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HADAMARD k-FRACTIONAL INTEGRAL AND ITS APPLICATION

BY GHULAM FARID

DOCTOR OF PHILOSPHY IN BUSINESS MATHEMATICS

February, 2015 GLOBAL INSTITUTE LAHORE

HADAMARD k-FRACTINAL INTEGRAL AND ITS APPLICATION

BY GHULAM FARID

A dissertation submitted to Faculty of Management Sciences

In Partial Fullfilment of the Requirements for the Degree of

DOCTOR OF PHILOSPHY IN BUSINESS MATHEMATICS

FEBRUARY, 2015

IN THE NAME OF ALLAH, THE MOST BENIFICIAL, THE MOST MERCIFUL

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GLOBAL INSTITUTE LAHORE

HADAMARD k-FRACTIONAL INTEGRAL AND ITS APPLICATION

BY GHULAM FARID

A dissertation submitted to Faculty of Management Sciences, in partial fullfilment of the requirements for the degree of DOCTOR OF PHILOSPHY IN BUSINESS MATHEMATICS

Dissertation Committee

DECLARATION This is to certify that this research work has not been submitted for obtaining similar degree from any other university/college.

DEDICATED TO

My Beloved Son Muhammad Yahya Khan

ACKNOWLEDGEMENT

Gratitude to almighty Allah for His unending blessings that has reinforced me to complete this humble piece of dissertation. I am also obliged to all the teachers and fellows at Global Institute, Lahore.

I am grateful to my supervisor Dr. Ghulam Mustafa Habibullah for his motivation, support and guidance to the right direction. I am highly obliged to Muhammad Kashif Khan, the Chairman Global Institute Lahore and all of the college officials including the Rector, Pro-Rector and all the faculty members of the institute for their guidance, support and favours.

The financial support of Higher Education Commission, Pakistan in facilitating me to take up this research work, is most gratefully acknowledged.

I am also thankful to my brother Professor Muhammad Zulqarnain Khan and my wife for their continuous support and encouragement.

At the end, I pay tribute to my parents and elders who always supported me and prayed for my success.

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RESEARCH COMPLETION CERTIFICATE

Certified that the research work contained in this thesis titled "HADAMARD k-FRACTIONAL INTEGRAL AND ITS APPLICATION" has been carried out and completed by Ghulam Farid under my supervision during his PhD Business Mathematics Programme.

(Prof. Dr. G. M. Habibullah) Supervisor

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SUMMARY The Fractional Calculus has been attractive and hot topic among the researchers since 18th century, because of its extensive application in differential and integral equations and other disciplines of mathematics, physics and economics. The motivation of this thesis is to extend the fractional integrals and derivatives, particularly Hadamard fractional integral, and to establish basic properties of the extended fractional integral operators. The application of the extended operators involving the formation of the fractional integral inequalities and solutions of fractional integral equations is focussed in the work. The first chapter includes the introductory background of the fractional calculus. The appropriate literature pertaining to the fractional calculus, involving the theoretical and practical aspects of fractional differential and fractional integral operators has been reviewed. In the second chapter, we have listed symbols, notations and the basic results that are used throughout the dissertation. A number of inequalities involving the Holder’s inequality and AM-GM inequality have been presented. We have defined an extended form of Hadamard fractional integral and have called it Hadamard k-fractional integral. We have also discussed a number of properties of the extended integral operator. In the third chapter, we have established numerous fractional integral inequalities involving the inequalities of Chebyshev functional using the notion of synchronous functions, asynchronous functions, and the like. In the fourth chapter, we have presented some inequalities involving the rearrangement inequalities. On the basis of AM-GM, Holder and the rearrangement inequalities, we have established many fractional integral inequalities related to the extended operator. In the fifth chapter, we have introduced a number of extensions of the fractional integral operators involving the Hadamard type fractional operators. We have discussed the properties of the extended operators involving the semigroup property and commutative law. We have also considered the Mellin transforms and boundedness of some of the extended operators. In the sixth chapter, we have introduced extended fractional derivatives related to the extended fractional integral operators and have discussed their compositions.

In the seventh chapter, we have presented some integral equations and have found their solutions using some of the extended fractional integral operators. We have also illustrated the use of some of the extended fractional calculus operators in finding solutions of fractional differential equations. ix

TABLE OF CONTENTS

Page DECLARATION v DEDICATION vi ACKNOWLEDGEMENT vii RESEARCH COMPLETION CERTIFICATE viii SUMMARY ix CHAPTER 1 1 INTRODUCTION 1.1 BACKGROUND 1 1.2 LITERATURE REVIEW 1 CHAPTER 2 11 EXTENSION OF HADAMARD FRACTIONAL INTEGRAL 2.1 PRELIMINARIES 11 2.2 HADAMARD k-FRACTIONAL INTEGRAL 19 2.2.1 Definition 19 2.3 PROPERITES OF HADAMARD k-FRACTIONAL INTEGRAL 19 2.3.1 Theorem 19 2.3.2 Corollary 20 2.3.3 Theorem (Commutative Law) 20 2.3.4 Theorem (Semigroup Property) 21 2.3.5 Definition 21 2.3.6 Theorem 21 2.3.7 Theorem 22 2.3.8 Theorem 22 2.3.9 Theorem 22 2.3.10 Theorem 23 2.3.11 Theorem 23 2.3.12 Theorem 24 k  2.3.13 Theorem (Boundedness of HI(,) a t ) 24 CHAPTER 3 25 FI INEQUALITIES RELATING THE EXTENDED OPERATOR 3.1 FI INEQUALITIES FOR CHEBYSHEV FUNCTIONAL 25 3.1.1 Theorem 26 3.1.2 Theorem 26 3.1.3 Note 26 3.1.4 Note 27 3.1.5 Note 27 3.1.6 Note 27 3.1.7 Theorem 27 x

3.1.8 Lemma 27 3.1.9 Corollary 28 3.1.10 Note 28 3.1.11 Lemma 28 3.1.12 Lemma 29 3.1.13 Note 29 3.1.14 Theorem 29 3.1.15 Theorem 30 3.1.16 Corollary 31 3.1.17 Theorem 31 3.1.18 Theorem 32 3.1.19 Corollary 32 3.1.20 Theorem 32 3.1.21 Corollary 33 3.1.22 Theorem 33 3.1.23 Theorem 34 3.1.24 Theorem 35 3.1.25 Theorem 35 3.1.26 Theorem 36 3.1.27 Theorem 37 3.1.28 Theorem 37 3.1.29 Theorem 38 3.1.30 Theorem 39 3.1.31 Note 39 3.1.32 Theorem 39 3.1.33 Theorem 40 3.1.34 Theorem 41 3.1.35 Theorem 42 CHAPTER 4 43 MORE FI INEQUALITIES FOR THE EXTENDED OPERATOR 4.1 GRUSS TYPE INEQUALITIES AND FI INEQUALITIES RELATING AM-GM AND HOLDER’S INEQUALITIES 43 4.1.1 Theorem 43 4.1.2 Corollary 44 4.1.3 Theorem 45 4.1.4 Theorem 45 4.1.5 Theorem 46 4.1.6 Corollary 47 4.1.7 Theorem 47 4.1.8 Example 47 4.1.9 Corollary 48 4.1.10 Theorem 48 4.1.11 Corollary 49 xi

4.1.12 Theorem 49 4.1.13 Example 50 4.1.14 Corollary 51 4.1.15 Theorem 51 4.1.16 Example 52 4.1.17 Corollary 53 4.1.18 Theorem 53 4.1.19 Corollary 54 4.1.20 Theorem 54 4.1.21 Corollary 55 4.1.22 Theorem 55 4.1.23 Corollary 56 4.1.24 Theorem 56 4.1.25 Corollary 57 4.1.26 Theorem 57 4.1.27 Theorem 57 4.1.28 Theorem 58 4.1.29 Corollary 59 4.2 FI INEQUALITIES BASED ON REARRANGEMENT INEQUALITIES 59 4.2.1 Lemma 59 4.2.2 Note 60 4.2.3 Example 60 4.2.4 Note 60 4.2.5 Theorem 60 4.2.6 Theorem 61 4.2.7 Corollary 62 4.2.8 Example 62 4.2.9 Theorem 62 4.2.10 Theorem 63 4.2.11 Lemma 65 4.2.12 Note 65 4.2.13 Lemma 65 4.2.14 Note 66 4.2.15 Example 66 4.2.16 Theorem 66 4.2.17 Theorem 67 4.2.18 Theorem 68 4.2.19 Example 69 4.2.20 Theorem 70 4.2.21 Lemma 70 4.2.22 Theorem 72 4.2.23 Example 73 xii

4.2.24 Example 73 4.3 MORE FI INEQUALITIES RELATING AM-GM AND HOLDER’S INEQUALITIES 74 4.3.1 Theorem 74 4.3.2 Example 75 4.3.3 Corollary 75 4.3.4 Corollary 76 4.3.5 Lemma 76 4.3.6 Corollary 77 4.3.7 Theorem 78 4.3.8 Theorem 78 4.3.9 Example 79 4.3.10 Theorem 81 4.3.11 Theorem 82 4.3.12 Corollary 83 4.3.13 Theorem 84 4.3.14 Theorem 87 4.3.15 Theorem 90 4.3.16 Theorem 91 CHAPTER 5 93 MORE EXTENSIONS OF FI OPERATORS 5.1 EXTENSIONS OF THE FRACTIONAL INTEGRAL OPERATOR 93 5.2 PROPERTIES OF THE EXTENDED OPERATORS 94 5.2.1 Theorem 94 5.2.2 Corollary 95 5.2.3 Theorem 95 5.2.4 Corollary 96 5.2.5 Theorem 96 5.2.6 Corollary 96 5.2.7 Theorem 97 5.2.8 Corollary 97 5.2.9 Lemma 97 5.2.10 Lemma 98 5.2.11 Lemma 99 5.2.12 Theorem 100 5.2.13 Theorem 100 5.2.14 Theorem 101 5.2.15 Theorem 101 5.2.16 Lemma 102 5.2.17 Theorem 102 5.2.18 Corollary 103 5.2.19 Theorem 103 xiii

5.2.20 Corollary 104 5.3 OTHER EXTENDED FRACTIONAL INTEGRALS 104 5.3.1 Theorem 104 5.3.2 Theorem 104 5.3.3 Theorem 105 5.3.4 Corollary 105 5.3.5 Theorem 105 5.3.6 Corollary 106 5.4 MELLIN TRANSFORM OF THE EXTENDED FRACTIONAL INTEGRALS 106 5.4.1 Lemma 106 5.4.2 Theorem 106 5.4.3 Theorem 107 5.4.4 Theorem 108 5.4.5 Theorem 109 5.5 BOUNDEDNESS OF THE EXTENDED FRACTIONAL INTEGRAL OPERATORS 110 5.5.1 Definition 110 5.5.2 Theorem 111 5.5.3 Corollary 113 5.5.4 Corollary 113 CHAPTER 6 115 THE EXTENDED FD OPERATORS 6.1 FRACTIONAL DERIVATIVE RELATED TO HADAMARD k- FRACTIONAL INTEGRAL 115 6.1.1 Definition 115 6.1.2 Theorem 115 6.1.3 Corollary 116 6.2 FRACTIONAL DERIVATIVES RELATED TO OTHER EXTENDED FRACTIONAL INTEGRALS 116 6.2.1 Definition 116 6.2.2 Theorem 117 6.2.3 Corollary 117 6.2.4 Definition 118 6.2.5 Theorem 118 6.2.6 Corollary 119 6.2.7 Definition 119 6.2.8 Theorem 119 6.2.9 Corollary 120 6.2.10 Definition 120 6.2.11 Theorem 120 6.2.12 Corollary 121

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CHAPTER 7 122 APPLICATION OF THE EXTENDED FC OPERATORS 7.1 SOLUTIONS OF INTEGRAL EQUATIONS USING HADAMARD k-FRACTIONAL INTEGRAL OPERATOR 122 7.1.1 Lemma 122 7.1.2 Corollary 123 7.1.3 Lemma 123 7.1.4 Theorem 124 7.1.5 Theorem 124 7.2 SOLUTIONS OF INTEGRAL EQUATIONS USING ANOTHER EXTENDED FI OPERATOR 125 7.2.1 Lemma 125 7.2.2 Lemma 125 7.2.3 Theorem 126 7.2.4 Theorem 126 7.3 SOLUTIONS OF FRACTIONAL DIFFERENTIAL EQUATIONS VIA EXTENDED FC OPERATORS 127 7.3.1 Example 127

CONCLUSION 129 RECENTLY PUBLISHED PAPERS OF THE AUTHOR 130 REFERENCES 131-138

CHAPTER 1 INTRODUCTION

1.1 BACKGROUND Calculus has a long historical background, however the revolution of modern calculus came in 17th century. Isaac Newton and Gottfried Wilhelm Leibnitz are known to be the founders of modern calculus. Meanwhile, in 1695, 1 Leibniz discussed derivative of order (see Ross (1977) and Samko, Kilbas, 2 and Marichev (1993)). Though the concept of fractional calculus was originated in 17th century, however, the subject promoted in 20th century. In the recent years, fractional calculus has become an interesting and hot topic among researchers due to its extensive applicatons in various areas of mathematics, physics, biology, economics and engineering. There is a list of definitions used in the literature; e.g. Riemann Liouville, Caputo, Hadamard, Grunwald- Letnikov, Erdelyi-Cober, Marchaud and Riesz, etc. The fractional integral of order   0, over the interval [,]at , a  0, defined as 11t t (1) I( f ( x ))  (log )1 f ( ) d ()a   is attributed to Hadamard (1892), which differs from Riemann-Liouville and Caputo’s definition in the sense that the kernel of the integral (1) contains logarithmic function of arbitrary exponent.

1.2 LITERATURE REVIEW The fractional version of calculus was introduced in 17th century. Large dedicated literature is available to study fractional integrals. We refer some of them beginning with Oldham (1974) followed by Samko et al. (1993), Podlubny (1998) and Hilfer et al. (2000). To study means and inequalities, we refer P. Bullen (1998) and P. S. Bullen (2003). And to learn about Lp -Spaces and Real Analysis, one may see Okikiolu (1971) and Douglass (1996) respectively. Now, we review chronologically, some of the significant works on fractional integrals and the related inequalities, investigated by the researchers, which are useful to our study. Chebyshev (1882) has studied synchronous functions. Hadamard (1892) has introduced fractional integral If as defined in (1).

Grüss (1935) has introduced an integral inequality involving two bounded functions. Kober (1941) has discussed a consequence of Schur’s theorem and fractional integrals of purely imaginary order. Erdelyi (1965) has given an application of integrals of fractional order by making a connection with some differential operators. Belward (1972) has found solutions of some Fredholm integral equations by connecting fractional integrals and the inverse operators related to the equations. Krantz (1979) has shown estimates for kernels of mixed type, fractional integral operators and optimal estimates for some operators. Sprinkhuizen-Kuyper (1979) has studied a class of integral operators containing negative fractional exponent of some second order operator. Faber and Wing (1986) have obtained upper bounds on the singular values of fractional integral operators of the form 1 x L f()()() x x y1 f y dy ,   0. () 0 Andersen and Sawyer (1988) have developed some weighted norm inequalities for each of the Riemann-Liouville and the Weyl fractional integral operators. Gupta and Agrawal (1989) have developed fractional integral formulae for the multivariable H function and for a general class of polynomials. Srivastava, Goyal, and Jain (1990b) have proved a theorem relating the classical Laplace transform, Weyl fractional integral and Whittaker transforms. Srivastava, Goyal, and Jain (1990a) have derived a number of expressions for the compositions of certain multidimensional fractional integral operators and have shown that the fractional integral operators can be recognized with elements of algebra of functions having multidimensional Mellin convolution as the product. Nigmatullin (1992) has established a relationship between fractional integral and Cantor’s fractal set and has demonstrated physical interpretation of the fractional integral. Dostanic (1993) has found the precise asymptotic of singular numbers of the fractional integral operator 1 x I f()()() x x y1 f y dy () 0 1 1 for the case 0  and has improved the known estimates for  1. 2 2 Friedrich (1993) has discussed and compared the rheological behaviors of certain fractional models for mechanical stress relaxation, one of integral type and the other of differential type. Gupta and Soni (1994) have derived three remarkable expressions for composition of certain fractional integral operators whose kernels involve a multivariable H function and a general class of polynomials. Rubin (1994) has introduced and investigated, using a special Fourier analysis, a new class of fractional integrals relating balls in Rn . Srivastava, Saxena, and Ram (1995) have investigated two novel families of multidimensional fractinal integral operators involving a general class of polynomials with essentially arbitrary coefficeints and have established various theorems involving compositons, convolutions, inversion formulae and

2 multidimensional Mellin transforms of the operators. Vanberkel and Vaneijndhoven (1995) have studied composition of fractional integral and fractional differential operators relating the Riemann-Liouville and the Weyl calculus and have derived Rodrigues-type formulae for Gegenbauer functions and certain relations for the operators. Gupta and Soni (1996) have developed the Mellin transforms, inversions, convolutions, the related Parseval-Goldstein theorem and the image of the multivariable H function, and have studied applications of the operators. Kim, Lee, and Srivastava (1996) have investigated various properties of a generalized ,,   fractional integral operator J0,z involving the Gaussian hypergeometric ,,   functions and have given relationships involving J0,z , the Ruscheweyh derivative D and the Carlson-Shaffer operator L(,) a c . Mandal, Chakrabarti, and Mandal (1996) have presented a method for obtaining closed form solution of a system of generalized Abel integral equation using fractional integral operators and have given its applications in classical theory of elasticity. Ren, Yu, and Su (1996) have proved that the fractional exponent of the fractional integral is not uniquely determined by the fractal dimension of the self-similar set or generalized self-similar set. Ren, Yu, Zhou, Mehaute, and Nigmatullin (1997) have shown that there is no direct relationship between the fractional exponent of fractional integral and the fractal structure of the memory set considered. Al-Saqabi and Kiryakova (1998) have found explicit solutions of Volterra integral equations of 2nd kind and fractional differential equations involving Erdelyi-Kober fractional integrals. Kalla, Virchenko, and Tsarenko (1998) have developed the notion of fractional order integral transforms and their generalization with kernels which are transcendental solutions of some differential equations. A. El-Sayed (1999) has discussed the properties involving the existance and uniqueness of the solutions of a class of evolutionary fractional integral equations. Elwakil and Zahran (1999) have achieved, using Riemann-Liouville fractional integral operator, the relationship between the continuous time random walk (CTRW) and fractional master equation (FME) by obtaining the corresponding waiting time density (WTD). Godoy and Urciuolo 1 11 (1999) have shown boundedness of TLRLR:()()p n q n , 1p , r , r qp for Tfx() kx (  aykx )(  ay )...( k x  ayfydy )() ,  1 1 2 2 mm jn m qi j 1 where ki( y )  2 i, j (2 y ) , 1qi  ,  1 r , and ij, satisfied jZ i1 qi suitable regularity conditions. Avetisian (2000) has established certain embedding theorems and integral representations in weighted classes of functions harmonic or holomorphic in the unit disk of the complex plane. Ding, Lu, and Zhang (2001) have introduced a kind of maximal operator of fractional order associated with the mean Luxemburg norm and have used the

3 techniques of sharp function to establish the week type LLlog estimates for the commutators of the fractional integral operator and the related maximal operator. Eilertsen (2001) has established certain integral inequalities for fractional derivatives within certain weighted L2 setting. Kilbas (2001) has studied fractional integration and differentiation on a finite interval [,]ab of the real axis in the frame of certain Hadamard setting. Malamud (2001) has discussed Jensen and Chebyshev inequalities and a problem of W. Walter. Ali, Kiryakova, and Kalla (2002) have found solutions of certain fractional multi- order integral and differential equations using a Poisson-type transform. Canqin, Dachun, and Pu (2002) have introduced (,) N -type fractional integral operator and have discussed its boundedness on the Hardy spaces, the week Hardy spaces and the Herz-type Hardy spaces. Bardina, Jolis, and A Tudor (2003) have shown convergence in law to certain multiple fractional integral. Kholpanov, Zakiev, and Fedotov (2003) have proposed a method, based on fractional differential- integral calculus, to solve boundary-value problems for parabolic equations with unknown boundaries, where the solutions join, moving according to time- dependent laws. Kilbas (2003) has studied the integral equation 11x ux  ()(log)1 f () u du g () x , 0 a  x  b, () a u x u with real  and   0 on a finite segment [,]ab of the real line, and has found conditions for existence of the solution under certain conditions. Seeger and pq Wainger (2003) have proved sharp LL endpoint bounds for singular fractional integral operators and the related Fourier integral operators, under the non-vanishing rotational curvature hypothesis. Zheng-you, Gen-guo, and Chang- jun (2003) have introduced a numerical method for the fractional integral that only stores part history data and have estimated its discretization error. L. Wang and Pan (2004) have discussed boundedness of certain multi-linear fractional integral commutators by introducing a kind of maximal operators of fractional order and technique of sharp function. Lin and Srivastava (2004) have presented a number of fractional derivative and other integral representations for several general families of the Hurwitz–Lerch Zeta functions. Stanislavsky (2004) has established a relation between the fractional integral and stable distributions in probability theory. W. G. El-Sayed and El-Sayed (2004) have discussed the existence of monotonic solution, in L1[0,1], for a specific functional integral equation of the mixed type. Darwish (2005) has presented a theorem for nonlinear quadratic integral equations of fractional orders in the Banach space of real functions defined and continuous on a closed and bounded interval.

