ≤ 3 (50) B1 B2 3 3
 2 , 2 ∈ 4 , 2 3B1−4B2 B1

142 /

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