Golubov (2005) has introduced a modified strong dyadic integral J and a modified strong dyadic derivative D() of fractional order   0, for functions in the Lebesgue space LR() . He and Shu (2005) have studied boundedness of the higher order commutators T,,bm on the classical Hardy spaces and the Herz- type Hardy spaces. Li, Han, and Wei Meng (2005) have established some new

4 delay integral inequalities which provide explicit bounds on unknown functions, and can be used as tools in the qualitative theory of certain delay integral equations and delay differential equations. Bardina and Jolis (2006) have constructed a multiple integral of Stratonovich type with respect to the fractional Brownian motion with Hurst 1 parameter H  . Belinsky and Linde (2006) have investigated compactness 2 properties of the integral operator 1 S f()()() x x t 1 f t dt   0 under certain conditions. Bliev (2006) has selected the Besov spaces that embed into the class of continuous functions and enjoyed the Fredholm theory of linear singular integral equations with Cauchy kernel and has specified basic results of the theory in the class of continuous functions in terms of Besov spaces. Darwish and El-Bary (2006) have studied the existence of solutions to an integral equation of fractional order with hysteresis. Ferdi (2006) has developed a new procedure for efficiently computing the fractional order derivatives and integrals of discrete-time signals by combining the three techniques; power series expansion (PSE), s -to- z transform and signal modeling. Pachpatte (2006) has established new Chebyshev-Gruss type integral inequalities by using integral representation for n -time differentiable functions. Z. Yang and Lan (2006) have discussed Lipschitz boundedness for a class of fractional multilinear operators with variable kernels and have shown that these operators are both

Lipschitz bounded from Lp to H q . Balderrama and Urbina (2007) have studied the fractional integral and the fractional derivative for Jacobi expansion by obtaining an analogous of P. A. Meyer's multipliers theorem for Jacobi expansions. Banaś and Rzepka (2007) have used the technique associated with the Hausdorff measure of non- compactness to find monotonic solutions of a quadratic integral equation of fractional order. Burman (2007) has shown that the j th singular value of the fractional integral operator of order   0, is ( j ) (1 O ( j 1 )) instead of the already known value. Dıaz and Pariguan (2007) have introduced the k - generalized gamma function  k , k -generalized beta function Bk and

Pochhammer k -symbol ()x nk, , and discussed their properties involving the integral representations of  k and Bk functions. Guliyev, Serbetci, and Ekincioglu (2007) have found the necessary and sufficient conditions for the boundedness of the rough B-fractional integral operators from the Lorentz spaces Lps,, to Lqr,, , 1pq    , 1rs    and from L1,r , to Lq,,  , 1q   , 1r  . Heo (2007) has studied LLpq estimates for some singular fractional integral operators. F. Jiang and Meng (2007) have established some new delay integral inequalities which provide explicit bounds on certain

5 unknown functions and can be used to investigate the qualitative properties of certain delay differential equations and delay integral equations. Sekine, Owa, and Tsurumi (2007) have determined by means of the subordination theorem, certain integral means inequalities with coefficients inequalities of certain analytic functions for the fractional derivatives and the fractional integral. Yürekli (2007) has introduced certain theorems relating the Riemann–Liouville fractional integral and the Weyl fractional integral to some well-known integral transforms including Laplace, Stieltjes, generalized Stieltes, Hankel, and K - transforms. Bernardis and Lorente (2008) have proved certain weighted inequalities for the commutators of the Riemann-Liouville and Weyl Fractional Integral Operators. Ding and Lan (2008) have discussed the boundedness of the commutators of a fractional integral operator T and a BMO function. Guliyev and Omarova (2008) have obtained necessary and sufficient conditions on the parameters for the boundedness of the fractional maximal operator and the fractional integral operator on the Laguerre hypergroup from the spaces LKp () to the spaces LKq () and from the spaces LK1() to the weak spaces WLq () K , KR[0,  )  . Ma and Pečarić (2008) have established some new weakly singular integral inequalities of Gronwall–Bellman type and have indicated some applications to fractional differential and integral equations. Marinković, Rajković, and Stanković (2008) have established new integral inequalities of q - Chebyshev type, q -Grüss type, q -Hermite-Hadamard type and Cauchy- Buniakowsky type. Momani and Ibrahim (2008) have studied, using the fixed point theorems due to Dhage, the existence of periodic solutions for a nonlinear integral equation of periodic functions involving Weyl–Riesz fractional integral operator under the mixed generalized Lipschitz, Carathéodory and monotonicity conditions. Ragusa (2008) has proved a sufficient condition for commutators of fractional integral operators to belong to Vanishing Morrey spaces VLp, . Shi and Tao (2008) have studied weighted Lp boundedness for a certain multilinear fractional integral on product spaces. Tarasov and Zaslavsky (2008) have derived fractional diffusion equation using generalization of the Kac integral and Kac method for paths measure based on the Levy distribution. Varlamov (2008) has obtained several new Wronskian relations that lead to the calculation of a number of indefinite integrals containing fractional derivatives of the Airy functions. Ahn, Bhambhani, and Chen (2009) have establishes a new strategy to tune a fractional order integral and derivative controller satisfying gain and phase margins based on Bode’s ideal transfer function as a reference model, for a temperature profile tracking. Anastassiou (2009) has presented very general weighted univariate and multivariate Hilbert-Pachpatte type integral inequalities involving Caputo and Riemann-Liouville fractional derivatives. Banaś and Zaja̧c (2009) have investigated a functional integral equation of fractional order in the space of real functions which are defined, continuous and bounded on the real

6 half-axis and have proved, using the technique of measures of non-compactness and the fixed point theorem of Darbo type that the equation has solutions in the function space. Belarbi and Dahmani (2009) have established some new integral inequalities for the Chebyshev functional in the case of two synchronous functions, using the Riemann-Liouville fractional integral. Darwish and Ntouyas (2009) have presented an existence theorem for monotonic solutions of a perturbed quadratic fractional integral equation in C[0,1]. Eridani, Kokilashvili, and Meskhi (2009) have discussed the boundedness of fractional integral operators in Morrey spaces defined on quasimetric measure spaces. Ferdi (2009) has presented an efficient method for the design of recursive digital fractional order integrator when the order of integration lies in the interval [0,1]. Hashim, Abdulaziz, and Momani (2009) have applied the homotopy analysis method and Caputo’s sense fractional derivatives to solve linear and nonlinear fractional initial-value problems. Lepik (2009) has applied Haar wavelets for the solution of fractional integral equations and has considered Fractional Volterra and Fredholm integral equations and used the proposed method to analyze fractional harmonic vibrations. Moen (2009) has presented a weighted theory for multilinear fractional integral operators and maximal functions and has found sufficient conditions for some weight inequalities of these operators. Mophou and N’guérékata (2009) have proved, by means of fixed point methods, the existence and uniqueness of integral solution for some nondensely defined fractional semilinear differential equation with nonlocal conditions. Srivastava and Tomovski (2009) have introduced and investigated a fractional calculus with an integral operator which contains a certain family of generalized Mittag– Leffler functions. Yuan, Yuan, Meng, and Zhang (2009) have established some new delay integral inequalities which provide explicit bounds on unknown functions and can be used as tools in the qualitative theory of certain delay differential and integral equations. Balachandran, Park, and Diana Julie (2010) have studied the existence of solutions of a nonlinear functional integral equation of fractional order with deviating arguments. Denton and Vatsala (2010) have developed fractional integral inequality results when 01q , the nonlinear term is increasing in u and satisfies a one sided Lipschitz condition. Jumarie (2010) has established Cauchy's integral formula via the modified Riemann-Liouville derivative for analytic functions of fractional order. Lacey, Moen, Pérez, and Torres (2010) have investigated the relationship between the operator norms of fractional integral operators acting on weighted Lebesgue spaces and the constant of the weights and have obtained sharp bounds for both the fractional integral operators and the associated fractional maximal functions. Muslim, Conca, and Nandakumaran (2010) have studied a fractional integral equation in an arbitrary Banach space X and established the existence and uniqueness of solutions of the given problem using the analytic semigroups theory of linear operators and the fixed point method. Nakai (2010) has proved boundedness of singular and fractional integral operators on Campanato spaces over X with variable growth

7 conditions, where X is a space of homogeneous type in the sense of Coifman and Weiss. Ortigueira (2010) has defined the quantum fractional derivative using formulations analogue to the common Grünwald–Letnikov derivatives and has deduced with the help of the Mellin transform, two integral formulations similar to the usual Liouville derivatives. Panahi and Zanjani (2010) have used Adomian decomposition method and Legendre polynomials to propose a method for finding approximations of solution of fractional differential equation. Pradolini (2010) has proved weighted norm inequalities and pointwise estimates between the multilinear fractional integral operator and the multilinear fractional maximal. Yu and Chen (2010) have obtained the weak type LLlog estimate for the multilinear fractional commutator by using the technique of sharp function and introducing a new kind of maximal operator of the multilinear fractional order associated with the mean Luxumburg norm. Ahmad and Nieto (2011) have investigated a boundary value problem of Riemann-Liouville fractional integro-differential equations with fractional nonlocal integral boundary conditions and have obtained some new results by applying standard fixed point theorems. Banaś and Zaja̧c (2011) have presented a new approach, depending on converting of the mentioned equations to the form of functional integral equations of Volterra–Stieltjes type, to the theory of functional integral. Darwish (2011) has presented a theorem using an equation containing the famous Chandrasekhar integral equation, for monotonic solutions of a perturbed quadratic fractional integral equation in C[0,1]. Jia and Liu (2011) have studied multiple nonnegative solutions for the fractional differential equation with certain integral boundary conditions, by means of Leggett– Williams fixed point theorem and have obtained some new results on the existence of at least three nonnegative solutions. Katugampola (2011) has presented a new fractional integration which generalizes the Riemann–Liouville and Hadamard fractional integrals into a single form and has proved semigroup property for the operator. Ma, Pečarić, and Zhang (2011) have generalized the results for the constructions of explicit bounds and the qualitative properties for the solutions of some two-dimensional fractional differential systems. Rami (2011) has studied the fractional variational problems from extended exponentially fractional integral and has explored both the Lagrangian and Hamiltonian formulations. Rida and Arafa (2011) have developed a new application of the Mittag-Leffler Function method that extended the application of the method to linear differential equations with fractional order. Y. Jiang (2011) has studied endpoint estimates for fractional integral associated to Schrodinger operators on the Heisenberg groups. Zhong and Chen (2011) have obtained the boundedness for the fractional integral operators from the modulation Hardy space  pq, to the modulation Hardy space rq, for all 0.p   Cabada and Wang (2012) have studied, using known Guo–Krasnoselskii fixed point theorem, the existence of positive solutions for a class of nonlinear

8 boundary-value problem of fractional differential equations with integral boundary conditions. Chinchane and Pachpatteb (2012) have used Hadamard fractional integral to establish some new result on fractional integral inequalities by considering the extended Chebyshev functional in case of synchronous function. Guezane-Lakoud and Khaldi (2012) have established sufficient conditions for the existence and uniqueness of solutions for boundary value problems for fractional differential equations with fractional integral condition, using Banach contraction principle and Leray–Schauder nonlinear alternative and Caputo fractional derivative. Iida, Sato, Sawano, and Tanaka (2012) have introduced a multi-Morrey norm which is strictly smaller than m -fold product of the Morrey norms and have obtained the boundedness property of the Adams type for multilinear fractional integral operators. Jarad, Abdeljawad, and Baleanu (2012) have introduced a Caputo-type modification of Hadamard fractional derivatives and have studied the properties of the modified derivatives. Loonker and Banerji (2012) have poroposed the investigation of the Sumudu transformation for certain distribution spaces with regard to the fractional integral and differential operators of the transform. Mubeen and Habibullah (2012) have introduced an extended fractional integral operator to be called k -fractional integral, based on k -gamma function and have illustrated the uses of the extended integral operator. Mubeen (2012) has presented a k  analogue of Kummer’s first formula. Romero, Cerutti, and Dorrego (2012) have presented a generalization to the k -calculus of the Weyl fractional integral and have shown the semigroup property and have calculated their fractional Fourier transform. W. Yang (2012) has employed a fractional q -integral on a certain time scale to establish some fractional q -integral Chebyshev type inequalities using one or two fractional parameters. Zhu, Yang, and Zhao (2012) have employed a fractional q -integral on a certain time scale to establish some new fractional q -integral Gruss-type inequalities, by using one or two fractional parameters. Anastassiou (2013) has presented vectorial general integral inequalities involving products of multivariate convex and increasing functions applied to vectors of functions. Chinchane and Pachpatte (2013) have established, using the Hadamard fractional integral, certain integral inequalities for the Chebyshev functional in case of synchronous function. Romero and Luque (2013) have defined a new fractional derivative in the k -calculus context, the k -Weyl fractional derivative and have studied the action of Laplace and Stieltjes transforms on the new fractional operator and the k -Weyl fractional integral operator. Sroysang (2013) has presented some new integral inequalities via Hadamard integral. Zhang and Wang (2013) have established some fundamental but important Riemann-Liouville fractional integral identities including a twice differentiable mapping. Ahmad and Ntouyas (2014) have studied the existence and uniqueness of solutions for a fractional integral boundary value problem involving Hadamard

9 type fractional differential equations and integral boundary conditions. Chinchane and Pachpatte (2014) have developed certain fractional integral inequalities based on Hadamard fractional integral operator. G. Wang, Agarwal, and Chand (2014) have established certain new Gruss type fractional integral inequalities involving the Gauss hypergeometric function. Gu and Cai (2014) have proved some sharp maximal function inequalities for the commutators related to certain generalized fractional singular integral operators. Katugampola (2014) has introduced a new fractional integral operator which generalizes the Riemann-Liouville and the Hadamard fractional integrals and has illustrated the results with example. Sudsutad, Ntouyas, and Tariboon (2014) have established new fractional integral inequalities via Hadamard fractional integral and have obtained several new integral inequalities using Young and weighted AM-GM inequalities. Zheng (2014) has presented some new Gronwall-type inequalities and has derived explicit bounds for the functions concerned and the properties of the modified Riemann-Liouville fractional derivative.

CHAPTER 2 EXTENSION OF HADAMARD FRACTIONAL INTEGRAL

In this chapter, we list basic definitions, symbols, notations and a number of results that will be used throught the dissertation. We introduce an extension of Hadamard fractional integral and call it Hadamard k fractional integral. The extension is based on k gamma function. We discuss a number of basic properties including the semigroup property, commutative law and boundedness of the extended integral operator. Moreover, we introduce some inequalities that will be used to establish the related fractional integral inequalities, for the extended integral operator, in the coming chapters.

2.1 PRELIMINARIES We start with some basic definitions, symbols, notations and results that will be used in later discussion. (i) The set of integers, positive integers, negative integers, real numbers, positive real numbers, non negative real number and the set of complex numbers are denoted by the formal notations Z , Z  , Z  , R, R , R and C respectively. (ii) Integer part of a real number q is denoted by q.

(iii) We usually write loga for loge a . n (iv) The sequence (,,,...,)a1 a 2 a 3 an is denoted by ()a jj1 . (v) We simply write ,  a for   a,   a , etc. Likewise if x(,) a b and y(,) a b , we write it simply by x,(,) y a b , etc. (vi) Interchange of x and y is symbolically written as xy . (vii) If range of a function f is subset of (0, ) , we say that f is a positive function. (viii) The set of all continuous functions from (0, ) to R is denoted by C  and the set of all strictly increasing functions which are continuously differentiable from to R {0} is denoted by D . n n n n n n (ix) We simple write fxj () for (fxj ( )) , a j for ()a j , k ()x for (k (x )) ,  etc. However, in the notation HI(,) a t ,  is not an exponent, it is, in fact, the order of the integral operator.    (x) We can denote I( f ( x )), defined in (1), by I f() t , HI(,) a t f() t ,    HI(,) a t ( f )( t ) , HIf(,) a t or HD(,) a t ( f ( x )) too, instead of the formal notation  HD(,) a t f() t . (xi) We, sometimes, use the abbrevitations FI for Fractional Integral, FD for Fractional Derivative and FC for Fractional Calculus.

d( f ( t )) (xii) We define the operators D and D as D( f ( t ))  t and t t t dt d( tf ( t )) D( f ( t ))  respectively. t dt (xiii) The Mellin transform of a function f is defined as  (2) M{ f }( s )  f ( t ) ts1 dt , 0 while the Mellin convolution of two functions and g is defined as  td (3) f g()()() t f g  . 0  (xiv) Dıaz and Pariguan (2007) have introduced k  gamma and k beta functions as x 1 n k n!() k nk  k (x ) lim , x C  kZ, k  0 , n ()knx where ()knx is the Pochhammer k  symbol for factorial function and is defined as n1 (4) ()()knx x jk , j0 1 xy 1 11 (5) B( x , y ) tkk (1 t ) dt . k  k 0 It is easy to see that  txk 1 x (6) ()x  ekk tx1 dt  k  (),Re()0, x  k  k 0

kk(x ) ( y ) 1 x y (7) Bk (,) x y  B (,), Re()0,Re()0, x  y  k ()x y k k k

(8) k (k ) 1,

(9) kk()()x  k  x  x ,

k ()x nk (10) ()knx  . k ()x Obviously, for k 1, we have the classical gamma and beta functions

(11) 1()()xx   ,

(12) B1(,)(,) x y B x y . (xv) The classical gamma function is defined as x (13) ()(,)(,) etdtt 1   etdt  t   1   etdt  t   1    x    x , 00x where

 (14) (,) x et t 1 dt , x x (15) (,)x  et t 1 dt 0 are called upper and lower incomplete gamma functions respectively. (xvi) If  is neither zero nor a negative integer, () n (16) ()  , n () where () n is the factorial function defined as

()()nn 1 . (xvii) For any positive integer n and pair (,)k  of positive real numbers; using (4), (6) and (16), it is easy to see that ()() (17) kk  ()  nk .     k knn( 1)(  2)...(  n ) k (  n ) k k k k n Likewise, for any positive integer and tuple (,,)k  of positive real numbers,  ; using (4), (6) and (16), we find that ()   nk (18) k  ()  .  k k n () k n If we replace  by nk , (18) is obvious from (17) too.

(xviii) The Pochhammer-Barnes confluent hypergeometric function 11F(;;) a b z is defined as  n ()azn (19) 11F(;;) a b z   . n0 ()!bnn

And the exponential function 00F is defined as  zn (20) z 00F( ;  ; z )   exp z  e . n0 n!

Moreover, we use the notation 11F(;;)kk a b z for  n ()knaz (21) 11F(;;)kk a b z   , n0 ()!knbn (xix) It is easy to see that z (22) 1F 1(k a ; k b ; z ) e 1 F 1 ( k ( b  a ); k b ;  z ) , (see Mubeen (2012)).   x (xx) Substituting y  , it is easy to see that tx t (23) ()()()(,)tp1   x q  1 d   t  x p  q  1 B p q , tx . x Likewise, let

t  1 t 11 (24) Id (log )kk (log )  , tu , k  u  u  t log log dt then if we put y  u , 1y  ,  log dy and y :0 1. Thus, (24) t t log log  u u u becomes  1     tt1  1  1  1  I(log )k (1  y ) k y k dy  (log ) k B ( , ) . k  u0 u k k By use of (7), it becomes t     1 tt1  1  1 (25) (log )k (log ) kd k (log ) k B (  ,  ) .  k u  uu Moreover, we can use (25) in another form u     1  1uu  1  1 (26) (log )k (log ) kd k (log ) k B (  ,  ) , ut ,  k t tt  log which can be obtained either by substituting y  t or directly by ut and u log t  in (25). (xxi) We call the real numbers p and p , Holder’s conjugate numbers, if the numbers are connected by the relation 11 (27) 1. pp (xxii) Let (,)X be a normed linear space, p be a finite real number and for

xj  X, j 1,2,3,...; x ( x1 , x 2 , x 3 ,...) be an infinite sequence of elements of X . Then the space lXp () , f , lX () and f are defined as p   p p l( X ) { x :  x j   }, j1 1  p p xx , p  j j1   l( X ) { x : sup x j   }, j1  xx sup  j . j1 (xxiii) Let p be a finite real number and (,,)Xm be a  -finite measure space. Then Lp (,,) X m is defined as the class of all  -measurable complex valued

p functions f such that f dm  and f is the positive real number defined  p X by 1 p p (28) f f dm , p  X while L (,,) X m is defined as the class of all  -measurable complex valued functions for which there exists   0 such that m({ x X : f ( x )  })  0 and f which is also called essential supremum of f , is defined as  finf{  R : m ({ x  X : f ( x )  })  0}.  (xxiv) Minkowski’s Integral Inequality

Let (,,)Xm X and (,,)Ym Y be  -finite measure spaces and f(,) x y be a . -measurable function such that f(., y ) Lp ( X ) for each yY , then for 1p  ; we have

1 1 p  p p p (29) fxydmydmx(,)()()(,)()()YXXY  fxy dmx dmy ,     XYYX  (see Theorem 3.3.5 of Okikiolu (1971) and Eq. 1.33, p. 9 of Samko et al. (1993)). 2.1.1.1 Lemma: 1 Let nZ  , p  and a  0. Then for all   0, one has n 1an 1np np 1 1 (30) na p p p . pp Proof: np 1 1 If a  0, the inequality (30) holds as  p is non-negative under the given p conditions. And if a  0, letting an 11np np 1 h() pp  pp we have np 1 1 (np 1) n1  p n1 j j h()()  a2   a p j0 which shows that h() is negative on (0,a ), zero at   a and positive on (,)a  . Hence, h() is minimum at   a. That is, for all  (0, ), we have h()()  h a .

This leads to the conclusion (30). 2.1.1.2 Corollary: Let nZ  , np q 0 and a  0. For all   0, we have qqan q np np q q (31) na p p p . pp Proof: If q  0, the result (31) holds in the form of the identity nn . p And if q  0, replacing p by , the inequality (31) follows from (30). q 2.1.1.3 Lemma:

If cpjj,0 , jn1,2,3,..., and n (32)  p j 1, j1 then for each  [ 1,1], we have n n p j 2 (33) cjexp(1 ) c j p j . j1 j1 Proof: For each fixed  [ 1,1], we have (34) exp(xx ) (  )  , for all xR . If   0 , (34) holds as 10 . n 1 And if   0 , we take xx j , where xj c j() c j p j . j1 Thus, using (32), it follows that n c p j n  j (35) p j j1 ((x j  ) ) n , j1 cpjj j1 n 2 (36) exp(pxjj ) exp(1 ), j1 the relation (34) leads to the inequality nn p j exp(pj x j ) (( x j  )  ) . jj11 This leads, by use of (35) and (36), to the inequality (33). 2.1.1.4 Corollary:

If cpjj,0 , jn1,2,3,..., and

n (37)  p j 1, j1 then we have n n p j (38) cj  c j p j . j1 j1 Proof: The result follows from (33) taking  1. 2.1.1.5 Example: 1 One may notice that for n  2 , if we take pp, we find, from 122 (38), the simplest form of the AM-GM inequality cc (39) 12 cc 2 12  which is obviously valid for all c12, c R .  Moreover, it follows, from (39), that for all c1,, c 2 c 3  R 2 2 2 2 2 2 (40) cc12 cc 12  cc 13  cc 13  cc 23  cc 23  6 ccc 123 , another form of which is

(c1 c 2 )( c 2  c 3 )( c 3  c 1 )  8 c 1 c 2 c 3 which holds, by (39), as c c c  c c  c 1 2 2 3 3 1 1. 2c1 c 2 2 c 2 c 3 2 c 3 c 1 2.1.1.6 Lemma:

p p Let x( x1 , x 2 , x 3 ,...) l and y( y1 , y 2 , y 3 ,...) l , where (,)pp is a pair of Holder’s conjugate numbers and p[1, ], then  (41) x y x y .  jj pp j1 Proof: (This is elaborated just to refer relevant inequalities contained in this proof.) Case-1: If p , p  1 and the result (41) holds because of the fact that  xj y j sup x j y j . jj111j  Case-2: If p 1, p  and the inequality (41) holds as  (42) xj y j sup y j x j . jj111j  Case-3:

If 1p  , we consider the function f() t t t , 01t , 01 . Then 1 ft( ) (  1)  0 on [0,1], t1 which implies that f is an increasing function on [0,1]. Therefore, we find that (43) 0ft ( )  1  . 1 a If we take   , t  , ab,0 then, by use of (27), (43) leads to the relation p b 11 ab (44) abpp . pp x y Particularly, if for each j , we set a  ()j p and b  ()j p then (44) leads to x y p p the relation

11xyjj x y x y {()()}pp . jj pp p x p y pp By summing over j from 1 to  , it leads to the relation (41) by use of (27). 2.1.1.7 Corollary: For any ab,0 , if (,)pp is a pair of Holder’s conjugate numbers and p[1, ], then abpp (45) ab . pp Proof: Replacing a and b by a p and b p respectively, (44) leads to the desired result. 2.1.1.8 Note: We can also establish (45), using (38) as: 1 1 mn If we replace a , b and p by c p , c p and respectively, then, by use of 1 2 m (27), we find that app b m n (46)  c  c  c p  c p , p p 1 m n 2 m n 1 1 2 2 1 1 where p  and p  . 1 p 2 p Since (37) is satisfied, using n  2 in (38), (46) leads to the relation pp ab pp12  c12 c  ab pp

18 which is (45).

2.2 HADAMARD k-FRACTIONAL INTEGRAL

For any tuple (,,)ka of positive real numbers, ta , we define Hadamard k fractional integral, of order  , of a function fx() as t  11t 1 (47) k I f( t ) (log )k f ( ) d . H(,) a t  kk ()a   k  k  k  We can denote HI(,) a t f() t which is HI(,) a t ( f ( x )) in fact, by HI(,) a t ( f )( t ), k I ( f ( x ))( t ) or k D f() t too. It is easy to see that H(,) a t H(,) a t  k  1 t k (48) HI(,) a t (1) (log ) . k () a Furthermore, we define k 0 (49) HI(,) a t f f .

1  One may notice that HI(,) a t f()() t I f t , the classical Hadamard k  fractional integral defined in (1). Clearly, the operator HI(,) a t is a linear operator. Moreover, if f is integrable on , k I f() t exists. [,]at H(,) a t 2.2.1 Definition: If a function f is integrable on [,]at for all ta , we say that f is integrable on [,)a  . And if f is integrable on [0, ) , we, sometimes, say simply that f is integrable.

2.3 PROPERITES OF HADAMARD k-FRACTIONAL INTEGRAL

k  Defining HI(,) a t f() t as in (47) for the function f which is continuous on the open interval (,)at for some ta subjecting to growth condition at a , we prove the properties, involving commutative law and semigroup property, of the k  extended integral operator. Moreover, we define HI(,) a t , for   0, as the inverse k  k  operation to HI(,) a t , that is, we define g()() t H I(,) a t f t to be the solution, if it k  exists, of the integral equation f()() t H I(,) a t g t . For imaginary  , Kober (1941) has investigated an extension of I  . 2.3.1 Theorem: For each fixed tuple (,,,)ka of positive real numbers, ta , we have k k  k   (50) HI(,)(,)(,) a t H I a t f( t ) H I a t f ( t ). Proof: Using (47) and applying Fubini’s theorem, we find that

tt  1 1 1 t 11 (51) kkI I f( t ) (log )kk (log ) d f ( u ) du . H(,)(,) a t H a t 2  kkk()() au u   u Using (25), it becomes t  Bt( , ) 1 1 kkI I f( t ) k (log )k f ( u ) du . H(,)(,) a t H a t k()() u u kka Eventually, using (7), we obtain t  11t 1 kIIft k ( ) (log )k fuduIft ( ) k   ( ). H(,)(,)(,) a t H a tk() u u H a t k a 2.3.2 Corollary: For each fixed pair (,)ka of positive real numbers, ta , and fixed pair (,) of non-negative real numbers, we have k k  k   (52) HI(,)(,)(,) a t H I a t f( t ) H I a t f ( t ). Proof: The result (52) follows from (49) and (50) directly. 2.3.3 Theorem: (Commutative Law) For each fixed pair (,)ka of positive real numbers, ta , and fixed pair (,) of real numbers, one has k k  k  k  (53) HatHatI(,)(,)(,)(,) I f( t ) HatHat I I f ( t ). Proof: Case-1: If ,0 , the relation (53) follows directly from (52) as  is commutative in R. Case-2: kk If ,0 , assuming that HI(,)(,) a t H I a t ( f ( x )) g ( x ) , using result of case-1, we k k   k   k   obtain f()()() xHatHat I(,)(,)(,)(,) I g x HatHat I I g x , implying that kkI I f()() x g x and consequently, (53) holds for H(,)(,) a t H a t ,0 too. Case-3: If one of , is positive and the other is negative, without loss of generality we assume that   0, . Now if we let k I f()() x g x , f()() x k I g x ,   0 H(,) a t H(,) a t implying, by use of result of case-1, that kIfx()()() k I  k Igx  k I  k Igx  Hat(,)(,)(,)(,)(,) HatHat HatHat k k  k  k  k  and eventually, we obtain HatHatI(,)(,)(,)(,)(,) Ifx()()() Hat Igx HatHat I Ifx . Hence, (53) holds for all , R .

2.3.4 Theorem: (Semigroup Property) For each fixed pair (,)ka of positive real numbers, ta , and fixed pair (,) of real numbers, we have k k  k   (54) HI(,)(,)(,) a t H I a t( f ( x )) H I a t ( f ( x )). Proof: Case-1: If ,0 , the relation (54) holds as we have already shown (see (52)). Case-2: kk If ,0 , assuming that HI(,)(,) a t H I a t ( f ( x )) g ( x ) , using (50) and (53), we obtain fx()()()()k I k Igx   k I   k Igx   k I     gx , implying that HatHat(,)(,)(,)(,)(,) HatHat Hat k  HI(,) a t f()() x g x , and thus (54) holds for ,0 too. Case-3: If one of , is positive and the other is negative, without loss of generality we assume that 0, 0. Then k  (i) if 0, assuming that HI(,) a t ( f ( x )) g ( x ) , k k   k    k  using (50), we have HatI(,)(,)(,)(,) g( x ) HatHat I I ( f ( x )) Hat I ( f ( x )) , kk implying that g( x ) H I k H I k ( f ( x )) , and consequently, (54) remain valid for this case, k  (ii) and if 0, assuming that HI(,) a t ( f ( x )) g ( x ) , we have k  f( x ) H I(,) a t ( g ( x )), using (50), we obtain kI f( x ) k I  k I    ( g ( x )) k I   ( g ( x )) and so Hat(,)(,)(,)(,) HatHat Hat kk HI(,)(,) a t H I a t f()() x g x , and eventually, (54) remain valid for this case too. Hence, (54) holds for all , R . 2.3.5 Definition:

We define the class C0 as

C0 ={ f : f is continuous on (0, t ) and f is integrable at 0} and the class C as C={ f C :  g in C such that k I g  f }.  0 0H ( a , t ) 2.3.6 Theorem: If , k If exists and belongs to C for each fixed pair (,)ka of fC 0 H(,) a t 0 positive real numbers, ta and fixed non-negative real number  . Proof: The result follows due to continuity of the function f (see Theorem 6.2.7 of Douglass (1996)).

2.3.7 Theorem: If , k I f C for each fixed pair (,)ka of positive real fC 0 H(,) a t  numbers, ta and fixed pair (,) of non-negative real numbers,  . Proof: Since ,0   and , it follows from Theorem 2.3.6 that k If and fC 0 H(,) a t k If exist and belong to C . Then letting k I f g , it follows, by use of H(,) a t 0 H(,) a t kk k  semigroup property, that HI(,)(,) a t f H I a t g implying that HI(,) a t f C . 2.3.8 Theorem: If n is a positive integer and  n, for each fixed tuple (,,)ka of positive real numbers, ta , k If exists and belongs to C if and only if H(,) a t 0

fC n . Proof: If we assume that k If exists and belongs to C , letting k I f g , we H(,) a t 0 H(,) a t kn have f H I(,) a t g which implies that fC n .

Conversely, if fC n , by definition there exists a function gC 0 such that k I g f which implies that g kn I f and so k If exists and belongs H(,) a t H(,) a t H(,) a t to C0 . 2.3.9 Theorem: Let n be a positive integer, 01p and   np  . Then for each fixed pair (,)ka of positive real numbers, ta , (i) if k If exists and belongs to C , , H(,) a t 0 fC n1 (ii) if , k If exists and belongs to C , fC n H(,) a t 0 (iii) the necessary and sufficient condition for existence of k If in C is H(,) a t 0 kp that HI(,) a t f C n . Proof: (i) If we assume that k If exists and belongs to C , letting k I f g , H(,) a t 0 H(,) a t kn1 kp1 we find, by semigroup property, that f H I(,) a t h , where h H I(,) a t g . But it follows by Theorem 2.3.6 that hC as 10p . Thus, we conclude 0 that fC n1.

(ii) Now if fC n , by definition there exists a function gC 0 such that kn k  HI(,) a t g f which, by semigroup property, leads to h H I(,) a t f , where h kp I g . But it follows by Theorem 2.3.6 that hC as p  0. Hence, H(,) a t 0 we accomplish that k If exists and belongs to C . H(,) a t 0

(iii) If we assume that k If exists and belongs to C , taking k I f g , H(,) a t 0 H(,) a t k p k n we obtain, by semigroup property, that HI(,)(,) a t f H I a t g , implying that kp HI(,) a t f C n . kp And if HI(,) a t f C n , by definition, there exists a function gC 0 such k p k n that HI(,)(,) a t f H I a t g which, by semigroup property, leads to g k I f . Thus, we find that k If exists and belongs to C . H(,) a t H(,) a t 0 2.3.10 Theorem:

If fC  , for each fixed pair (,)ka of positive real numbers, ta , we have k  (i) HI(,) a t f C for each  R , k  (ii) HI(,) a t f C   for each , R. Proof:

Since fC  , by definition, there exists a function gC 0 such that k  f H I(,) a t g which, by semigroup property, gives kk   k  (i) HI(,)(,) a t g H I a t f , implying that HI(,) a t f C , kk       k  (ii) HI(,)(,) a t g H I a t f , implying that HI(,) a t f C   . 2.3.11 Theorem:

kg,(,) For any f, g C0 , if we define HI(,) a t as   t  (g ( t ))k 1 t 1 (55) kgI,(,) f( t ) (log )kk ( g ( )) f (  ) d  H(,) a t k ()   k a then k,(,),(,),(,) g   k  g  k   g  (56) HI(,)(,)(,) a t H I a t f()() t H I a t f t . Proof: Using (55) and applying Fubini’s theorem, we find that k,(,),(,) g   k  g  HI(,)(,) a t H I a t f() t     tt   (g ( t ))k 1 1  11 t  (g ( u ))k f ( u ) (log ) k (log ) k d du . 2  kkk()() au u  u   log( ) Substitution y  u , and using (7), it becomes t log( ) u     t    (g ( t ))k 1 t 1 kI,(,),(,) g   k I  g  f( t ) (log )kk ( g ( u )) f ( u ) du H(,)(,) a t H a t  kk ()a u u

kg ,(,)   HI(,) a t f( t ). 2.3.12 Theorem:

kk If HI(,)(,) a t f H I a t g , fg . Proof: kk Since HI(,)(,) a t f H I a t g , it follows by (47), due to linearity of integral, that t  1 t 1  (log )k (f ( ) g ( t )) d 0 a  implying that fg (see p. 105 of Okikiolu (1971)).

k  2.3.13 Theorem: (Boundedness of HI(,) a t ) Let p 1. Then, for each fixed tuple (,,)ka of positive real numbers, k  ta , the operator HI(,) a t , defined in (47), is bounded in the sense k p p HILL(,) a t :  such that k I f K f , H(,) a t p p logt   1  1 K ep ().k d kk () log a Proof: In (47), for any fixed ta if   11 t 1  (log )k ,   (at , ), (,)t  kk ()    0,   (at , ) then (,)(,)ct c  c1  t  , c  0 . Therefore,   0 and by Theorem 4.2.6 of Okikiolu (1971), we conclude that k p p HILL(,) a t :  such that k I f K f , K K(,,,,) p k a t , H(,) a t p p 1  ttp 11 1  1 d Kd11 (1,  )   p (  log  ) k aakk () logt   1  1  edp ()k . kk () log a

CHAPTER 3

Fi INEQUALITIES RELATING THE EXTENDED OPERATOR

Mathematical inequalities play important role in differential and integral equations and so have applications in several fields of mathematics. Fractional inequalities are much helpful to study properties, involving the existence and uniqueness of solutions, of fractional differential equations. In this chapter, we establish a number of inequalities regarding Hadamard k fractional integral involving the inequalities for Chebyshev functional. We find Hadamard k fractional integrals for polynomials and exponents of natural logarithmic function so that we can use them in formation of the related fractional integral inequalities. For our convenience, we, sometimes, use the p p p notation f , fg and f (or fx()) for the scalars fx(), f()() x g x and (fx ( )) k  respectively, etc. Moreover, we use the simple notation HI(,) a t fg() t instead of k  HI(,) a t ( fg )( t ), etc.

3.1 FI INEQUALITIES FOR CHEBYSHEV FUNCTIONAL Many researchers have given considerable attention to the Chebyshev functional 1b 1 b  1 b  (57) Tfg(,)()()()() fxgxdx  fxdx   gxdx  , b aa b  a a  b  a a  where f and g be integrable functions which are synchronous on [,]ab , that is, for all x,[,] y a b , (58) (()f x f ())(() y g x  g ()) y  0 , (see Chebyshev (1882)), or equivalently, (59) fxgx()()()()()()()() fygy  fxgy  fygx . A number of inequalities for the functional, defined in (57), have appeared in the literature, see Malamud (2001), Pachpatte (2006) and Marinković et al. (2008). We establish certain integral inequalities, involving the inequalities for the Chebyshev functional, using the Hadamard k fractional integral and the fact that for each fixed tuple (,,)ka of positive real number, ta , we have  1 t 1 (log )k  0 xkk () x for all x(,) a t .

3.1.1 Theorem:

Let f and g be two integrable functions on [0, ) . If f and g are synchronous on [0, ) , for each fixed tuple (,,)ka of positive real numbers, ta , we have  t  (60) kI fg() t ()(log)k k I  f () t k I  g (). t H(,)(,)(,) a t ka H a t H a t Proof: Since f and g are synchronous functions on [0, ) , for all xy,0 , we have (61) fxgx()()()()()()()() fygy  fxgy  fygx .  1 t 1 Multiplying both sides by (log ) k and then integrating with respect xkk () x to x over (,)at , the inequality (61), by use of (47), becomes k k  k  k  (62) HatI(,)(,)(,)(,) fg() t f ()() y g y Hat I (1)  g () y Hat I f () t  f () y Hat I g (). t Integrating with respect to y over (,)at after multiplying both sides by  1 t 1 (log ) k , by use of (47) and (48), the inequality (62) leads to the ykk () y relation (60). 3.1.2 Theorem:

If two integrable functions f and g are synchronous on [0, ) , for each fixed tuple (,,,)ka of positive real numbers, ta , one has  11ttkkkk (63) (log )HI(,)(,) a t fg ( t ) (log ) H I a t fg ( t ) kk()() aa   k k  k  k  HatIftIgt(,)(,)(,)(,)( ) Hat ( ) Hat IftIgt ( ) Hat ( ). Proof: Since f and g are synchronous functions on [0, ) , for all xy,0 , we have (64) fxgx()()()()()()()() fygy  fxgy  fygx .  1 t 1 Multiplying both sides by (log ) k and then integrating with respect xkk () x to x over (,)at , the inequality (64), by use of (47), becomes k k  k  k  (65) HatI(,)(,)(,)(,) fg() t f ()() y g y Hat I (1)  g () y Hat I f () t  f () y Hat I g (). t Integrating with respect to y over (,)at after multiplying both sides by  1 t 1 (log ) k , by use of (47) and (48), the inequality (65) leads to the ykk () y relation (63). 3.1.3 Note: For  , the relation (63) leads to the result (60).

3.1.4 Note: The inequality (63) and hence (60) will be reversed if the functions f and g are asynchronous functions on [0, ) , that is, for all xy, [0, ) , (()f x f ())(() y g x  g ()) y  0 . 3.1.5 Note: If the functions f and g are both increasing (or both decreasing) functions on [0, ) , (58) is satisfied. So (63) (and hence (60) too) is satisfied. 3.1.6 Note: If one of the functions f and g is increasing and the other is decreasing on [0, ), the inequality (58) is reversed. So the inequality (63) (and hence (60) too) is reversed. 3.1.7 Theorem:

If {fj : j 1,2,3,..., n } is a collection of n integrable, positive and increasing functions on [0, ) , for each fixed tuple (,,)ka of positive real numbers, ta , we have nn 1n (66) k k  k  HatI(,)(,)(,) f j( t )  Hat I (1) Hatj I f ( t ) . jj11 Proof: For n 1, (66) trivially holds for equality. For n  2, (66) remains valid by (60). Assume that (66) is satisfied for nl2. That is, ll 1l (67) k k  k  HatI(,)(,)(,) f j( t )  Hat I (1) Hatj I f ( t ) . jj11 l Since  f j and f j1 are integrable and increasing functions, it follows by use j1 of Note 3.1.5, (60) and (67) that ll1 1 k k  k  k  HatI(,) f j( t )  Hat I (,) (1) Hat I (,) f j ( t ) Hatl I (,)1 f ( t ) jj11 l1 l kk  HI(,)(,) a t(1) H I a t f j ( t ). j1 Hence, it follows by induction that the inequality (66) is satisfied for all integral positive values of n . 3.1.8 Lemma: For each fixed tuple (,,)k  of positive real numbers, t 1, we have

   11 k  kkk () (68) HtI(1, ) (log x ) ( t ) (log t ) . k () Proof: The result follows, by using (47) with (7) and (25) for u 1, as t   111t  1  1 k I (log x )k ( t ) (log ) k (log ) k d Ht(1, )  kk ()1    1 1 k  (logtB )k ( , ) k ()   () 1  k (logt )k . () k 3.1.9 Corollary: For each fixed pair (,)k  of positive real numbers, t 1, one has  1 k  k (2k ) k (69) HtI(1, ) log x ( t ) (log t ) , k ( 2k ) and generally, for each  1, we have   k k (( 1)k ) k (70) HtI(1, ) (log x ) ( t ) (log t ) . k (  (  1)k ) Proof: The results (69) and (70) follow directly from (68) by using   2k and ( 1)k respectively. 3.1.10 Note: One may verify (48) from (70), for   0, with collaboration of (8) and (9). 3.1.11 Lemma: For each fixed tuple (,,)ka of positive real numbers, ta , we have

k  tt (71) HIx(,) a t ( )  ( ,log( )). kk () k a Proof: Using f() x x in (47), we find that tt 11t11 d t (72) k I ( x ) (log )kk (log ) d . H(,) a t  kkkk()()aa     t t t log( ) t If we put y  log( ) , y:log( ) 0 as  : a t and eey  ,  a  implying that   te y and thus d  te y dy . Thus, (72) gives

t log( ) a  t 1 kyI () x e yk dy H(,) a t  kk () 0 which implies (71) by use of (15). 3.1.12 Lemma: For each fixed tuple (,,)ka of positive real numbers, ta , and fixed  R {0}, we have    k k  tt  (73) HIx(,) a t ( )  ( ,log( ) ). kk () k a Proof: Using f() x x in (47), we find that t  1 t 1 (74) k I( x  ) (log ) k  1 d . H(,) a t  kk ()a t t t log( ) t If we put y  log( ) , y:log( ) 0 as  : a t and eey  ,  a  implying that   te y and thus d  te y dy . Thus, (74) gives t log( )  a  t 1 (75) kyI() x  e  yk dy . H(,) a t  kk () 0 1 tt If we substitute uy  , dy du and uy:0 log( ) as :0 log( ) .  aa Thus, (75) becomes  t   log( )  a  t  k 1 kuI() x e uk du H(,) a t  kk () 0 which implies (73) by use of (15). 3.1.13 Note: One may notice that for  1, we find (71) from (73). 3.1.14 Theorem: If two integrable functions f and g are such that, for all xy, [0, ) , (76) (()f x f ())(() y g x  g ()) y  0, then for each fixed tuple (,,,)ka of positive real numbers, ta , we have

 1 t k k k  k  (77) (log )HI(,)(,)(,) a t fg ( t ) H I a t f ( t ) H I a t g ( t ) k () a  k k 1 t k k  HI(,)(,)(,) a t f( t ) H I a t g ( t ) (log ) H I a t fg ( t ). k () a Proof:

We rewrite (76) as (78) fxgx()()()()()()()() fxgy fygx fygy .  1 t 1 Multiplying both sides by (log ) k and then integrating with respect xkk () x to x over (,)at , the inequality (78), by use of (47), becomes k k  k  k  (79) HatI(,)(,)(,)(,) fg() t g () y Hat I f () t f () y Hat I g () t f ()() y g y Hat I (1). Integrating with respect to y over (,)at after multiplying both sides by  1 t 1 (log ) k , by use of (47) and (48), the inequality (79) leads to the ykk () y relation (77). 3.1.15 Theorem: If three integrable functions f , g and h are such that, for all xy, [0, ), (80) (()fx fygx ())(()  gyhx ())(()  hy ())  0, then for each fixed tuple (,,,)ka of positive real numbers, ta , we have

 1 t k k k  k  (81) (log )HI(,)(,)(,) a t fgh ( t ) H I a t f ( t ) H I a t gh ( t ) k () a  k k 1 t k k  HI(,)(,)(,) a t gh( t ) H I a t f ( t ) (log ) H I a t fgh ( t ) k () a k k  k  k  HatI(,)(,)(,)(,) fh()()()() t Hat I g t Hat I fg t Hat I h t k k  k  k  HatI(,)(,)(,)(,) h( t ) Hat I fg ( t ) Hat I g ( t ) Hat I fh ( t ). Proof: We rewrite (80) as (82) fxgxhx()()()()()()()()()()()() fxgyhy  fygxhx  fygyhy fxgyhx()()()()()()()()()()()()  fxgxhy  fygyhx  fygxhy .  1 t 1 Multiplying both sides by (log ) k and then integrating with respect xkk () x to x over (,)at , the inequality (82), by use of (47), becomes kk (83) HI(,)(,) a t fgh()()()() t H I a t f t g y h y kk HI(,)(,) a t ghtfy()() H I a t (1)()()() fygyhy kk HI(,)(,) a t fh()()()() t g y H I a t fg t h y kk HI(,)(,) a t htfygy()()() H I a t gtfyhy ()()(). Integrating with respect to y over (,)at after multiplying both sides by  1 t 1 (log ) k , by use of (47) and (48), the inequality (83) leads to the ykk () y relation (81).

3.1.16 Corollary: If three integrable functions f , g and h are such that, for all xy, [0, ) (84) (()fx fygx ())(()  gyhx ())(()  hy ())  0, then for each fixed tuple (,,)ka of positive real numbers, ta , we have

 1 t k k k  k  (85) (log )HI(,)(,)(,) a t fgh ( t ) H I a t f ( t ) H I a t gh ( t ) k () a k k  k  k  HatI(,)(,)(,)(,) fh( t ) Hat I g ( t ) Hat I fg ( t ) Hat I h ( t ). Proof: Replacing  by  , the result (85) follows immediately from the inequality (81).

3.1.17 Theorem: If three integrable functions f , g and h are such that, for all xy, [0, ), (86) (()fx fygx ())(()  gyhx ())(()  hy ())  0 , then for each fixed tuple (,,,)ka of positive real numbers, ta , one has

 1 t k k k  k  (87) (log )HI(,)(,)(,) a t fgh ( t ) H I a t fh ( t ) H I a t g ( t ) k () a k k  k  k  HatI(,)(,)(,)(,) gh()()()() t Hat I f t Hat I h t Hat I fg t k k  k  k  HatI(,)(,)(,)(,) fg()()()() t Hat I h t Hat I f t Hat I gh t  k k 1 t k k  HI(,)(,)(,) a t g( t ) H I a t fh ( t ) (log ) H I a t fgh ( t ). k () a Proof: We rewrite (86) as (88) fxgxhx()()()()()()()()()()()() fxgyhx fygxhx fygyhx fxgxhy()()()()()()()()()()()()  fxgyhy  fygxhy  fygyhy .  1 t 1 Multiplying both sides by (log ) k and then integrating with respect xkk () x to x over (,)at , the inequality (88), by use of (47), becomes kk (89) HI(,)(,) a t fgh()()() t H I a t fh t g y kk HI(,)(,) a t ghtfy()()()()() H I a t htfygy kk HI(,)(,) a t fg()()()()() t h y H I a t f t g y h y kk HI(,)(,) a t gtfyhy()()() H I a t (1)()()() fygyhy . Integrating with respect to y over (,)at after multiplying both sides by  1 t 1 (log ) k , by use of (47) and (48), the inequality (89) leads to the ykk () y relation (87).

3.1.18 Theorem:

Let f j :[0, )  [0,  ) be an integrable function for each j 1,2. If f1 and f2 are increasing or both decreasing on [0, ) , for each fixed tuple

(,,,,,)ka  12  of positive real numbers, ta , one has  11ttkkkk1  2  1  2 (90) (log )HI( a , t ) f 1 f 2 ( t ) (log ) H I ( a , t ) f 1 f 2 ( t ) kk()() aa   kIftIft1( ) k   2 ( ) k IftIft   1 ( ) k   2 ( ). Hat(,)1 Hat (,)2 Hat (,)1 Hat (,)2 Proof: Since the positive functions and are both increasing or both decreasing on , for all x,(,) y a t , we have 2 jj (91) (fjj ( x ) f ( y )) 0 j1 for each fixed (,) . We rewrite it as

1  2  1  2  1  2  1  2 (92) fxfxfyfy1() 2 () 1 () 2 ()  fxfy 1 () 2 ()  fyfx 1 () 2 ().  1 t 1 Multiplying both sides by (log ) k and then integrating with respect xkk () x to x over (,)at , the inequality (92), by use of (47), becomes

kk1  2  1  2 (93) HI( a , t ) f 1 f 2( t ) f 1 ( y ) f 2 ( y ) H I ( a , t ) (1)

2kk  1  1  2 f2( y )H I ( a , t ) f 1 ( t ) f 1 ( y ) H I ( a , t ) f 2 ( t ). Integrating with respect to y over (,)at after multiplying both sides by  1 t 1 (log ) k , by use of (47) and (48), the inequality (93) leads to the ykk () y relation (90).

3.1.19 Corollary: Let be an integrable function for each . If and are increasing or both decreasing on , for each fixed tuple

(,,,,)ka 12  of positive real numbers, , we have 22 t  (94) kkjjk HI(,)(,) a t f j( t ) k ( ) (log ) H I a t f j ( t ). jj11a Proof: If we replace  by  , the desired result (94) follows directly from (90).

3.1.20 Theorem: Let be an integrable function for each . If one of the functions and is increasing and the other is decreasing on , for each fixed tuple of positive real numbers, , one has

 11ttkkkk1  2  1  2 (95) (log )HI( a , t ) f 1 f 2 ( t ) (log ) H I ( a , t ) f 1 f 2 ( t ) kk()() aa   kIftIft1( ) k   2 ( ) k IftIft   1 ( ) k   2 ( ). Hat(,)1 Hat (,)2 Hat (,)1 Hat (,)2 Proof:

Since one of the positive functions f1 and f2 is increasing and the other is decreasing on [0, ) , for all x,(,) y a t , we have 2 jj (96) (fjj ( x ) f ( y )) 0 j1 for each fixed (,) . We rewrite it as

1  2  1  2  1  2  1  2 (97) fxfxfyfy1() 2 () 1 () 2 ()  fxfy 1 () 2 ()  fyfx 1 () 2 ().  1 t 1 Multiplying both sides by (log ) k and then integrating with respect xkk () x to x over (,)at , the inequality (97), by use of (47), becomes

kk1  2  1  2 (98) HI( a , t ) f 1 f 2( t ) f 1 ( y ) f 2 ( y ) H I ( a , t ) (1)

2kk  1  1  2 f2( y )H I ( a , t ) f 1 ( t ) f 1 ( y ) H I ( a , t ) f 2 ( t ). Integrating with respect to y over (,)at after multiplying both sides by  1 t 1 (log ) k , by use of (47) and (48), the inequality (98) leads to the ykk () y relation (95).

3.1.21 Corollary:

Let f j :[0, )  [0,  ) be an integrable function for each j 1,2. If one of the functions and is increasing and the other is decreasing on , for each fixed tuple (,,,,)ka 12  of positive real numbers, ta , we have 22 t  (99) kkjjk HI(,)(,) a t f j( t ) k ( ) (log ) H I a t f j ( t ). jj11a Proof: Replacing  by  , the desired result (99) follows immediately from (95).

3.1.22 Theorem:

If f j :[0, )  (0,  ) is integrable function for each such that fx() fx() m  inf 1 and M  sup 1 then for each fixed tuple (,,)ka of 0x  fx2 () 0x  fx2 () positive real numbers, ta , we have 2 2 ()mM 2 (100) kk2  HI( a , t ) f j()(). t  H I ( a , t ) f 1 f 2 t  j1 4mM Proof:

fx() fx() Since m  inf 1 and M  sup 1 , for all x(,) a t , we have 0x  fx2 () 0x  fx2 () f()() x  f x  11mM     0 f22()() x  f x  implying that 22 (101) fx1() mMf 2 ()( x  mMfxfx  )()(). 1 2  1 t 1 Multiplying both sides by (log ) k and then integrating with respect xkk () x to x over (,)at , the inequality (101), by use of (47), becomes k22 k  k  (102) HI(,)1 a t ft( ) mMI H (,)2 a t ft ( )  ( mMI  ) H (,)12 a t fft ( ). As mM  0, we have 2 kkI f22( t ) mM I f ( t ) 0  H( a , t ) 1 H ( a , t ) 2  implying that k2 k  2 k  2 k  2 (103) HatIftmMIft(,)1( ) Hat (,)2 ( ) 2 Hat IftmMIft (,)1 ( ) Hat (,)2 ( ). The desired result (100) follows from the inequalities (102) and (103). 3.1.23 Theorem: Let f and g be two integrable functions defined on (1, ). If f is increasing and g is differentiable such that (104) m inf ( xg ( x )) x1 then for each fixed pair (,)k  of positive real numbers, t 1, we have

  kkk (105) HIfgt(1, t )()  k ()(log)  t  () tmIfxxt  H (1, t ) (()log)(), where  1 k k k (2k ) k k  (106) ()tH IftIgtm(1, t ) () H (1, t ) () (log) t H Ift (1, t ) (). k ( 2k ) Proof: It follows from (104) that the function h( x ) g ( x ) m log x is an increasing function on . Using (60), (69) and (106), it follows from Note 3.1.5 that k  HtI(1, ) ( f ( x ) h ( x ))( t )   k kk  k( ) (logt ) H I(1, t ) f ( t ) H I (1, t ) ( g ( x )  m log x )( t )   k k ( ) (logt ) k k  k  k  HIftIgtmIftI(1, t )( ) H (1, t ) ( ) H (1, t ) ( ) H (1, t ) (log xt )( )   k k (  ) (logtt )  ( ).

This leads to the result (105). 3.1.24 Theorem: Let f and g be two integrable functions defined on (1, ). If f is decreasing and g is differentiable such that (107) M sup( xg ( x )) x1 then for each fixed pair (,)k  of positive real numbers, t 1, one has

  kkk (108) HIfgt(1, t )()  k ()(log)  t  () tMIfxxt  H (1, t ) (()log)() , where  1 k k k (2k ) k k  (109)  (t )H IftIgtM(1, t ) ( ) H (1, t ) ( ) (log t ) H Ift (1, t ) ( ). k ( 2k ) Proof: It follows from (107) that the function H( x ) g ( x ) M log x is a decreasing function on . Using (60), (69) and (109), it follows from Note 3.1.5 that k  HtI(1, ) ( f ( x ) H ( x ))( t )   k kk  k( ) (logt ) H I(1, t ) f ( t ) H I (1, t ) ( g ( x )  M log x )( t )   k k ( ) (logt ) k k  k  k  HIftIgtMIftI(1, t )( ) H (1, t ) ( ) H (1, t ) ( ) H (1, t ) (log xt )( )   k k (  ) (logtt )  ( ). This leads to the conclusion (108). 3.1.25 Theorem: Let f and g be two integrable functions defined on . If f and g are differentiable and

(110) m12inf ( xf ( x )), m inf ( xg ( x )). xx11 then for each fixed pair (,)k  of positive real numbers, t 1, we have

 2 k  k (3k ) k (111) HtI(1, ) fg( t ) m 1 m 2 (log t ) k ( 3k )    (  ) (logt )k  ( t )   ( t )   ( t ) , k  1 2 3 where 2  2( 1) kk k (2k ) k (112) 1()tH I (1, t ) f () t H I (1, t ) g () t m 1 m 2 2 (log) t , k ( 2k )   (2k ) 1 (113) k k kk 2(t ) (log t ) m 2H I (1, t ) f ( t ) m 1 H I (1, t ) g ( t ) , k ( 2k )

kk (114) 3()tmI 1H (1, t ) (()log)() gx xtmI 2 H (1, t ) (()log)() fx xt . Proof:

It follows from (110) that h11( x ) f ( x ) m log x and h22( x ) g ( x ) m log x are increasing functions on (1, ). Using (60), (69), (112) and (113) , it follows from Note 3.1.5 that k  HtI(1, )( h 1 ( x ) h 2 ( x ))( t )   k kk  k( ) (logt ) H I(1, t ) ( fxm ( )  1 log xtIgxm )( ) H (1, t ) ( ( )  2 log xt )( )   k  k (  ) (logt ) ( 12 ( t )   ( t )) . By use of (70) and (114), it leads to the inequality (111).

3.1.26 Theorem: Let f and g be two integrable functions defined on . If f and g are differentiable and

(115) M12sup( xf ( x )), M sup( xg ( x )). xx11 then for each fixed pair (,)k  of positive real numbers, t 1, we have

 2 k  k (3k ) k (116) HtI(1, ) fg( t ) M 1 M 2 (log t ) k ( 3k )    (  ) (logt )k  ( t )   ( t )   ( t ) , k  1 2 3 where 2  2( 1) kk k (2k ) k (117) 1(t )H I (1, t ) f ( t ) H I (1, t ) g ( t ) M 1 M 2 2 (log t ) , k ( 2k )   (2k ) 1 (118) k k kk  2(t ) (log t ) M 2H I (1, t ) f ( t ) M 1 H I (1, t ) g ( t ) , k ( 2k ) kk (119)  3()tMI 1H (1, t ) (()log)() gx xtMI 2 H (1, t ) (()log)() fx xt . Proof:

It follows from (115) that H11( x ) f ( x ) M log x and H22( x ) g ( x ) M log x are decreasing functions on . Using (60), (69), (117) and (118) , it follows from Note 3.1.5 that k  HtI(1, )( H 1 ( x ) H 2 ( x ))( t )   k kk  k( ) (logt ) H I(1, t ) ( fxM ( )  1 log xtIgxM )( ) H (1, t ) ( ( )  2 log xt )( )   k  k (  ) (logt ) ( 12 ( t )   ( t )) It, thus leads to the conclusion (116) by use of (70) and (119).

3.1.27 Theorem: Let f and g be two integrable functions defined on (0, ) . If f is increasing and g is differentiable such that (120) m inf ( g ( x )) x0 then for each fixed tuple (,,)ka of positive real numbers, ta , we have

 t  (121) kkI fg() t  ()(log) k  () t  m I (())() xf x t , H(,)(,) a t ka H a t where

k k mt t k  (122) (t )H I(,)(,)(,) a t f ( t ) H I a t g ( t ) ( ,log( )) H I a t f ( t ). kk () k a Proof: It follows from (120) that the function h()() x g x mx is an increasing function on (0, ) . Using (60), (71) and (122), it follows from Note 3.1.5 that k  HI(,) a t ( f ( x ) h ( x ))( t )  t   ()(log)k kkI f () t I (() g x  mx )() t ka H(,)(,) a t H a t   t k k ( ) (log ) a k k  k  k  HatI(,)(,)(,)(,) ft( ) Hat IgtmI ( ) Hat ft ( ) Hat I ( xt )( )   t k k (  ) (log )  (t ). a This leads to the result (121).

3.1.28 Theorem: Let and be two integrable functions defined on . If is decreasing and is differentiable such that (123) M sup( g ( x )) x0 then for each fixed tuple (,,)ka of positive real numbers, , we have

 t  (124) kkI fg() t  ()(log) k  () t  M I (())() xf x t , H(,)(,) a t ka H a t where

k k Mt t k  (125) (t )H I(,)(,)(,) a t f ( t ) H I a t g ( t ) ( ,log( )) H I a t f ( t ). kk () k a Proof: It follows from (123) that the function H()() x g x Mx is an decreasing function on .

Using (60), (71) and (125), it follows from Note 3.1.5 that k  HI(,) a t ( f ( x ) H ( x ))( t )  t   ()(log)k kkI f () t I (() g x  Mx )() t ka H(,)(,) a t H a t   t k k ( ) (log ) a k k  k  k  HatI(,)(,)(,)(,) ftIgtMI( ) Hat ( ) Hat ftI ( ) Hat ( xt )( )   t k k (  ) (log )  (t ). a This leads to the result (124).

3.1.29 Theorem: Let f and g be two integrable functions defined on (0, ) . If f is increasing and g is differentiable such that (126) m inf ( x1 g ( x )) , x0 for some  R {0}, then for each fixed tuple (,,)ka of positive real numbers, ta , we have  tm (127) kkI fg() t  ()(log) k  () t  I  ( x  f ())() x t , H(,)(,) a t ka  H a t where    k k k m t t  k  (128) ()tH I(,)(,)(,) a t f () t H I a t g () t (,log()) H I a t f (). t kk () k a Proof: m It follows from (126) that the function h()() x g x x is an increasing  function on (0, ) . Using (60), (73) and (128), it follows from Note 3.1.5 that k  HI(,) a t ( f ( x ) h ( x ))( t )   tmk kk    k()(log) HI(,)(,) a t f () t H I a t (() g x  x )() t a 

m kIftIgt( ) k  ( ) k IftI  ( ) k  ( xt  )( ) Hat(,)(,)(,)(,) Hat Hat Hat    t k k (  ) (log )  (t ). a This leads to the result (127).

3.1.30 Theorem: Let f and g be two integrable functions defined on (0, ) . If f is decreasing and g is differentiable such that (129) M sup( x1 g ( x )) , x0 for some  R {0}, then for each fixed tuple (,,)ka of positive real numbers, ta , we have  tM (130) kkI fg() t  ()(log) k  () t  I  ( x  f ())() x t , H(,)(,) a t ka  H a t where    k k k M t t  k  (131) ()tH I(,)(,)(,) a t f () t H I a t g () t (,log()) H I a t f (). t kk () k a Proof: M It follows from (129) that the function H()() x g x x is an decreasing  function on (0, ) . Using (60), (73) and (131), it follows from Note 3.1.5 that k  HI(,) a t ( f ( x ) H ( x ))( t )  tM  ()(log)k kkI f () t I  (() g x  x  )() t ka H(,)(,) a t H a t    t k k ( ) (log ) a M kIftIgt( ) k  ( ) k IftI  ( ) k  ( xt  )( ) Hat(,)(,)(,)(,) Hat Hat Hat    t k k (  ) (log )  (t ). a This leads to the result (130).

3.1.31 Note: For  1, the inequalities (121) and (124) follow directly from (127) and (130) respectively. 3.1.32 Theorem:

Let f1 and f2 be two integrable functions defined on (0, ) . If and are differentiable and  (132) mjj inf ( f ( x )) , j 1,2, x0 then for each fixed tuple (,,)ka of positive real numbers, , we have

  2 k k  tt2  2 (133) HI( a , t ) f 1 f 2( t ) m 1 m 2  ( ,log( ) ) kk () k a  t   ()(log) k   ()t   () t   () t , k a 1 2 3 where 2 m m t2  t (134) (t )k I f ( t )12 2 ( ,log( )), 1 H ( a , t ) j 22 j1 kk () k a tt (135) kk 2(t ) ( ,log( )) m 2(,)1H I a t f ( t ) m 1(,)2 H I a t f ( t ) , kk () k a kk (136) 3()t mI 1(,)2H a t (())() xfxt mI 2(,)1 H a t (())() xfxt . Proof:

It follows from (132) that for each j 1,2; hj()() x f j x m j x is increasing function on (0, ) . Using (60), (73), (134) and (135) , it follows from Note 3.1.5 that   2 kkt k HIhxhxt( a , t )(()())() 1 2  k ()(log) H Ifxmxt ( a , t ) (() j  j )() a j1   t k  k (  ) (log ) ( 12 (tt )   ( )) a which leads to the inequality (133) by use of (73) and (136).

3.1.33 Theorem:

Let f1 and f2 be two integrable functions defined on (0, ) . If and are differentiable and  (137) Mjj sup( f ( x )) , , x0 then for each fixed tuple (,,)ka of positive real numbers, ta , we have

  2 k k  tt2  2 (138) HI( a , t ) f 1 f 2( t ) M 1 M 2  ( ,log( ) ) kk () k a  t   ()(log) k   ()t   () t   () t , k a 1 2 3 where 2 M M t2  t (139) (t )k I f ( t )12 2 ( ,log( )) , 1 H ( a , t ) j 22 j1 kk () k a tt (140) kk 2(t ) ( ,log( )) M 2(,)1H I a t f ( t ) M 1(,)2 H I a t f ( t ) , kk () k a kk (141)  3()t MI 1(,)2H a t (())() xfxt MI 2(,)1 H a t (())() xfxt . Proof:

It follows from (137) that for each j 1,2; Hj()() x f j x M j x is decreasing function on (0, ) . Using (60), (73), (139) and (140) , it follows from Note 3.1.5 that   2 kkt k HIHxHxt( a , t )( 1 () 2 ())()  k ()(log) H IfxMxt ( a , t ) (() j  j )() a j1   t k  k (  ) (log ) ( 12 (tt )   ( )) a which leads to the inequality (138) by use of (73) and (141).

3.1.34 Theorem:

Let f1 and f2 be two integrable functions defined on (0, ) and  be a non zero real number. If and are differentiable and 1  (142) mjj inf ( x f ( x )) , , x0 then for each fixed tuple (,,)ka of positive real numbers, ta , we have

  2 k k m12 m t(2 ) t 2 (143) HI( a , t ) f 1 f 2 ( t ) 2  ( ,log( ) ) kk () k a  t   ()(log) k   ()t   () t   () t , k a 1 2 3 where 2  2 m m t2k t (144) (t )k I f ( t )12 2 ( ,log( ) ) 1 H ( a , t ) j 2 2 2 , j1 kk () k a  tt kk   (145) 2()t (,log()) m 2(,)1H I a t f () t m 1(,)2 H I a t f () t , 1  k ka kk () mm (146)  ()t12kk Ixfxt (  ())() Ixfxt  (  ())(). 3H ( a , t ) 2 H ( a , t ) 1 Proof: m It follows from (142) that for each ; h()() x f xj x is increasing jj function on . Using (60), (73), (144) and (145), it follows from Note 3.1.5 that   2 m kkt k j  HIhxhxt( a , t )(()())() 1 2  k ()(log) H Ifx ( a , t ) (() j  xt )() a j1    t k  k (  ) (log ) ( 12 (tt )   ( )) a which leads to the inequality (143) by use of (73) and (146).

3.1.35 Theorem:

Let f1 and f2 be two integrable functions defined on (0, ) and  be a non zero real number. If and are differentiable and 1  (147) Mjj sup( x f ( x )), j 1,2, x0 then for each fixed tuple (,,)ka of positive real numbers, ta , we have

  M M t2 (2 ) k t (148) k 12 2 HI( a , t ) f 1 f 2 ( t ) 2  ( ,log( ) ) kk () k a  t   ()(log) k   ()t   () t   () t , k a 1 2 3 where 2  2 M M t2k t (149) (t )k I f ( t )12 2 ( ,log( ) ) 1 H ( a , t ) j 2 2 2 , j1 kk () k a  tt kk   (150) 2()t (,log()) M 2(,)1H I a t f () t M 1(,)2 H I a t f () t , 1  k ka kk () MM (151)  ()t12kk Ixfxt (  ())() Ixfxt  (  ())() . 3H ( a , t ) 2 H ( a , t ) 1 Proof: M It follows from (147) that for each ; H()() x f xj x is decreasing jj function on (0, ) . Using (60), (73), (149) and (150), it follows from Note 3.1.5 that   2 M kkt k j  HIHxHxt( a , t )(() 1 2 ())()  k ()(log) H Ifx ( a , t ) (() j  xt )() a j1    t k  k (  ) (log ) ( 12 (tt )   ( )) a which leads to the inequality (148) by use of (73) and (151).

CHAPTER 4 MORE FI INEQUALITIES FOR THE EXTENDED OPERATOR

In this chapter, we establish fractional integral inequalities, relating the Hadamard k  fractional integral, involving the inequalities based on AM-GM inequalities, Holder’s inequality and inequalities of rearrangement. The fractional integral inequalities relating the bounded functions are also explored.

If f1 and f2 are two integralbe functions on [,]ab such that for all

x[,] a b , m1 f 1() x M 1, m2 f 2() x M 2 for some mjj, M R , j 1,2, then 11bb2 2 2 Mm (152) f()(). x dx f x dx jj jj   b aaaj1 j  1 b a j  1 2 The inequality (152) was introduced by G. Gruss in 1935 and has been studied by many researchers thereafter. A number of its extensions and generalization have appeared in the literature. In this chapter, we also establish some Gruss type inequalities using the extended Hadamard fractional integral operator.

4.1 GRUSS TYPE INEQUALITIES AND FI INEQUALITIES RELATING AM-GM AND HOLDER’S INEQUALITIES In this section, we discuss fractional integral inequalities involving the inequalities based on AM-GM and Holder’s inequalities. We also discuss the fractional integral inequalities relating the functions with some sense of boundedness. 4.1.1 Theorem: Let f and g be two integrable and positive functions on [0, ) . If there are two real numbers m and M such that for all x  0, fx() (153) 0 mM     , gx() then for each fixed tuple (,,,,)k p p a of positive real numbers, ta , p 1, we have 1 1 1 11 kp pm pp k pp k   (154) HI(,)(,)(,) a t f g()()()() t  H I a t f t  H I a t g t  , M where (,)pp is a pair of Holder’s conjugate numbers. Proof: It follows from (153) that 1 1 1  (g ( x ))p M p  ( f ( x )) p  implying, by use of (27), that 1 1 1  (155) (())(())f xp g x p M p f (). x

 1 t 1 Multiplying both sides by (log ) k and then integrating with respect xkk () x to x over (,)at , the inequality (155) leads to the relation 1 1 1  kkp p p HI(,)(,) a t f g()() t M H I a t f t implying that 1 1 1 1 p  1 (156) kkI fp g p()(). t M pp I f t p H(,)(,) a t H a t   It, also, follows from (153) that 1 1 1 (f ( x ))p m p ( g ( x )) p implying, by use of (27), that 1 1 1 (157) (())(())f xp g x p  m p g (). x  1 t 1 Multiplying both sides by (log ) k and then integrating with respect xkk () x to x over (,)at , the inequality (157) leads to the relation 1 1 1 kkp p p HI(,)(,) a t f g()() t m H I a t g t implying that 1 1 1p 1 1 (158) kkI fp g p()(). t m pp I g t p H(,)(,) a t H a t   Multiplication of (156) and (158) leads to the conclusion (154) by use of (27).

4.1.2 Corollary: Let f and g be two integrable and positive functions on [0, ) . If there are two real numbers m and M such that for all x  0, fxp () (159) 0 mM    , gxp () for some fixed pair (,)pp of Holder’s conjugate numbers, p 1, then for each fixed tuple (,,)ka of positive real numbers, ta , we have 1 11 km pp k  ppp k  p  (160) HI(,)(,)(,) a t fg()()()() t  H I a t f t  H I a t g t  . M Proof: The result follows directly from the inequality (154), by replacing (fx ( )) p for fx() and (gx ( )) p for gx().

4.1.3 Theorem: Let f and g be two integrable and positive functions on [0, ) . If there are two real numbers m and M such that for all x  0, fx() (161) 0 mM     , gx() then for each fixed tuple (,,,)k p a of positive real numbers, ta , we have 1 1 1 k pp k  p pMm( 2) 1 k  p p (162)  HI(,)(,)(,) a t f()()()() t  H I a t g t  H I a t f  g t  . (mM 1)( 1) Proof: Since f()() x Mg x , for all , we have (163) Mp(() f x g ()) x p  ( M  1) p f p () x .  1 t 1 Multiplying both sides by (log ) k and then integrating with respect xkk () x to x over (,)at , the inequality (163) leads to the relation 11 M k ppp k p (164)  HI(,)(,) a t()()() f g t  H I a t f t  . M 1 Also, as f()() x mg x , for all , we have 11 (165) (()f x g ()) xp  (1  ) p g p () x . mmp Multiplying both sides by and then integrating with respect to over , the inequality (165) leads to the relation 11 1 k ppp k p (166)  HI(,)(,) a t()()() f g t  H I a t g t  . m 1 Adding (164) and (166), we find the inequality (162). 4.1.4 Theorem: Let and be two integrable and positive functions on . If there are two real numbers and such that for all , (167) , then for each fixed tuple (,,)ka of positive real numbers, and fixed p 1, we have 22 k ppp k p (168)  HI(,)(,) a t f()() t   H I a t g t  11 (mM 1)( 1) k ppp k p 2 HI(,)(,) a t f ( t )  H I a t g ( t ) . M Proof: Since , for all , we have

(169) Mp(() f x g ()) x p  ( M  1) p f p () x .  1 t 1 Multiplying both sides by (log ) k and then integrating with respect xkk () x to x over (,)at , the inequality (169) leads to the relation 11 M k ppp k p (170)  HI(,)(,) a t()()() f g t  H I a t f t  . M 1 Also, as f()() x mg x , for all x  0, we have 11 (171) (()f x g ()) xp  (1  ) p g p () x . mmp Multiplying both sides by and then integrating with respect to over , the inequality (171) leads to the relation 11 1 k ppp k p (172)  HI(,)(,) a t()()() f g t  H I a t g t  . m 1 Multiplying (170) and (172), and using Minkowski inequality, we find that 1 1 2 (mM 1)( 1) k pp k  p p k  p p  HI(,)(,)(,) a t f()()()() t  H I a t g t  H I a t f g t  M 2 11 k ppp k p  HI(,)(,) a t f()() t  H I a t f t   2 2 1 1 k pp k  p p k  p p k  p p  HatIft(,)(,)(,)(,)( )  Hat Ift ( )  2 Hat Ift ( )  Hat Igt ( ) which leads to the inequality (168). 4.1.5 Theorem:

If {fj : j 1,2,3,..., n } is a collection of n positive and integrable functions defined on [0, ) , then for each fixed tuple (,k , p12 , p ,..., pn ,) a of n positive real numbers, ta ,  p j 1, and each fixed  [ 1,1], we have j1 n n 2 kkp j (173) exp(1 ) pj H I(,)(,) a t f j ( t ) H I a t f j ( t ) . j1 j1 Proof:

Taking cfjj , jn1,2,3,..., , the result (173) follows from (33) due to linearity k  of the operator HI(,) a t .

4.1.6 Corollary:

If {fj : j 1,2,3,..., n } is a collection of n positive and integrable functions defined on [0, ) , then for each fixed tuple (,k , p12 , p ,..., pn ,) a of n positive real numbers, ta ,  p j 1, we have j1 n n kkp j (174)  pj H I(,)(,) a t f j()() t H I a t f j t . j1 j1 Proof: Taking  1, the inequality (174) follows from (173).

4.1.7 Theorem: Let f be an integrable function on [0, ) such that (175) ()()()t f t t on [0, ), for some integrable functions  and  , then for each fixed tuple (,,,)ka of positive real numbers, ta , one has k k  k  k  (176) HatI(,)(,)(,)(,) f()()()() t Hat I t Hat I t Hat I f t k k  k  k  HatI(,)(,)(,)(,) f()()()() t Hat I f t Hat I t Hat I t . Proof: It follows from (175) that for all x,(,) y a t , we have (177) fx()()()()()()()() y  xfy  fxfy   x  y .  1 t 1 Multiplying both sides by (log ) k and then integrating with respect xkk () x to x over (,)at , the inequality (177), by use of (47), becomes k k  k  k  (178) HatIft(,)(,)(,)(,)()() y Hat I  ()() tfy  Hat Iftfy ()()  Hat I  ()(). t  y Integrating with respect to y over (,)at after multiplying both sides by  1 t 1 (log ) k , by use of (47), the inequality (178) leads to the relation ykk () y (176).

4.1.8 Example:

Let f be an integrable function on [1, ) and 0 be a positive real number such that

(179) logt f ( t ) 0  log t on [1, ), then for each fixed pair (,)k  of positive real numbers, t 1, we have  1 k  0 kk2k (2k ) (180) HtI(1, ) f( t ) (log t ) (log t ) kk(  )  (   2k )

 2 1 k  k (2k ) k  HtI(1, ) f( t ) (log t ) ( t ) , k ( 2k ) where    (2k ) 1 (t )0 (log t )kkk (log t ) . kk(  )  (   2k ) Solution: The relation (180) is found from (176) by replacing  by  and using

( (x ),  ( x )) (log x , 0 log x ) with collaboration of (48) and (69). 4.1.9 Corollary: Let f be an integrable function on [0, ) satisfying (181) m f() t M on [0, ), for some m, M R, then for each fixed tuple (,,,)ka of positive real numbers, ta , we have  M tkkkk m t (182) (log )HI(,)(,) a t f ( t ) (log ) H I a t f ( t ) kk()() aa    mM t k kk (log )HI(,)(,) a t f ( t ) H I a t f ( t ). kk()()   a Proof: For (,)(,)  mM , the result (182) follows from (176) by use of (48). 4.1.10 Theorem: Let f be an integrable function on [0, ) such that (183) f()() t g t c on [0, ), for some integrable function g and some scalar cR , then for each fixed tuple (,,,)ka of positive real numbers, ta , we have  k k ctk k  k  k  (184) HatIftIgt(,)(,)(,)(,)(,)( ) Hat ( ) (log ) Hat Ift ( ) Hat IgtIft ( ) Hat ( ) k () a 2   c tkkk  c t (log )HI(,) a t g ( t ) (log ) k()()() aa   k   k  k k  k  k  HatI(,)(,)(,)(,) ftI()()()() Hat ft Hat IgtIgt Hat  c tkkkk c t (log )HI(,)(,) a t g ( t ) (log ) H I a t f ( t ). kk()() aa   Proof: It follows from (183) that for all x,(,) y a t , we have 2 (185) fxgy()()()()()() cfx  fygx  cgy  c fxfy()()()()()()  gxgy  cgx  cfy .

 1 t 1 Multiplying both sides by (log ) k and then integrating with respect xkk () x to x over (,)at , the inequality (185), by use of (47), becomes k k  k  k 2 k  (186) HatI(,)(,)(,)(,)(,) ftgycI()() Hat ft ()  Hat IgtfycgyI ()()  () Hat (1)  cI Hat (1) kI ftfy()()  k I  gtgy ()()  cI k  gt ()  cfy () k I  (1). Hat(,)(,)(,)(,) Hat Hat Hat Integrating with respect to y over (,)at after multiplying both sides by  1 t 1 (log ) k , by use of (47) and (48), the inequality (186) leads to the ykk () y relation (184). 4.1.11 Corollary: Let f be an integrable function on [0, ) satisfying (187) f()() t g t c on [0, ), for some integrable function g and some scalar cR , then for each fixed tuple (,,)ka of positive real numbers, ta , one has ct2 2 (188) 2kkI f ( t ) I g ( t ) (log ) k H(,)(,) a t H a t 22 k () a kk22  HI(,)(,) a t f()(). t  H I a t g t  Proof: Replacing  by  , the desired result (188) follows from (184). 4.1.12 Theorem: Let f be an integrable function on [0, ) such that (189) ()()()t f t t on [0, ), for some integrable functions  and  , then for each fixed tuple (,,,,,)k p p a of positive real numbers, ta , p 1, we have tt (log )kk (log ) aak p k p (190) HI(,)(,) a t()()()() f t  H I a t  f t ppkk()()    k k  k  k  HatI(,)(,)(,)(,) f()()()() t Hat I f t Hat I t Hat I t kI f()()()() t k I  t k I  t k I  f t , Hat(,)(,)(,)(,) Hat Hat Hat where (,)pp is a pair of Holder’s conjugate numbers. Proof: For any x,(,) y a t , if we set a f()() x x and b ()() y f y , (45) gives 11 (191) (()fx ()) xpp  (()  yfy  ())  (() fx   ())(() xyfy   ()). pp

 1 t 1 Multiplying both sides by (log ) k and then integrating with respect xkk () x to x over (,)at , the inequality (191), by use of (47), becomes 11 (192) kI( f )() p t  (() y  f ()) y p k I (1) ppH(,)(,) a t H a t k  HI(,) a t ( f  )()(() t y  f ()). y Integrating with respect to y over (,)at after multiplying both sides by  1 t 1 (log ) k , by use of (47), the inequality (192) leads to the relation ykk () y 11 (193) kI( f )() p t k I  (1)  k I  (  f )() p t k I  (1) ppHat(,)(,)(,)(,) Hat Hat Hat kk HI(,)(,) a t( f  )( t ) H I a t (  f )( t ) which leads to (190) by use of (48).

4.1.13 Example:

Let f be an integrable function on [1, ) and 0 be a positive real number such that

(194) logt f ( t ) 0  log t on [1, ), then for each fixed pair (,)k  of positive real numbers, t 1, one has t  (log ) k k  2 a (195)  HtI(1, ) f()() t   t k ()    (2kk )11 (2 ) kk(logt )k0 (log t ) k (log t ) k k(  2kk )   k (  )  k (   2 )  1 k  0 kkk (2k ) 2HtI(1, ) f ( t ) (log t ) (log t ) kk(  )  (   2k )  2 k k   (logt )Ht I(1, ) ( f ( x )log x )( t ) , k () where 2  k  2 0 k  (t )Ht I(1, ) f ( t ) (log t ) 2k ( )   (2kk )12 (3 ) 0 kk(logtt )kk (log ) . kk(  2kk )  (  3 ) Solution:

If we take ( (t ),  ( t )) (log t , 0 log t ), it follows by use of (48), (69) and (70) that

 1 k  k (2k ) k (196) HtI(1, )( t ) (log t ) , k ( 2k )  2 k  2 k (3k ) k (197) HtI(1, ) ( t ) (log t ) , k ( 3k )  1 k  0 kkk (2k ) (198) HtI(1, ) ( t ) (log t ) (log t ) , kk(  )  (   2k ) k  2 (199) HtIt(1, ) () 2    21 0 kkk(3kk ) k (2 ) k (logt )  (log t )  20 (log t ) . k(  )  k (   3kk )  k (   2 ) The relation (195) is found from (190) using  ( (t ),  ( t ),  , p , p ) (log t , 0 log t ,  ,2,2) and (196)-(199). 4.1.14 Corollary: Let f be an integrable function on [0, ) such that (200) m f() t M on [0, ), for some m, M R, then for each fixed tuple (,,,)ka of positive real numbers, ta , we have tt (log )kk (log ) aakk22 (201) HI(,)(,) a t f()() t H I a t f t kk()()    t  (log ) k 2kkI f ( t ) I f ( t )  ( M  m )2 a H(,)(,) a t H a t kk()()   tt (log )kk (log ) 2(M  m )aakk I f ( t )  I f ( t ) . H(,)(,) a t H a t kk()()      Proof: Using (, ,,p p ) (, m M ,2,2) and then rearranging the terms of the resulting inequality, we find (201) from (190) by use of (48). 4.1.15 Theorem: Let f be an integrable function on [0, ) such that (202) ()()()t f t t on [0, ), for some integrable functions  and  , then for each fixed  [ 1,1] and fixed tuple (,,,,,)k p12 p a of positive real numbers, ta , pp121, we have

 ttkk pp12(log ) (log ) 2 aakk (203) exp(1 )HI(,)(,) a t f ( t ) H I a t ( t ) kk()()     

kkpp12 HI(,)(,) a t()()()() f  t H I a t  f t  ttkk pp12(log ) (log ) 2 aakk exp(1  )HI(,)(,) a t ( t )  H I a t f ( t ) . kk()()      Proof:

For any x,(,) y a t , if we use c1  f()() x x and c2  ()() y f y , (33) leads to 2 (204) exp(1 ) p12 f ( x ) p ( y )

pp122 (()fxx  ())(()  yfy  ())  exp(1   ) pxpfy12  ()  ().  1 t 1 Multiplying both sides by (log ) k and then integrating with respect xkk () x to x over (,)at , the inequality (204), by use of (47), becomes 2 kk (205) exp(1 ) p1H I ( a , t ) f ( t ) p 2 ( y ) H I ( a , t ) (1)

k  pp12 HI(,) a t ( f  ) ( t ) ( ( y )  f ( y )) 2 kk exp(1  ) p1H I ( a , t ) ( t )  p 2 f ( y ) H I ( a , t ) (1) . Integrating with respect to y over (,)at after multiplying both sides by  1 t 1 (log ) k , by use of (47), the inequality (205) leads to the relation ykk () y 2 k k  k  k  exp(1 ) p1(,)Hat I f ( t ) Hat I (,) (1) p 2(,) Hat I ( t ) Hat I (,) (1) kkI()()()() f pp12 t I  f t H(,)(,) a t H a t exp(1 2 )pk I ( t ) k I  (1)  p k I  f ( t ) k I  (1)  1(,)Hat Hat (,) 2(,) Hat Hat (,)  which leads to (203) by use of (48).

4.1.16 Example:

Let f be an integrable function on [1, ) and 0 be a positive real number such that

(206) logt f ( t ) 0  log t on [1, ), then for each fixed  [ 1,1] and fixed pair (,)k  of positive real numbers, t 1, we have 2 2 exp(1  ) k (207) 0 22 (logt ) k ()

kk 2HI(1, t ) f ( x )  log x ( t ) H I (1, t ) 0  log x  f ( x )( t ). Solution: The relation (207) is obtained from (203), using 11 ( (x ),  ( x ),  , p , p ) (log x ,  log x ,  , , ) 1 2 0 22 with collaboration of (48) and (69).

4.1.17 Corollary: Let f be an integrable function on [0, ) such that (208) m f() t M on [0, ), for some m, M R, then for each fixed  [ 1,1] and fixed tuple (,,,)ka of positive real numbers, ta , we have tt   (log )kk (log ) 2 aak  (209) exp(1 )HI(,) a t f ( t ) M k()()()    k   k    kk 2HI(,)(,) a t f  m ( t ) H I a t M  f ( t ) tt   m(log )kk (log ) 2 aak  exp(1  )  HI(,) a t f ( t ) . k()()()   k    k    Proof: 1 Taking   m,   M and pp, the result (209) follows from (203) by 122 use of (48). 4.1.18 Theorem:

For each j  0,1; let f j :[0, )  [0,  ) be an integrable function, then for each fixed tuple (,,,,,)k p p a of positive real numbers, ta , p 1, we have tt pp(log )kk (log ) aak p k p (210) HI( a , t ) f 1()() t H I ( a , t ) f 2 t kk()()    kk  pp H I( a , t ) f 1()() t H I ( a , t ) f 2 t , where (,)pp is a pair of Holder’s conjugate numbers. Proof:

For any x,(,) y a t , if we take (a , b ) ( f12 ( x ), f ( y )), it follows from (45) that 11 (211) fpp( x ) f ( y ) f ( x ) f ( y ). pp1 2 1 2

 1 t 1 Multiplying both sides by (log ) k and then integrating with respect xkk () x to x over (,)at , the inequality (211), by use of (47), becomes k p p k  k  (212) pIftH(,)1 a t() pfyI 2 () H (,) a t (1) ppIftfy H (,)1 a t ()(). 2 Integrating with respect to y over (,)at after multiplying both sides by  1 t 1 (log ) k , by use of (47), the inequality (212) leads to the relation ykk () y k p k  k  p k  k  k  pIftIHat(,)1() Hat (,) (1) pIftI Hat (,)2 () Hat (,) (1) ppIftIft Hat (,)1 () Hat (,)2 () which leads to (210) by use of (48). 4.1.19 Corollary:

For each j  0,1; let f j :[0, )  [0,  ) be an integrable function, then for each fixed tuple (,,,,)k p p a of positive real numbers, ta , p 1, we have  11 t  (213) kIft p() k Ift  p ()   ()(log)k k IftIft  () k  () , pHat(,)1 p Hat (,)2 k a Hat (,)1 Hat (,)2 where (,)pp is a pair of Holder’s conjugate numbers. Proof: The inequality (213) follows directly from (210) by using  for  . 4.1.20 Theorem: For each ; let be an integrable function, then for each fixed tuple (,,,,,)k p p a of positive real numbers, , , we have 11 (214) kIftIft p()()()() k  p k IftIft  p k  p ppHat(,)1 Hat (,)2 Hat (,)1 Hat (,)2 kk  HI( a , t ) f 1 f 2()() t H I ( a , t ) f 1 f 2 t , where is a pair of Holder’s conjugate numbers. Proof:

For any x,(,) y a t , if we take (,)a b (() f1 x f 2 (), y f 1 () y f 2 ()) x , it follows from (45) that 11 (215) fxfyp() p () f p () yf p () x fxfxfyfy ()()()(). pp1 2 1 2 1 2 1 2

Multiplying both sides by and then integrating with respect

to over , the inequality (215), by use of (47), becomes 11 (216) kIftfy p()() p fyIft p () k  p () k Ifftfyfy  ()()(). ppH(,)1 a t 2 1 H (,)2 a t H (,)12 a t 1 2

Integrating with respect to y over (,)at after multiplying both sides by  1 t 1 (log ) k , the inequality (216), by use of (47), leads to the relation ykk () y (214). 4.1.21 Corollary:

For each j  0,1; let f j :[0, )  [0,  ) be an integrable function, then for each fixed tuple (,,,,)k p p a of positive real numbers, ta , p 1, we have

11k p k  p k  p k  p k  2 (217) HatIftIft(,)1()()()()() Hat (,)2 Hat IftIft (,)1 Hat (,)2 Hat Ifft (,)12  , pp where (,)pp is a pair of Holder’s conjugate numbers. Proof: Replacing  by  , the inequality (217) follows directly from (214). 4.1.22 Theorem:

For each ; let f j :[0, )  (0,  ) be an integrable function, then for each fixed tuple (,,,,,)k p p a of positive real numbers, , , we have 11 (218) kIftIft p()()()() k  p k IftIft  p k  p ppHat(,)1 Hat (,)2 Hat (,)1 Hat (,)2 k p11 k p  HI( a , t ) f 1 f 2()() t H I ( a , t ) f 1 f 2 t , where is a pair of Holder’s conjugate numbers. Proof: f()() x f y For any x,(,) y a t , if we take (,)(,)ab  11, it follows from (45) that f22()() x f y 11 (219) fxfyp() p () fyfx p  ()() p fxf () p11 ()() xfyf p  (). y pp1 2 1 2 1 2 1 2  1 t 1 Multiplying both sides by (log ) k and then integrating with respect xkk () x to x over (,)at , the inequality (219), by use of (47), becomes 11 (220) kI f p()()()() t f p y f p y k I f p t ppH( a , t ) 1 2 1 H ( a , t ) 2 k p11 p  HI( a , t ) f 1 f 2( t ) f 1 ( y ) f 2 ( y ). Integrating with respect to over after multiplying both sides by

, the inequality (220), by use of (47), leads to the relation

(218).

4.1.23 Corollary:

For each j  0,1; let f j :[0, )  (0,  ) be an integrable function, then for each fixed tuple (,,,,)k p p a of positive real numbers, ta , p 1, we have 11 (221) kIftIft p()()()() k  p k IftIft  p k  p ppHat(,)1 Hat (,)2 Hat (,)1 Hat (,)2 k p11 k p  HI( a , t ) f 1 f 2()() t H I ( a , t ) f 1 f 2 t , where (,)pp is a pair of Holder’s conjugate numbers. Proof: If we replace  by  , the inequality (221) follows directly from (218). 4.1.24 Theorem:

For each ; let f j :[0, )  (0,  ) be an integrable function, then for each fixed tuple (,,,,,)k p p a of positive real numbers, , , we have 11 (222) kIftIft p()()()() k  p k IftIft  p k  p ppHat(,)1 Hat (,)2 Hat (,)1 Hat (,)2 k k p11 p  HI( a , t ) f 1 f 2()() t H I ( a , t ) f 1 f 2 t , where is a pair of Holder’s conjugate numbers. Proof: f()() x f x For any x,(,) y a t , if we take (,)(,)ab  12, it follows from (45) that f12()() y f y 11 (223) fxfyp() p () fyfx p () p  () fxfxf ()() p11 () yf p  (). y pp1 2 1 2 1 2 1 2  1 t 1 Multiplying both sides by (log ) k and then integrating with respect xkk () x to x over (,)at , the inequality (223), by use of (47), becomes 11 (224) kI f p()()()() t f p y f p y k I f p t ppH( a , t ) 1 2 1 H ( a , t ) 2 k p11 p  HI( a , t ) f 1 f 2( t ) f 1 ( y ) f 2 ( y ). Integrating with respect to y over (,)at after multiplying both sides by  1 t 1 (log ) k , the inequality (224), by use of (47), leads to the relation ykk () y (222).

4.1.25 Corollary:

For each j  0,1; let f j :[0, )  (0,  ) be an integrable function, then for each fixed tuple (,,,,)k p p a of positive real numbers, ta , p 1, we have k p k  p k  k  p11 p (225) HatIftIft(,)1()()()() Hat (,)2 Hat IfftIff (,)12 Hat (,)1 2 t , where (,)pp is a pair of Holder’s conjugate numbers. Proof: If we replace  by  , the inequality (225) follows from (222) with use of (27). 4.1.26 Theorem:

For each ; let f j :[0, )  [0,  ) be an integrable function, then for each fixed tuple (,,,,,)k p p a of positive real numbers, , and fixed real number c , we have 11 (226) kIftIft p()()()() k  c k IftIft  c k  p ppHat(,)1 Hat (,)2 Hat (,)1 Hat (,)2 cc kkpp  HI( a , t ) f 1 f 2()() t H I ( a , t ) f 2 f 1 t , where is a pair of Holder’s conjugate numbers. Proof: cc pp For any x,(,) y a t , if we take (,)a b (() f1 x f 2 (), y f 1 ()()) y f 2 x , it follows from (45) that 11 cc (227) fxfyp()() c fyf c () p () x fxfxf ()()pp () yf (). y pp1 2 1 2 1 2 2 1  1 t 1 Multiplying both sides by (log ) k and then integrating with respect xkk () x to x over (,)at , the inequality (227), by use of (47), becomes 11 cc (228) kIftfy p()() c fyIft c () k  p () k Ifftfyfy  ()()().pp ppH(,)1 a t 2 1 H (,)2 a t H (,)12 a t 2 1 Integrating with respect to y over (,)at after multiplying both sides by  1 t 1 (log ) k , the inequality (228), by use of (47), leads to the relation ykk () y (226). 4.1.27 Theorem:

For each ; let f j :[0, )  (0,  ) be an integrable function, then for each fixed tuple (,,,,,)k p p a of positive real numbers, , and fixed real number , we have

11 (229) kIftIft c()()()() k  p  k IftIft  c k  p ppHat(,)1 Hat (,)2 Hat (,)2 Hat (,)1 cc kpp k p11 p  HI( a , t ) f 1 f 2()() t H I ( a , t ) f 1 f 2 t , where (,)pp is a pair of Holder’s conjugate numbers. Proof: cc fpp()() x f x For any x,(,) y a t , if we take (,)(,)ab  12, it follows from (45) f12()() y f y that 11 cc (230) fxfyc() p () fxfyfxfxf c ()() ppp () () p11 () yf p (). y pp1 2 2 1 1 2 1 2  1 t 1 Multiplying both sides by (log ) k and then integrating with respect xkk () x to x over (,)at , the inequality (230), by use of (47), becomes 11 (231) kI f c()()()() t f p y k I f c t f p y ppH( a , t ) 1 2 H ( a , t ) 2 1 cc k pp p11 p  HI( a , t ) f 1 f 2( t ) f 1 ( y ) f 2 ( y ). Integrating with respect to y over (,)at after multiplying both sides by  1 t 1 (log ) k , the inequality (231), by use of (47), leads to the relation ykk () y (229). 4.1.28 Theorem:

For each j  0,1; let f j :[0, )  (0,  ) be an integrable function, then for each fixed tuple (,,,,,)k p p a of positive real numbers, ta , p 1 and fixed real number c , we have tt (log )kk (log ) aak c p k c p (232) HI( a , t ) f 1 f 2()() t H I ( a , t ) f 1 f 2 t ppkk()()    cc kkpp  HI( a , t ) f 1 f 2()() t H I ( a , t ) f 1 f 2 t , where is a pair of Holder’s conjugate numbers. Proof: cc pp f11()() x f y For any , if we take (,)(,)ab  pp , by use of (27), it pp f22()() y f x follows from (45) that

11 cc (233) fxfxc() p () fyf c () p () y fxfpp () ()()(). yfxfy pp1 2 1 2 1 1 2 2  1 t 1 Multiplying both sides by (log ) k and then integrating with respect xkk () x to x over (,)at , the inequality (233), by use of (47), becomes 11 cc (234) kIfft c p() fyfyI c ()() p k  (1) k Ifftfyfy  pp () ()(). ppH(,)12 a t 1 2 H (,) a t H (,)12 a t 1 2 Integrating with respect to y over (,)at after multiplying both sides by  1 t 1 (log ) k , the inequality (234), by use of (47) and (48), leads to the ykk () y relation (232). 4.1.29 Corollary:

For each j  0,1; let f j :[0, )  (0,  ) be an integrable function, then for each fixed tuple (,,,,)k p p a of positive real numbers, ta , p 1 and fixed real number c , we have 11 (235) kI f c f p()() t k I f c f p t ppH( a , t ) 1 2 H ( a , t ) 1 2  cc t  ( )(log )k kkI fpp f ( t ) I f f ( t ) , ka H( a , t ) 1 2 H ( a , t ) 1 2 where (,)pp is a pair of Holder’s conjugate numbers. Proof: If we replace  by  , the inequality (235) follows from (232).

4.2 FI INEQUALITIES BASED ON REARRANGEMENT INEQUALITIES In this section, we present a number of rearrangement inequalities and then based on these inequalities, we establish the related fractional integral inequalities for the extended integral operator. 4.2.1 Lemma:

n n * n Let ()a jj1 and ()bjj1 be sequences of positive real numbers and ()bjj1 n be any permutation of terms of the sequence ()bjj1 . If (i) the sequences are either increasing or both decreasing, we have n n n * (236) aj b j  a j b j  a j b n j 1 , j1 j  1 j  1 (ii) one of the sequences is increasing and the other is decreasing, we have n n n * (237) aj b j  a j b j  a j b n j 1 . j1 j  1 j  1

Proof: Let n S  ajj b , j1

S a1 b 1  a 2 b 2 ...  ak 1 bk 1  akbl

ak1 b k  1 ...  a l  1b l  1 alkb  a l  1 b l  1  ...  a n b n . Then

S S ( ak  a l )( b k  b l ) .

(i) If the sequences are either increasing or both decreasing, aakl and

bbkl have same signs. This implies that SS . Repeated use of the same argument leads to the relation (236). (ii) And if one of the sequences is increasing and the other is decreasing,

aakl and bbkl have opposite signs. So, SS  and hence, repeated use of the same argument, leads to the relation (237). 4.2.2 Note:

n n If ()bjj1 is decreasing (increasing), ()bn j 11 j  is increasing (decreasing). Hence, (237) follows directly from (236). In fact, both represent the same under their conditions. 4.2.3 Example:

For all a12, a R, we have 22 (238) a1 a 22 a 1 a 2 . Solution: 22*2 Taking ()(,)()aj j1 a 1 a 2 b j j 1 and ()(,)bjj1 a 2 a 1 , the result (238) follows,  directly from (236), for all a12, a R . Since the greater side of the inequality

(238) is independent of the signs of asj ' , the inequality holds for all a12, a R. 4.2.4 Note:

2 Note that (238) is a simple fact otherwise too, as (aa12 ) 0. It also follows directly from AM-GM inequality (38). 4.2.5 Theorem:

Let f and g be real valued and integrable functions on [0, ) . Then for each fixed tuple (,,,)ka of positive real numbers, ta , one has tt (log )kk (log ) aak22 k  k  k  (239) HatIft(,)(,)(,)(,)( ) Hat Igt ( ) 2 Hat IftIgt ( ) Hat ( ). kk()()    Proof:

For any x,(,) y a t , if we take (a12 , a ) ( f ( x ), g ( y )), (238) becomes

(240) f22( x ) g ( y ) 2 f ( x ) g ( y ).  1 t 1 Multiplying both sides by (log ) k and then integrating with respect xkk () x to x over (,)at , the inequality (240), by use of (47), becomes k22 k  k  (241) HIftgyI(,)(,)(,) a t() () H a t (1)2 H Iftgy a t ()(). Integrating with respect to y over (,)at after multiplying both sides by  1 t 1 (log ) k , by use of (47), the inequality (241) leads to the relation ykk () y k22 k  k  k  k  k  HatIftI(,)(,)(,)(,)(,)(,)() Hat (1) Hat IgtI () Hat (1)2 Hat IftIgt () Hat () which leads to (239) by use of (48). 4.2.6 Theorem: Let f be an integrable function on [0, ) such that (242) ()()()t f t t on [0, ), for some integrable functions  and  . Then for each fixed tuple (,,,)ka of positive real numbers, ta , we have tt (log )kk (log ) aakk22 (243) HI(,)(,) a t()()()() f t  H I a t  f t 2kk (  ) 2  (  ) k k  k  k  HatI(,)(,)(,)(,) f()()()() t Hat I f t Hat I t Hat I t k k  k  k  HatI(,)(,)(,)(,) f( t ) Hat I ( t ) Hat I ( t ) Hat I f ( t ). Proof: Since the inequality (238) remains valid for positive real numbers too, for any

x,(,) y a t , in context of (242), if we use a1  f()() x x and

a2  ()() y f y , (238) gives (244) (()fx ()) x22  (()  yfy  ())  2(() fx   ())(() x  yfy  ()).  1 t 1 Multiplying both sides by (log ) k and then integrating with respect xkk () x to x over (,)at , the inequality (244), by use of (47), becomes kk22 (245) HI(,)(,) a t( f )()(() t  y  f ()) y H I a t (1) k  2HI(,) a t ( f  )()(() t y  f ()). y Integrating with respect to y over (,)at after multiplying both sides by  1 t 1 (log ) k , by use of (47), the inequality (245) leads to the relation ykk () y k22 k  k  k  HatI(,)(,)(,)(,)( f )() t Hat I (1)  Hat I (  f )() t Hat I (1) kk 2HI(,)(,) a t ( f  )( t ) H I a t (  f )( t )

61 which leads to (243) by use of (48). 4.2.7 Corollary: Let f be an integrable function on [0, ) such that (246) m f() t M on [0, ) , for some m, M R, then for each fixed tuple (,,)ka of positive real numbers, ta , we have t  (log ) k a kk22 (247)  HI(,)(,) a t()()()() f m t  H I a t M  f t  2k ( ) t 2 (log ) k k  2 a  HI(,) a t f() t mM 22 k () t  (log ) k a k  ()().m MH I(,) a t f t k () Proof: The result (247) follows from (243) by taking (,,)(,,)   mM  with use of (48).

Now, we establish, by use of (236), a general form of (238) in the following example. 4.2.8 Example:

For all a1,, a 2 a 3  R , one has 3 3 2 (248) aj  a j a k . j1 jk, 1, jk Proof:

If we assume that a1 a 2  a 3  0 , the desired result (248) follows, for all  *3 a1,, a 2 a 3  R , from (236) by taking ()(,,)bjj1 b 2 b 3 b 1 and bajj , j 1,2,3.

Since the greater side of the inequality (248) is independent of the signs of asj ' , the inequality (248) holds for all a1,, a 2 a 3  R . 4.2.9 Theorem:

Let fRj :[0, )  be integrable function for each j 1,2,3. Then for each fixed tuple (,,,,)ka   of positive real numbers, ta , we have tt    (log )kk (log ) aakk22 (249) HI( a , t ) f 1()() t H I ( a , t ) f 2 t k()()()()   k    k   k 

t  (log ) k a k  2  HI( a , t ) f 3 () t kk()()   tt (log )kk (log ) aak k  k  k  HatIftIft(,)1()()()() Hat (,)2 Hat IftIft (,)2 Hat (,)3 kk()()    t  (log ) k a kk  HI( a , t ) f 3()(). t H I ( a , t ) f 1 t k () Proof: For any x,,(,) y z a t , it follows from (248) that 2 2 2 (250) fx1() fy 2 ()  fz 3 ()  fxfy 1 ()() 2  fyfz 2 ()() 3  fzfx 3 ()(). 1  1 t 1 Multiplying both sides by (log ) k and then integrating with respect xkk () x to x over (,)at , the inequality (250), by use of (47), becomes k2 2 k  k  2 (251) HI(,)1 a t f( t ) f 2 ( y ) H I (,) a t (1) H I (,) a t (1) f 3 ( z ) k k  k  HIftfy(,)1 a t()() 2  fyfzI 2 ()() 3 H (,) a t (1)  fzIft 3 () H (,)1 a t (). Integrating with respect to y over (,)at after multiplying both sides by  1 t 1 (log ) k , by use of (47), the inequality (251) leads to the relation ykk () y k2 k  k  2 k  k  k  2 (252) HatIftI(,)1( ) Hat (,) (1) Hat IftI (,)2 ( ) Hat (,) (1) Hat I (,) (1) Hat I (,) (1) fz 3 ( ) k k  k  k  HatIftIft(,)1() Hat (,)2 () Hat IftfzI (,)2 ()() 3 Hat (,) (1) kk  HI( a , t )(1) f 3 ( z ) H I ( a , t ) f 1 ( t ) . Integrating with respect to z over (,)at after multiplying both sides by  1 t 1 (log ) k , by use of (47) and (48), the inequality (252) leads to the zkk () z relation (249). 4.2.10 Theorem: Let f and g be an integrable functions on [0, ) such that (253) ()()()()t f t  g t  t on [0, ), for some integrable functions  and  . Then for each fixed tuple (,,,,)ka   of positive real numbers, ta , we have tt    (log )kk (log ) aakk22 (254) HI(,)(,) a t()()()() f t  H I a t g  f t k()()()()   k    k   k 

t  (log ) k a k  2 HI(,) a t ()() g t kk()()   t  (log ) k a k k  k  k   HatIftIft(,)(,)(,)(,)()()()() Hat Hat I tIgt Hat  k () t  (log ) k a k k  k  k   HatIgtIgt(,)(,)(,)(,)()()()() Hat Hat IftI Hat  t  k () t  (log ) k a k k  k  k   HatI(,)(,)(,)(,)()()()() t Hat I t Hat I g t Hat I f t  k () t  (log ) k a k k  k  k   HatIftIgt(,)(,)(,)(,)()()()() Hat Hat I tIft Hat  k () t  (log ) k a k k  k  k   HatIgtI(,)(,)(,)(,)()()()() Hat t Hat IftIgt Hat  k () t  (log ) k a k k  k  k   HatI(,)(,)(,)(,)()()()(). t Hat I f t Hat I g t Hat I t  k () Proof: Considering (253), for any x,,(,) y z a t , if we take

(,,)(()aaa1 2 3  fx  (),() xgy  fy (),() zgz  ()) , the inequality (248) leads to the relation (255) (()f x ()) x2  (() g y  f ()) y 2  (() z  g ()) z 2 (()f x  ())(() x g y  f ()) y (()gyfy  ())(() zgz  ())  (()  zgzfx  ())(()   ()). x  1 t 1 Multiplying both sides by (log ) k and then integrating with respect xkk () x to x over (,)at , the inequality (255), by use of (47), becomes k2 2 k  2 k  (256) HIf(,)(,)(,) a t( )()(() tgyfyI   ()) H a t (1)(()  zgzI  ()) H a t (1) k  HI(,) a t ( f  )( t ) ( g ( y )  f ( y )) kk (()gyfy  ())(() zgzI  ())H(,)(,) a t (1)(()   zgzIf  ()) H a t (   )(). t  1 t 1 Multiplying both sides by (log ) k and then integrating with respect ykk () y to y over (,)at , the inequality (256), by use of (47), becomes

k22 k  k  k  (257) HatI(,)(,)(,)(,)( f )() t Hat I (1)  Hat I ( g  f )() t Hat I (1) 2 kk ( (z ) g ( z ))H I(,)(,) a t (1) H I a t (1) kk HI(,)(,) a t( f  )( t ) H I a t ( g  f )( t ) kk HI(,)(,) a t( g  f )( t ) ( ( z )  g ( z )) H I a t (1) kk ( (z )  g ( z ))H I(,)(,) a t ( f  )( t ) H I a t (1).  1 t 1 Multiplying both sides by (log ) k and then integrating with respect to zkk () z z over (,)at , the inequality (257), by use of (47), becomes k22 k  k  k  k  k  (258) HatIf(,)(,)(,)(,)(,)(,)( ) ( tI ) HatHat (1) I (1)  Hat IgftI (  ) ( ) HatHat (1) I (1) k2 k  k  HI(,)(,)(,) a t( g ) ( t ) H I a t (1) H I a t (1) k k  k  HI(,)(,)(,) a t( f  )( t ) H I a t ( g  f )( t ) H I a t (1) k k  k  HI(,)(,)(,) a t( g  f )( t ) H I a t (  g )( t ) H I a t (1) k k  k  HI(,)(,)(,) a t(  g )( t ) H I a t ( f  )( t ) H I a t (1) which leads to (254) by use of (48). 4.2.11 Lemma

For all a1, a 2 , a 3  [1, ) , we have 33 aajj1 (259) aajj . jj1 1, aa41 Proof:

Without loss of generality, we can assume that a1 a 2  a 3 1, then the desired 3 result (259) follows from (236) by taking ()(,,)ajj1 a 1 a 2 a 3 , 3 *3 (bjj )1 (log a 1 ,log a 2 ,log a 3 ) and (bjj )1 (log a 3 ,log a 1 ,log a 2 ) . 4.2.12 Note:

a1 a 2 a 1 a 2 For a1 a 2  a 3 1, one may notice that aa13 and

a2 a 3 a 2 a 3 aa23 which, after multiplication, imply the result (259). Thus, it is an alternative way to see the same result. Now, we see a general form of the inequality (236) in the following lemma. 4.2.13 Lemma:

n n n Let (a1,j ) j 1 ,( a 2, j ) j  1 ,...,( a m , j ) j  1 be increasing or all decreasing sequences ***n n n of positive real numbers and (a2,j ) j 1 ,( a 3, j ) j  1 ,...,( a m , j ) j  1 be any permutations of n n n terms of the sequences (a2,j ) j 1 ,( a 3, j ) j  1 ,...,( a m , j ) j  1 respectively. Then for each mZ  , we have

nnn *** (260) aaaa1,2,3,jjjmj... , aaaa 1,2,3, jjjmj ... , aa 1,2, jnj  13, a nj   1 ... a mnj ,   1 , jjj111 Proof: We know that if ab and cd , ac bd for all a,,, b c d R . From this fact,  n we induce that for each kZ , 1km, the sequence (a1,j a 2, j a 3, j ... a k , j ) j 1 is an n n increasing (decreasing) sequence if the individual sequences ()a1,jj 1 , ()a2,jj 1 , n ...,()ak,1 j j are all increasing (decreasing) sequences. Hence, the result (260) follows by induction on m . 4.2.14 Note:

n One may notice that the sequence ()aa1,j 2, j j 1 may not be a monotonic n n sequence if one of the sequences ()a1,jj 1 and ()a2,jj 1 is increasing and the other 3 one is decreasing. For example, (aa1,j 2, j ) j 1  (8,14,3) is not a monotonic 3 3 sequence while (a1,jj ) 1  (1,2,3) is increasing and (a2,jj ) 1  (8,7,1) is decreasing. 4.2.15 Example:

For all a1, a 2 , a 3  [0, ) , one has 3 3 3 (261) aajj 3 . j1 j1 Proof: The result (261) follows from (260) for m  3 by taking 3 3 3 ()(,,)()()a1,j j 1 a 1 a 2 a 3  a 2, j j  1  a 3, j j  1 , *3 ()(,,)a2,jj 1 a 2 a 3 a 1 , *3 ()(,,)a3,jj 1 a 3 a 1 a 2 . 4.2.16 Theorem:

For each j 1,2,3; let f j be positive and integrable function defined on [0, ) . Then for each fixed tuple (,,,,)ka   of positive real numbers, ta , one has tt    (log )kk (log ) aakk33 (262) HI( a , t ) f 1()() t H I ( a , t ) f 2 t k()()()()   k    k   k  t  (log ) k a k  3  HI( a , t ) f 3 () t kk()()   k k  k   3HI(,)1 a t f ( t ) H I (,)2 a t f ( t ) H I (,)3 a t f ( t ). Proof:

For any x,,(,) y z a t , if we take (,a1 a 2 , a 3 ) ((), f 1 x f 2 (), y f 3 ()) z , it follows from (261) that 3 3 3 (263) (())fx1 ( fy 2 ())  ( fz 3 ())  3() fxfyfz 1 2 () 3 ().  1 t 1 Multiplying both sides by (log ) k and then integrating with respect xkk () x to x over (,)at , the inequality (263), by use of (47), becomes k3 k  3 k  3 (264) HI(,)1 a t f( t ) H I (,) a t (1) ( f 2 ( y )) H I (,) a t (1) ( f 3 ( z )) k   3HI( a , t ) f 1 ( t ) f 2 ( y ) f 3 ( z ).  1 t 1 Multiplying both sides by (log ) k and then integrating with respect ykk () y to y over (,)at , the inequality (264), by use of (47), becomes k3 k  k  k  3 k  k  3 (265) HatIftI(,)1( ) Hat (,) (1) Hat I (,) (1) Hat IftI (,)2 ( ) Hat (,) (1) Hat I (,) (1) ( fz 3 ( )) kk  3HI( a , t ) f 1 ( t ) H I ( a , t ) f 2 ( t ) f 3 ( z ).  1 t 1 Multiplying both sides by (log ) k and then integrating with respect to zkk () z z over (,)at , the inequality (265), by use of (47), becomes k33 k  k  k  k  k  HatI(,)1 f( t ) Hat I (,) (1) Hat I (,) (1) Hat I (,) (1) Hat I (,) (1) Hat I (,)2 f ( t ) k k  k  3  HI( a , t )(1) H I ( a , t ) (1) H I ( a , t ) f 3 ( t ) k k  k   3HI(,)1 a t f ( t ) H I (,)2 a t f ( t ) H I (,)3 a t f ( t ) which leads to (262) by use of (48). 4.2.17 Theorem: Let f and g be an integrable functions on [0, ) such that (266) ()()()()t f t  g t  t on [0, ), for some integrable functions  and  . Then for each fixed tuple (,,,,)ka   of positive real numbers, ta , we have tt    (log )kk (log ) aakk33 (267) HI(,)(,) a t()()()() f t  H I a t g  f t 3k (  )  k (  ) 3   k (  )  k (  ) t  (log ) k a k  3 HI(,) a t ()() g t 3kk (  ) (  ) k k  k  k  k  k  HatIftIgtIgt(,)(,)(,)(,)(,)(,)()()()()()() Hat Hat Hat I tIftIgt Hat Hat k k  k  k  k  k  HatIftIftI(,)(,)(,)(,)(,)(,)()()()()()() Hat Hat t Hat I  tIgtI Hat Hat  t k k  k  k  k  k  HatIftIgtI(,)(,)(,)(,)(,)(,)()()()()()() Hat Hat t Hat I  tIftI Hat Hat  t k k  k  k  k  k  HatIftIftIgt(,)(,)(,)(,)(,)(,)() Hat () Hat () Hat I () tIgtIgt Hat () Hat ().

Proof: Considering (266), for any x,,(,) y z a t , if we use

(,,)(()aaa1 2 3  fx  (),() xgy  fy (),() zgz  ()) , the inequality (261) gives (268) (()f x ()) x3  (() g y  f ()) y 3  (() z  g ()) z 3 3(f ( x )  ( x ))( g ( y )  f ( y ))( ( z )  g ( z )).  1 t 1 Multiplying both sides by (log ) k and then integrating with respect xkk () x to x over (,)at , the inequality (268), by use of (47), becomes k3 3 k  3 k  (269) HIf(,)(,)(,) a t( )()(() tgyfyI   ()) H a t (1)(()  zgzI  ()) H a t (1) k  3HI(,) a t ( f  )()(() t g y  f ())(() y z  g ()). z  1 t 1 Multiplying both sides by (log ) k and then integrating with respect ykk () y to y over (,)at , the inequality (269), by use of (47), becomes k33 k  k  k  (270) HatI(,)(,)(,)(,)( f )() t Hat I (1)  Hat I ( g  f )() t Hat I (1) 3 kk ( (z ) g ( z ))H I(,)(,) a t (1) H I a t (1) kk 3HI(,)(,) a t ( f  )() t H I a t ( g  f )()(() t z  g ()). z  1 t 1 Multiplying both sides by (log ) k and then integrating with respect to zkk () z z over (,)at , the inequality (270), by use of (47), becomes k33 k  k  k  k  k  (271) HatIf(,)(,)(,)(,)(,)(,)( ) ( tI ) HatHat (1) I (1)  Hat IgftI (  ) ( ) HatHat (1) I (1) k3 k  k  HI(,)(,)(,) a t( g ) ( t ) H I a t (1) H I a t (1) k k  k  3HI(,)(,)(,) a t ( f  )() t H I a t ( g  f )() t H I a t (  g )() t which leads to (267) by use of (48). 4.2.18 Theorem:

For each j 1,2,3; let f j be positive and integrable function defined on [0, ) . Then for each fixed tuple (,,,,)ka   of positive real numbers, ta , one has t  (log ) k a k k 22 k  k  (272)  HatIftIft(,)1()()()() Hat (,)2 Hat IftIft (,)1 Hat (,)2  k () t  (log ) k a k k 22 k  k   HatIftIft(,)1()()()() Hat (,)3 Hat IftIft (,)1 Hat (,)3  k ()

t  (log ) k a k k 22 k  k   HatIftIft(,)2()()()() Hat (,)3 Hat IftIft (,)2 Hat (,)3  k () k k  k   6HI(,)1 a t f ( t ) H I (,)2 a t f ( t ) H I (,)3 a t f ( t ). Proof:

For any x,,(,) y z a t , if we take (,c1 c 2 , c 3 ) ((), f 1 x f 2 (), y f 3 ()) z , it follows from (40) that 2 2 2 2 (273) fxfy1()()()()()()()() 2 fxfy 1 2  fxfz 1 3  fxfz 1 3 22 f2()()()() y f 3 z f 2 y f 3 z

 6f1 ( x ) f 2 ( y ) f 3 ( z ).  1 t 1 Multiplying both sides by (log ) k and then integrating with respect xkk () x to x over (,)at , the inequality (273), by use of (47), becomes kkk2 2 2 (274) HIftfy(,)1 a t()()()()()() 2 H Iftfy (,)1 a t 2 H Iftfz (,)1 a t 3 k2 k  2 k  2 HIftfz(,)1 a t()() 3  H I (,) a t (1)() fyfz 2 3 ()  H I (,) a t (1) fyfz 2 ()() 3 k   6HI( a , t ) f 1 ( t ) f 2 ( y ) f 3 ( z ).  1 t 1 Multiplying both sides by (log ) k and then integrating with respect ykk () y to y over (,)at , the inequality (274), by use of (47), becomes k k 22 k  k  (275) HatIftIft(,)1()()()() Hat (,)2 Hat IftIft (,)1 Hat (,)2 k k 22 k  k  HatI(,)(1) Hat Iftfz (,)1 () 3 () Hat I (,) (1) Hat Iftfz (,)1 ()() 3 k k 22 k  k  HatI(,)(1) Hat Iftfz (,)2 () 3 () Hat I (,) (1) Hat Iftfz (,)2 ()() 3 kk  6HI( a , t ) f 1 ( t ) H I ( a , t ) f 2 ( t ) f 3 ( z ).  1 t 1 Multiplying both sides by (log ) k and then integrating with respect to zkk () z z over (,)at , the inequality (275), by use of (47) and (48), leads to (272). 4.2.19 Example:

 For all a1,, a 2 a 3  R , we have 3 3 32 (276) aj  a j a j1 . j1 j1, aa41 Solution: 2 2 2 3 3 *3 Using (,,)a1 a 2 a 3 for ()a jj1 , ()(,,)bjj1 a 1 a 2 a 3 and ()(,,)bjj1 a 2 a 3 a 1 , the result (276) follows directly from (236).

4.2.20 Theorem:

 Let fRj :[0, )  be integrable function for each j 1,2,3. Then for each fixed tuple (,,,,)ka   of positive real numbers, ta , we have tt    (log )kk (log ) aakk33 (277) HI( a , t ) f 1()() t H I ( a , t ) f 2 t k()()()()   k    k   k  t  (log ) k a k  3  HI( a , t ) f 3 () t kk()()   tt (log )kk (log ) aak22 k  k  k  HatIftIft(,)1()()()() Hat (,)2 Hat IftIft (,)2 Hat (,)3 kk()()    t  (log ) k a kk2  HI( a , t ) f 3()(). t H I ( a , t ) f 1 t k () Proof:

For any x,,(,) y z a t , if we take (,a1 a 2 , a 3 ) ((), f 1 x f 2 (), y f 3 ()) z , it follows from (276) that 3 3 3 2 2 2 (278) fx1() fy 2 ()  fz 3 ()  fxfy 1 ()() 2  fyfz 2 ()() 3  fzfx 3 ()(). 1  1 t 1 Multiplying both sides by (log ) k and then integrating with respect xkk () x to x over (,)at , the inequality (278), by use of (47), becomes k3 3 k  k  3 (279) HI(,)1 a t f( t ) f 2 ( y ) H I (,) a t (1) H I (,) a t (1) f 3 ( z ) k2 2 k  2 k  HIftfyfyfzI(,)1 a t()() 2  2 ()() 3 H (,) a t (1)  fzIft 3 () H (,)1 a t (). Integrating with respect to y over (,)at after multiplying both sides by  1 t 1 (log ) k , by use of (47), the inequality (279) leads to the relation ykk () y k3 k  k  3 k  k  k  3 (280) HatIftI(,)1( ) Hat (,) (1) Hat IftI (,)2 ( ) Hat (,) (1) Hat I (,) (1) Hat I (,) (1) fz 3 ( ) k22 k  k  k  HatIftIft(,)1() Hat (,)2 () Hat IftfzI (,)2 ()() 3 Hat (,) (1) kk2  HI( a , t )(1) f 3 ( z ) H I ( a , t ) f 1 ( t ) . Integrating with respect to z over (,)at after multiplying both sides by  1 t 1 (log ) k , by use of (47) and (48), the inequality (280) leads to the zkk () z relation (277). 4.2.21 Lemma: The inequality

m m11 m  m  j j (281) (m 1)( a1  a 2 )  ( a 1  a 2 ) a 1 a 2 j0 holds for all

(i) a12, a R, if m is an odd positive integer,  (ii) a12, a R , if m is an even positive integer. Proof: After simplification, we can rewrite (281) as m m1 m  1 m  j  1 j (282) m( a1 a 2 ) 2 a 1 a 2 . j1 (i) If m is an odd positive integer, (282) can be written as mm11 m11m mm11 22jj m1 m  1 m  1  j j 22 22 (283) m( a1 a 2 )  2 a 1a 2  2aa1 2  2 a 1 a 2 . j11j

Since the greater side of (283) is independent of signs of asj ' , in order to prove  the result (283) for all a12, a R, it is sufficient to prove it for all a12, a R

only. So, without loss of generality, we can assume that aa120 . mm11 mm11 2222 *2 22 Taking ()(,)()aj j1 a 1 a 2 b j j 1 and ()(,)bjj1 a 2 a 1 , we find by (236) that mm11 mm11 22 (284) a1 a 22 a 1 a 2 . Then, it follows from (283) that we have to show only that mm11 mm11 22jj m 1 m1 m  1 m  1  j j 22 (285) ().a1 a 2  a 1 a 2  a 1 a 2 2 jj11  2 2n 1  k 2 n  1  k Let mn21 , nZ . Then, taking ()(,)ajj1 a 1 a 2 , 2kk 1 1 * 2kk 1 1 ()(,)bjj1 a 1 a 2 and ()(,)bjj1 a 2 a 1 , it follows, for each kn0,1,2,..., 1, from (236) that 2n 2 2 n  2 2 n  1  k k  1 2 n  1  k k  1 a1 a 2  a 1 a 2  a 2 a 1 . That is, m1 m  1 m  k k  1 m  k k  1 a1 a 2  a 1 a 2  a 2 a 1 . m 1 After adding these inequalities, we find (285). This completes the proof 2 of (281) for this case. (ii) If m is an even positive integer, (282) can be expressed as m m 11mm m m mm 221 1 jj 1  m1 m  1 m  1  j j 2 2 2 2 22 (286) m( a1 a 2 )  2 a 1 a 2  2(a1a 2  a 1 a 2 )  2aa 1 2 . j11j Using mn 2 , nZ  , it becomes

nn11 2n 1 2 n  1 2 n  1  j jn11 n n n n  j n 1 j (287) n(). a1 a 2  a 1aa 2 a1 a2  a 1 a 2  1a 2 jj11  For any a12, a R , we can assume, without loss of generality, that aa120 . 2nn 1 1 2 nn *2 nn Taking ()(,)ajj1 a 1 a 2 , ()(,)bjj1 a 1 a 2 and ()(,)bjj1 a 2 a 1 , we find by (236) that 2n 1 2 n  1 n  1 n n  1 n (288) a1 a 2  a 1 a 2  a 2 a 1 . Then, it follows from (287) that we have to show only that nn11 2n 1 2 n  1 2 n  1  j j n  j n  1  j (289) (n 1)( a1  a 2 )  a 1 a 2  a 1 a 2 . jj11 2 2n l 2 n l 2ll 1 1 * 2ll 1 1 Taking ()(,)ajj1 a 1 a 2 , ()(,)bjj1 a 1 a 2 and ()(,)bjj1 a 2 a 1 , it follows, for each ln0,1,2,..., 2, from (236) that 2n 1 2 n  1 2 n  l l  1 2 n  l l  1 a1 a 2  a 1 a 2  a 2 a 1 . After adding these n 1 inequalities, we find (289). This completes the proof of (281) for this case too. 4.2.22 Theorem:

Let f1 and f2 be positive (real valued) and integrable functions defined on [0, ) . Then for each fixed tuple (,,,)ka of positive real numbers, ta and each fixed even (odd) positive integer m , one has tt (log )kk (log ) aak m11 k m (290) HI( a , t ) f 1()() t H I ( a , t ) f 2 t kk()()    m 2 k m j1 k j   HI( a , t ) f 1( t ) H I ( a , t ) f 2 ( t ). m j1 Proof:

For any x,(,) y a t if we take (a1 , a 2 ) ( f 1 ( x ), f 2 ( y )) , it follows from (282) that m m1 m  12 m  j  1 j (291) f1( x ) f 2 ( y ) f 1 ( x ) f 2 ( y ). m j1  1 t 1 Multiplying both sides by (log ) k and then integrating with respect xkk () x to x over (,)at , the inequality (291), by use of (47), becomes m k m1 k  m  12 k  m  j  1 j (292) HIftI(,)1 a t() H (,)2 a t (1) fy () H Iftfy (,)1 a t ()(). 2 m j1  1 t 1 Multiplying both sides by (log ) k and then integrating with respect ykk () y to y over (,)at , the inequality (292), by use of (47) and (48), leads to (290).

4.2.23 Example:

Let f1 and f2 be positive and integrable functions defined on [0, ) . Then for each fixed tuple (,,,)ka of positive real numbers, ta , one has tt (log )kk (log ) aakk33 (293) HI( a , t ) f 1()() t H I ( a , t ) f 2 t kk()()    k22 k  k  k  HatIftIft(,)1( ) Hat (,)2 ( ) Hat IftIft (,)1 ( ) Hat (,)2 ( ). Proof:

For any x,(,) y a t if we take (a1 , a 2 ) ( f 1 ( x ), f 2 ( y )) , it follows from (281), using m  2 , that 3 3 2 2 (294) fx1() fy 2 ()  fxfy 1 ()() 2  fxfy 1 () 2 ().  1 t 1 Multiplying both sides by (log ) k and then integrating with respect xkk () x to x over (,)at , the inequality (294), by use of (47), becomes k3 3 k  k  2 k  2 (295) HatIftfyI(,)1() 2 () Hat (,) (1)  Hat Iftfy (,)1 ()() 2  Hat Iftfy (,)1 ()(). 2  1 t 1 Multiplying both sides by (log ) k and then integrating with respect ykk () y to y over (,)at , the inequality (295), by use of (47) and (48), leads to (293). 4.2.24 Example: Let f be an integrable function on [0, ) such that (296) ()()()t f t t on [0, ), for some integrable functions  and  , then for each fixed tuple (,,,)ka of positive real numbers, ta , one has tt (log )kk (log ) aakk33 (297) HI(,)(,) a t()()()() f t  H I a t  f t kk()()    k22 k  k  k  HatIf(,)(,)(,)(,)()()()()()()  tIftItI Hat  Hat  Hat   ft k22 k  k  k  HatIf(,)(,)(,)(,)(  )() tI Hat  () tIftI  Hat () Hat (   ft )(). Proof:

For any x,(,) y a t , in context of (296), if we take a1  f()() x x and

a2  ()() y f y , it follows from (281) , using m  2 , that (298) (()f x ()) x33  (() y  f ()) y (()fx  ())(() x22  yfy  ())  (() fx   ())(() x  yfy  ()).  1 t 1 Multiplying both sides by (log ) k and then integrating with respect xkk () x to x over (,)at , the inequality (298), by use of (47), becomes

kk33 (299) HI(,)(,) a t( f )()(() t  y  f ()) y H I a t (1) kk22 HIf(,)(,) a t(  )()(() tyfy   ())  H If a t (   )()(() tyfy   ()).  1 t 1 Multiplying both sides by (log ) k and then integrating with respect ykk () y to y over (,)at , the inequality (299), by use of (47) and (48), leads to (297).

4.3 MORE FI INEQUALITIES RELATING AM-GM AND HOLDER’S INEQUALITIES In this section, we discuss more fractional integral inequalities involving the inequalities for bounded functions and the inequalities based on AM-GM inequality, Holder’s inequality and the other inequalities introduced in chapter 2. 4.3.1 Theorem: Let f be an integrable function on [0, ) such that (300) ()()()t f t t on [0, ), for some integrable functions  and  . Then for each fixed tuple (,,,,,)k p q a of positive real numbers, ta and fixed nZ  , np q , we have the following results q np qqp  kppq k n np q t k (301) nH I(,)(,) a t( f )() t  H I a t ( f   )() t  (log)  , p pk () a q np qqp  kppq k n np q t k (302) nH I(,)(,) a t( f )() t  H I a t (   f )() t  (log)  . p pk () a Proof: In view of (300), for any x(,) a t ; the relation (31) with a f()() x x becomes qq( f ( x ) ( x ))n q np np q q (303) n( f ( x ) ( x )) p   p   p . pp  1 t 1 Multiplying both sides by (log ) k and then integrating with respect xkk () x to x over (,)at , the inequality (303), by use of (47), becomes q np qqq p np q (304) nk I()() fpp t  k I  ()() f   n t   k I  (1) H(,)(,)(,) a tpp H a t H a t which leads to (301) by use of (48).

Also considering (300), for any x(,) a t ; the relation (31) with a ()() x f x becomes

qq( ( x ) f ( x ))n q np np q q (305) n( ( x ) f ( x )) p   p   p . pp  1 t 1 Multiplying both sides by (log ) k and then integrating with respect xkk () x to x over (,)at , the inequality (305), by use of (47), becomes q np qqq p np q (306) nk I()() fpp t  k I  ()()   f n t   k I  (1) H(,)(,)(,) a tpp H a t H a t which leads to (302) by use of (48). 4.3.2 Example:

Let f be an integrable function on [1, ) and 0 be a positive real number such that

(307) logt f ( t ) 0  log t on [1, ), then for each fixed pair (,)k  of positive real numbers, t 1, we have   (2k ) 1 (308) 2k I f ( x ) log x ( t )k (log t ) k Ht(1, ) ( 2k ) k  k  1 k HtI(1, ) f( t ) (log t ) , k () kk (309) 2HI(1, t ) 0 log x  f ( x )( t )  H I (1, t ) f ( t )   1 (2k ) 1 0 (logtt )kkk (log ) . kk(  )  (   2k ) Solution: The relation (308) and (309) are obtained from (301) and (302) respectively, using

( (x ),  ( x ),  , p , q , n , a ) (log x , 0 log x ,1,2,1,1,1) with collaboration of (48) and (69).

4.3.3 Corollary: Let f be an integrable function on [0, ) such that (310) ()()()t f t t on [0, ), for some integrable functions  and  . Then for each fixed tuple (,,,,,)k p q a of positive real numbers, ta , pq , we have the following results qp p k  (311) q H I(,) a t f() t q q p q kkp pq p t k p pH I(,)(,) a t( f  )() t  q  H I a t  () t  (log)  , k () a

q p q ppkk (312) qH I(,)(,) a t f()()() t p H I a t f t q p q ppk  p q t k qH I(,) a t ( t ) (log )  . k () a Proof: For n 1, the relations (301) and (302) lead to the conclusions (311) and (312) respectively.

4.3.4 Corollary: Let f be an integrable function on [0, ) such that (313) m f() t M on [0, ) , for some m, M R. Then for each fixed tuple (,,)ka of positive real numbers, ta , we have the following results 1  kk2 mt1 k (314) HI(,)(,) a t f( t ) 2 H I a t ( f  m ) ( t )  (log ) , k () a 1  kk2 1 Mtk (315) HI(,)(,) a t f()2 t H I a t ( M  f )() t  (log). k () a Proof: Setting   m,   M and (pq , , ) (2,1,1) , the results (314) and (315) follow from (311) and (312) respectively. 4.3.5 Lemma: Let f be an integrable function on [0, ) such that (316) ()()()t f t t on [0, ), for some integrable functions  and  , then for each fixed tuple (,,)ka of positive real numbers, ta , we have t  (log ) k a kk2 2 (317) HI(,)(,) a t f()() t  H I a t f t  k () t  (log ) k a k  h(,,) f  H I(,) a t (   f )( f   )(), t k () where k k  k  k  hf(,,)()()()()  Hat I(,)(,)(,)(,)  tIftIftIt  Hat Hat  Hat   t  (log ) k a k k  k   HI(,)(,)(,) a t()()() t  H I a t  f t  H I a t f  t  k () k k  k  k  k  k  HatI(,)(,)(,)(,)(,)(,)() tIft Hat ()  Hat I  () tI Hat  () t  Hat IftI () Hat  (). t Proof:

For all x,(,) y a t , it is easy to see that (318) (()yfyfx ())(()   ()) x  (()  xfxfy  ())(()   ()) y (()xfxfx  ())(()   ()) x  (()  yfyfy  ())(()   ()) y ()()()()()()()()()()()xxfxfx  2   y   xfy   xfyfxy   ()()2()()xy   fxfy   ()() xy    ()() xfxfxx  ()()  ()()y f y  f ()() y  y   ()() y  y  f2 (). y  1 t 1 Multiplying both sides by (log ) k and then integrating with respect xk () x k to x over (,)at , the identity (318), by use of (47), becomes kk (319) (()yfyIf ())H(,)(,) a t (   )() t  H I a t (   ftfy )()(()   ()) y kk HI(,)(,) a t(  ff )(   )( t )  (  ( yfyfy )  ( ))( ( )   ( yI )) H a t (1) k k 2 k  k  HatI(,)(,)(,)(,)()()()()()() t  Hat Ift  Hat Ifty   Hat I  tfy k k  k  HI(,)(,)(,) a t()()()()()() tfy  H Ifty a t   H I a t  ty  k k  k  k  2HatIftfy(,)(,)(,)(,) ()()  Hat I ()() ty   Hat I  ft ()  Hat Ift  () 2 k   fy()  ()() yfyfyy  ()()    ()() yyI   H(,) a t (1). Integrating with respect to y over (,)at after multiplying both sides by  1 t 1 (log ) k , by use of (47), the relation (319) leads to the relation ykk () y k k  k  k  (320) HatI(,)(,)(,)(,)( ftIftI )() Hat (   )()  Hat (   ftIft )() Hat (   )() k k  k  k  HatI(,)(,)(,)(,)(  f )( f   )( t ) Hat I (1)  Hat I (   f )( f   )( t ) Hat I (1) k k  k 2 k  k  k  HatI(,)(,)(,)(,)(,)(,)() tI Hat (1)  Hat IftI () Hat (1)  Hat IftIt () Hat  () k k  k  k  k  k  HatI(,)(,)(,)(,)(,)(,)()()()()()() tIft Hat  Hat I  tIft Hat  Hat IftI Hat  t k k  k  k  k  k  HatI(,)(,)(,)(,)(,)(,)() tI Hat  ()2 t  Hat IftIft () Hat ()  Hat I  () tI Hat  () t k k  k  k  HatI(,)(,)(,)(,) f( t ) Hat I (1) Hat I f ( t ) Hat I (1) k2 k  k  k  k   HatIft(,)(,)(,)(,)(,)()  Hat I ft ()  Hat Ift  ()  Hat I  () tI Hat (1). After simplification and by use of (48), it leads to (317). 4.3.6 Corollary: Let f be an integrable function on [0, ) such that (321) m f() t M on [0, ) , for some m, M R, then for each fixed tuple (,,)ka of positive real numbers, ta , we have t  (log ) k a kk2 2 (322) HI(,)(,) a t f()() t  H I a t f t  k ()

tt   Mm(log )kk  (log )  aakk  HI(,)(,) a t f()() t  H I a t f t   kk()()            t  (log ) k a k  HI(,) a t ( M  f )( f  m )( t ). k () Proof: Taking   m,   M and using (48), the result (322) follows from (317). 4.3.7 Theorem: Let f be an integrable function on [0, ) such that (323) ()()()t f t t on [0, ), for some integrable functions  and  , then for each fixed tuple (,,)ka of positive real numbers, ta , one has t  (log ) k a kk2 2 (324) HI(,)(,) a t f( t ) H I a t f ( t ) h ( f , , ), k () where k k  k  k  hf(,,)()()()()  Hat I(,)(,)(,)(,)  tIftIftIt  Hat Hat  Hat   t  (log ) k a k k  k   HI(,)(,)(,) a t()()() t  H I a t  f t  H I a t f  t  k () k k  k  k  k  k  HatI(,)(,)(,)(,)(,)(,)() tIft Hat ()  Hat I  () tI Hat  () t  Hat IftI () Hat  (). t Proof: For all xa , it follows from (323) that (()f x ())(() x x  f ()) x  0 implying, for each ta , that  1 t k k  (log )HI(,) a t ( f )(  f )( t )  0. k () a Hence, the desired result (324) follows from (317). 4.3.8 Theorem:

For each j 1,2, let f j be an integrable function on [0, ) , such that

(325) j()()()t f j t j t , on [0, ), for some integrable functions  j and  j . Then for each fixed tuple (,,)ka of positive real numbers, ta , one has the following result

 t k (log ) 22 (326) a kk HI( a , t ) f 1 f 2()()(,,), t H I ( a , t ) f j t h f j j j k () jj11

where k k  k  k  hf(,,)()()()()  Hat I(,)(,)(,)(,)  tIftIftIt  Hat Hat  Hat   t  (log ) k a k k  k   HI(,)(,)(,) a t()()() t  H I a t  f t  H I a t f  t  k () k k  k  k  k  k  HatI(,)(,)(,)(,)(,)(,)() tIft Hat ()  Hat I  () tI Hat  () t  Hat IftI () Hat  (). t Proof: For any x,(,) y a t , let 2 (327) g(,,) fj j j (() f j x f j ()). y j1  11 1 ttkk Multiplying both sides by 22 (log ) (log ) and then integrating xykk () x y with respect to x and y over (,)at , it follows from (327) that kIfftI( ) k  (1) k IftIft  ( ) k  ( ) Hat(,)12 Hat (,) Hat (,)1 Hat (,)2 tt  11 tt11  g( f , , )(log )kk (log ) dxdy . 22  j j j 2kk ( ) aa xy x y It follows, by applying Cauchy-Schwarz inequality for integrals and (324), that k k  k  k  2  HatIfftI(,)12( ) Hat (,) (1) Hat IftIft (,)1 ( ) Hat (,)2 ( ) 2 2 kI(1) k I  f2 ( t ) k I  f ( t )  Hat(,)(,)(,) Hatj Hatj   j1 2  hf(,,)j j j j1 which leads to (326) by use of (48).

4.3.9 Example:

Let f1 and f2 be an integrable functions on [1, ) and 0 be a positive real number such that

(328) logt f10 ( t )   log t ,

(329) logt02  f ( t )  log t on [1, ), then for each fixed pair (,)k  of positive real numbers, t 1, we have

 (logt ) k 2 (330) kk HI(1, t ) f 1 f 2()() t  H I (1, t ) f j t k () j1

h( f1 ,log t ,log t  0 ) h ( f 2 ,log t  0 ,log t ) , where  (logt ) k h(,log,log f t t )   ()() t  t   () t 1 0 1 2 3 k ()  1 k (2k ) k k   (logt )Ht I(1, ) f 1 ( t ) k ( 2k )    (2kk )11 (2 ) kk(logt )k0 (log t ) k (log t ) k k(  2kk )   k (  )  k (   2 )    (2k ) 1 k I f( t )0 (log t )kkk (log t ) , Ht(1, ) 1  kk(  )  (   2k ) while    (2k ) 1  (t )0 (log t )kk k (log t )  k I f ( t ), 1Ht (1, ) 1 kk(  )  (   2k )   (2k ) 1  (t )k I f ( t )k (log t )k , 2Ht (1, ) 1 k ( 2k )   (2kk )12 (3 )  (t )0 kk (log t )kk (log t ) 3 kk(  2kk )  (  3 ) kk 0HI (1, t ) f 1() t 2 H I (1, t ) ( f 1 ()log)() x x t , and  (logt ) k h(,log f t ,log) t   ()() t  t   () t 2 0 1 2 3 k ()   (2k ) 1  k (logt )kk0 (log t )k I f ( t ) Ht(1, ) 2 kk(  2k )   (  )    (2kk )11 (2 ) kk(logt )k0 (log t ) k (log t ) k  k(  2kk )   k (  )  k (   2 )   (2k ) 1  k (logt )k k I f ( t ), Ht(1, ) 2 k ( 2k ) while   (2k ) 1  (t )k (log t )k k I f ( t ), 1Ht (1, ) 2 k ( 2k )

  (2k ) 1   (t )k I f ( t ) k (log t )kk  0 (log t ) , 2Ht (1, ) 2 kk(  2k )   (  )  (3kk )21 (2 )  (t )kk (log t )kk0 (log t ) 3 kk(  3kk )  (  2 ) kk 2HI(1, t ) ( f ( x )log x )( t ) 0 H I (1, t ) f ( t ). Solution: The relation (330) is obtained from (326), using

(1 (x ),  2 ( x ),  1 ( x ),  2 ( x ), a ) (log x ,log x   0 ,log x   0 ,log x ,1) with collaboration of (48), (69) and (70).

4.3.10 Theorem: Let f and f be integrable functions on [0, ) such that 1 2 (331) mj f j() t M j , j 1,2, on [0, ) , for some mjj, M R . Then for each fixed tuple (,,)ka of positive real numbers, ta , one has 2 ttkk (log )22 (log ) (332) aakkI f f( t ) I f ( t )  ( M  m ). H( a , t ) 1 2 H ( a , t ) j22 j j kk(  )jj11 4  (  )

Proof: Let k k  k  k  hf(,,)()()()()  Hat I(,)(,)(,)(,)  tIftIftIt  Hat Hat  Hat   t  (log ) k a k k  k   HI(,)(,)(,) a t()()() t  H I a t  f t  H I a t f  t  k () k k  k  k  k  k  HatI(,)(,)(,)(,)(,)(,)() tIft Hat ()  Hat I  () tI Hat  () t  Hat IftI () Hat  (). t Then, using (48), it is easy to see that tt   (log )kk  (log )  aakk hfmM(,,)()().jjj M j  Hatj Ift(,)(,)  Hatj Iftm  j  kk()()           

kk11 Taking ccpp1,,, 2 1 2 H IMftI ( a , t ) ( j  j )(), H ( a , t ) ( fmt j  j )(),, , it implies, 22 by use of (38), that 2 t  (log ) k a Mmjj (333) h(,,)(). fj m j M j   k ( ) 2  

Moreover, taking (,,,)(,,,)1  2  1  2 m 1 m 2 M 1 M 2 , it follows from (326) that  t k (log ) 22 a kk HIfft( a , t ) 1 2()()(,,). H Ift ( a , t ) j hfmM j j j k () jj11

This leads to (332) by use of (333). 4.3.11 Theorem:

Let for each j 1,2; f j be an integrable function on [0, ) such that

(334) j()()()t f j t j t ,

on [0, ), for some integrable functions  j and  j . Then for each fixed tuple (,,,)ka of positive real numbers, ta , one has the following results k k  k  k  (335) HatI(,)1 f()()()() t Hat I (,)2 t Hat I (,)1 t Hat I (,)2 f t k k  k  k  HatI(,)1 f()()()() t Hat I (,)2 f t Hat I (,)1 t Hat I (,)2 t , k k  k  k  (336) HatI(,)2 f()()()() t Hat I (,)1 t Hat I (,)2 t Hat I (,)1 f t k k  k  k  HatI(,)2 f()()()() t Hat I (,)1 f t Hat I (,)2 t Hat I (,)1 t , k k  k  k  (337) HatI(,)1 f()()()() t Hat I (,)2 f t Hat I (,)1 t Hat I (,)2 t k k  k  k  HatI(,)1 f()()()() t Hat I (,)2 t Hat I (,)1 t Hat I (,)2 f t , k k  k  k  (338) HatI(,)1()()()() t Hat I (,)2 t Hat I (,)1 f t Hat I (,)2 f t k k  k  k  HatI(,)1()()()() t Hat I (,)2 f t Hat I (,)1 f t Hat I (,)2 t . Proof: For all x,(,) y a t , it follows from (334) that

(339) fx1()()()()()()()() 2 y  1 xfyfxfy 2  1 2   1 x  2 y ,

(340) fx2()()()()()()()() 1 y  2 xfyfxfy 1  2 1   2 x  1 y ,

(341) fxfy1()()()()()()()() 2 1 x  2 yfx  1  2 y   1 xfy 2 ,

(342) 1()()()()()()()()x  2 yfxfy 1 2   1 xfyfx 2  1  2 y .  1 t 1 Multiplying both sides by (log ) k and then integrating with respect xkk () x to x over (,)at , the inequality (339), by use of (47), leads to k k  k  k  (343) HatIft(,)1()() 2 y Hat I (,)1  ()() tfy 2  Hat Iftfy (,)1 ()() 2  Hat I (,)1  ()(). t  2 y Integrating with respect to y over (,)at after multiplying both sides by  1 t 1 (log ) k , by use of (47), the inequality (343) leads to the relation ykk () y (335).

 1 t 1 Multiplying both sides by (log ) k and then integrating with respect xkk () x to x over (,)at , the inequality (340), by use of (47), becomes k k  k  k  (344) HatIft(,)2()() 1 y Hat I (,)2  ()() tfy 1  Hat Iftfy (,)2 ()() 1  Hat I (,)2  ()(). t  1 y Integrating with respect to y over (,)at after multiplying both sides by  1 t 1 (log ) k , by use of (47), the inequality (344) leads to the relation ykk () y (336).  1 t 1 Multiplying both sides by (log ) k and then integrating with respect xkk () x to x over (,)at , the inequality (341), by use of (47), gives k k  k  k  (345) HatIftfy(,)1()() 2 Hat I (,)1 ()() t  2 y  Hat Ift (,)1 ()()  2 y  Hat I (,)1  ()(). tfy 2 Integrating with respect to y over (,)at after multiplying both sides by  1 t 1 (log ) k , by use of (47), the inequality (345) leads to the relation ykk () y (337).  1 t 1 Multiplying both sides by (log ) k and then integrating with respect xkk () x to x over (,)at , the inequality (342), by use of (47), provides k k  k  k  (346) HatI(,)1()() t  2 y Hat Iftfy (,)1 ()() 2  Hat I (,)1  ()() tfy 2  Hat Ift (,)1 ()().  2 y Integrating with respect to y over (,)at after multiplying both sides by  1 t 1 (log ) k , by use of (47), the inequality (346) leads to the relation ykk () y (338). 4.3.12 Corollary:

Let f j be an integrable function on [0, ) , for each j 1,2, such that

(347) mj f j() t M j , on [0, ), for some mjj, M R . Then for each fixed tuple (,,,)ka of positive real numbers, ta , one has  M21 tkkkk m t (348) (log )HI( a , t ) f 1 ( t ) (log ) H I ( a , t ) f 2 ( t ) kk()() aa    kk m12 M t k HI( a , t ) f 1( t ) H I ( a , t ) f 2 ( t ) (log ) , kk()()   a  M12 tkkkk m t (349) (log )HI( a , t ) f 2 ( t ) (log ) H I ( a , t ) f 1 ( t ) kk()() aa  

 kk m21 M t k HI( a , t ) f 2( t ) H I ( a , t ) f 1 ( t ) (log ) , kk()()   a  kk m12 m t k (350) HI( a , t ) f 1( t ) H I ( a , t ) f 2 ( t ) (log ) kk()()   a  m21 tkkkk m t (log )HI( a , t ) f 1 ( t ) (log ) H I ( a , t ) f 2 ( t ) , kk()() aa    M12 M t k kk (351) (log ) HI( a , t ) f 1 ( t ) H I ( a , t ) f 2 ( t ) kk()()   a  M12 tkkkk M t (log )HI( a , t ) f 2 ( t ) (log ) H I ( a , t ) f 1 ( t ). kk()() aa   Proof:

If we take (,,,)(,,,)1  2  1  2 m 1 m 2 M 1 M 2 , by use of (48), the results (348)- (351) follow from the inequalities (335)-(338) respectively. 4.3.13 Theorem:

Let for each j 1,2; f j be an integrable function on [0, ) such that

(352) j()()()t f j t j t ,

on [0, ) , for some integrable functions  j and  j , then for each fixed tuple (,,,,,)k p p a of positive real numbers, ta , p 1, we have kk (353) HI( a , t )( f 1 1 )( t ) H I ( a , t ) ( f 2 2 )( t ) tt (log )kk (log ) aak p k p HI( a , t )()()()() f 1  1 t  H I ( a , t ) f 2  2 t , ppkk()()    kk (354) HI(,)1 a t( f 12 )( f   2 )( t ) H I (,)1 a t ( f   12 )( f   2 )( t ) 1 kI()()()() f  p t k I f  p t p H( a , t ) 1 1 H ( a , t ) 2 2 1 kI()()()() f  p t k I f  p t , p H( a , t ) 2 2 H ( a , t ) 1 1 kk (355) HI( a , t )( 1 f 1 )( t ) H I ( a , t ) ( 2 f 2 )( t ) tt (log )kk (log ) aak p k p HI( a , t )()()()() 1  f 1 t  H I ( a , t ) 2  f 2 t , ppkk()()    kk (356) HI(,)1 a t( f 1 )(  2  f 2 )( t ) H I (,)1 a t (   f 1 )(  2  f 2 )( t ) 1 kI()()()()  f p t k I  f p t p H( a , t ) 1 1 H ( a , t ) 2 2 1 kI()()()()  f p t k I  f p t , p H( a , t ) 2 2 H ( a , t ) 1 1

84 where (,)pp is a pair of Holder’s conjugate numbers. Proof: (i) For any x,(,) y a t , in context of (352), if we take

(,)(()a b f1 x  1 (),() x f 2 y  2 ()) y , using (45), it follows that 11 (357) (()fx ())(() xfy   ()) y  (() fx   ()) xpp  (() fy   ()). y  1 1 2 2pp 1 1 2 2  1 t 1 Multiplying both sides by (log ) k and then integrating with respect xkk () x to x over (,)at , the inequality (357), by use of (47), becomes k  (358) HI( a , t )( f 1 1 )( t ) ( f 2 ( y ) 2 ( y )) 11 kI( f  )() p t  (() f y  ()) y p k I (1). ppH( a , t ) 1 1 2 2 H ( a , t ) Integrating with respect to y over (,)at after multiplying both sides by  1 t 1 (log ) k , by use of (47), the inequality (358) leads to the relation ykk () y kkI( f )( t ) I ( f )( t ) H( a , t ) 1 1 H ( a , t ) 2 2 11 kI( f  )() p t k I  (1)  k I  ( f  )() p t k I  (1) ppHat(,)11 Hat (,) Hat (,)2 2 Hat (,) which gives (353) by use of (48).

(ii) For any x,(,) y a t , in perspective of (352), if we take , (,)ab ((() fx1  1 ())( xfy 2 ()   2 ()),(() yfy 1   1 ())( yfx 2 ()   2 ())) x using (45), it follows that

(359) (()fx1 1 ())( xfy 2 ()   2 ())(() yfy 1   1 ())( yfx 2 ()   2 ()) x 11pp (()fx  ())(() xfy   ()) y  (() fy   ())(() yfx   ()). x  pp1 1 2 2 1 1 2 2  1 t 1 Multiplying both sides by (log ) k and then integrating with respect xkk () x to x over (,)at , the inequality (359), by use of (47), becomes k  (360) HI(,)1122 a t ( f )( f   )()(() t f 1 y   1 ())(() y f 2 y   2 ()) y 1 kI ( f  ) p ( t ) ( f ( y )  ( y )) p p H( a , t ) 1 1 2 2 1 kI ( f  )()(() p t f y  ()). y p p H( a , t ) 2 2 1 1

Integrating with respect to y over (,)at after multiplying both sides by  1 t 1 (log ) k , by use of (47), the inequality (360) leads to the relation ykk () y (354). (iii) For any x,(,) y a t , in context of (352), if we take , (,)(()a b1 x  f 1 (), x 2 () y  f 2 ()) y using (45), it follows that 11 (361) (()xfx ())(()  yfy  ())  (()  xfx  ())pp  (()  yfy  ()). 1 1 2 2pp 1 1 2 2  1 t 1 Multiplying both sides by (log ) k and then integrating with respect xkk () x to x over (,)at , the inequality (361), by use of (47), becomes k  (362) HI( a , t )( 1 f 1 )( t ) ( 2 ( y ) f 2 ( y )) 11 kI(  f )() p t  (() y  f ()) y p k I (1). ppH( a , t ) 1 1 2 2 H ( a , t ) Integrating with respect to y over (,)at after multiplying both sides by  1 t 1 (log ) k , by use of (47), the inequality (362) leads to the relation ykk () y kkI( f )( t ) I ( f )( t ) H( a , t ) 1 1 H ( a , t ) 2 2 11 kI(  f )() p t k I  (1)  k I  (  f )() p t k I  (1) ppHat(,)11 Hat (,) Hat (,)2 2 Hat (,) which gives (355) by use of (48).

(iv) For any x,(,) y a t , in perspective of (352), if we take , (,)ab ((1 () xfx  1 ())(  2 () yfy  2 ()),(  1 () yfy  1 ())(  2 () xfx  2 ())) using (45), it follows that

(363) (1 ()xfx 1 ())(  2 () yfy  2 ())(  1 () yfy  1 ())(  2 () xfx  2 ()) 11pp (()xfx  ())(()  yfy  ())  (()  yfy  ())(()  xfx  ()). pp1 1 2 2 1 1 2 2  1 t 1 Multiplying both sides by (log ) k and then integrating with respect xkk () x to x over (,)at , the inequality (363), by use of (47), becomes k  (364) HI(,)1122 a t ( f )(   f )()(() t  1 y  f 1 ())(() y  2 y  f 2 ()) y 1 kI (  f ) p ( t ) ( ( y )  f ( y )) p p H( a , t ) 1 1 2 2 1 kI (  f )()(() p t y  f ()). y p p H( a , t ) 2 2 1 1

Integrating with respect to y over (,)at after multiplying both sides by  1 t 1 (log ) k , by use of (47), the inequality (364) leads to the relation ykk () y (356). 4.3.14 Theorem:

Let for each j 1,2; f j be an integrable function on [0, ) such that

(365) j()()()t f j t j t ,

on [0, ) , for some integrable functions  j and  j , then for each fixed tuple

(,,,,,)k p12 p a of positive real numbers, ta , 2 (366)  p j 1, j1 and each fixed  [ 1,1], we have  ttkk pp12(log ) (log ) aa22kk (367) exp(1 )HI( a , t ) f 1 ( t )  exp(1  ) H I ( a , t ) f 2 ( t ) kk()()   

kkpp12 HI( a , t )()()()() f 1  1 t H I ( a , t ) f 2  2 t  ttkk pp12(log ) (log ) aa22kk exp(1  )HI( a , t )  1 ( t )  exp(1   ) H I ( a , t )  2 ( t ) , kk()()    2 k k  k  k  (368) p1exp(1 ) Hat IftIftItI (,)1 () Hat (,)2 () Hat (,)1  () Hat (,)2  () t  2 k k  k  k  p2exp(1  ) Hat IftIftI (,)2 () Hat (,)1 ()  Hat (,)2  () tIt Hat (,)1  ()

kkp1 p 2 p 1 p 2 HI(,)11 a t()()()()()() f  f 2   2 t H I (,)2 a t f   2 f 11   t 2 k k  k  k  p1exp(1  ) Hat IftI (,)1 () Hat (,)2  () tItIft  Hat (,)1  () Hat (,)2 ()  2 k k  k  k  p2exp(1  ) Hat IftItI (,)2 () Hat (,)1  ()  Hat (,)2  () tIft Hat (,)1 (),  ttkk pp12(log ) (log ) aa22kk (369) exp(1 )HI( a , t )  1 ( t )  exp(1   ) H I ( a , t )  2 ( t ) kk()()   

kkpp12 HI( a , t )()()()() 1  f 1 t H I ( a , t ) 2  f 2 t  ttkk pp12(log ) (log ) aa22kk exp(1  )HI( a , t ) f 1 ( t )  exp(1  ) H I ( a , t ) f 2 ( t ) , kk()()    2 k k  k  k  (370) p1exp(1 ) Hat I (,)1  () tI Hat (,)2  () tIftIft Hat (,)1 () Hat (,)2 () 2 k k  k  k  p2exp(1  ) Hat I (,)2  () tI Hat (,)1  () tIftIft  Hat (,)2 () Hat (,)1 ()

kkp1 p 2 p 1 p 2 HI(,)11 a t()()()()()()  f  22  f t H I (,)22 a t   f  11  f t

2 k k  k  k  p1exp(1  ) Hat I (,)1  () tIftIftI Hat (,)2 ()  Hat (,)1 () Hat (,)2  () t  2 k k  k  k  p2exp(1  ) Hat I (,)2  () tIftIftI Hat (,)1 ()  Hat (,)2 () Hat (,)1  () t  . Proof: (i) For any x,(,) y a t , in the perspective of (365), if we take

(,)(()c1 c 2 f 1 x  1 (),() x f 2 y  2 ()) y and n  2, (33) gives 22 (371) p1exp(1 ) f 1 ( x )  p 2 exp(1  ) f 2 ( y ) (()f x  ())(() xpp12 f y  ()) y 1 1 2 2 22 p1exp(1  )  1 ( x )  p 2 exp(1   )  2 ( y ).  1 t 1 Multiplying both sides by (log ) k and then integrating with respect xkk () x to x over (,)at , the inequality (371), by use of (47), becomes 22kk (372) p1exp(1 )H I ( a , t ) f 1 ( t )  p 2 exp(1  ) f 2 ( y ) H I ( a , t ) (1)

k  pp12 HI( a , t )( f 1  1 ) ( t ) ( f 2 ( y )  2 ( y )) 22kk p1exp(1  )H I ( a , t )  1 ( t )  p 2 exp(1   )  2 ( y ) H I ( a , t ) (1). Integrating with respect to y over (,)at after multiplying both sides by  1 t 1 (log ) k , by use of (47), the inequality (372) leads to the relation ykk () y 22k k  k  k  p1exp(1 )Hat I (,)1 f ( t ) Hat I (,) (1)  p 2 exp(1  ) Hat I (,)2 f ( t ) Hat I (,) (1)

kkpp12 HI( a , t )()()()() f 1  1 t H I ( a , t ) f 2  2 t 22k k  k  k  p1exp(1  )Hat I (,)1  ( t ) Hat I (,) (1)  p 2 exp(1   ) Hat I (,)2  ( t ) Hat I (,) (1) which gives (367) by use of (48).

(ii) For any x,(,) y a t , in the perception of (365), if we take

(,cc1 2 ) ((() fx 1  1 ())( xfy 2 ()   2 ()),(() yfy 1   1 ())( yfx 2 ()   2 ())) x and n  2, (33) gives 2 (373) p1exp(1 )( f 1 ( x )   1 ( x ))( f 2 ( y )   2 ( y )) 2 p2exp(1  )( f 1 ( y )   1 ( y ))( f 2 ( x )   2 ( x ))

p1 p 2 p 1 p 2 (()fx1  1 ())(() xfx 2   2 ())(() xfy 2   2 ())(() yfy 1   1 ()). y  1 t 1 Multiplying both sides by (log ) k and then integrating with respect xkk () x to x over (,)at , the inequality (373), by use of (47), becomes 2 k  (374) p1exp(1 )H I ( a , t ) ( f 1   1 )( t ) ( f 2 ( y )   2 ( y )) 2 k  p2exp(1  )( f 1 ( y )   1 ( y ))H I ( a , t ) ( f 2   2 )( t )

k  p1 p 2 p 1 p 2 HI(,)1122 a t ( f  )( f   )()(() t f 2 y   2 ())(() y f 1 y   1 ()). y

Integrating with respect to y over (,)at after multiplying both sides by  1 t 1 (log ) k , by use of (47), the inequality (374) leads to the relation ykk () y (368). (iii) For any x,(,) y a t , in the perspective of (365), if we take

(,)(()c1 c 2 1 x  f 1 (), x 2 () y  f 2 ()) y and n  2, (33) gives 22 (375) p1exp(1 )  1 ( x )  p 2 exp(1   )  2 ( y ) (()x  f ())( xpp12 () y  f ()) y 1 1 2 2 22 p1exp(1  ) f 1 ( x )  p 2 exp(1  ) f 2 ( y ).  1 t 1 Multiplying both sides by (log ) k and then integrating with respect xkk () x to x over (,)at , the inequality (375), by use of (47), becomes 22kk (376) p1exp(1 )H I ( a , t )  1 ( t )  p 2 exp(1   )  2 ( y ) H I ( a , t ) (1)

k  pp12 HI( a , t )( 1  f 1 ) ( t ) ( 2 ( y )  f 2 ( y )) 22kk p1exp(1  )H I ( a , t ) f 1 ( t )  p 2 exp(1  ) f 2 ( y ) H I ( a , t ) (1). Integrating with respect to y over (,)at after multiplying both sides by  1 t 1 (log ) k , by use of (47), the inequality (376) leads to the relation ykk () y 22k k  k  k  p1exp(1 )Hat I (,)1  ( t ) Hat I (,) (1)  p 2 exp(1   ) Hat I (,)2  ( t ) Hat I (,) (1)

kkpp12 HI( a , t )()()()() 1  f 1 t H I ( a , t ) 2  f 2 t pexp(1 22 )k I f ( t ) k I  (1)  p exp(1  ) k I  f ( t ) k I  (1) 1Hat (,)1 Hat (,) 2 Hat (,)2 Hat (,) which gives (369) by use of (48). (iv) For any x,(,) y a t , in the view of (365), if we take

(,cc1 2 ) (( 1 () xfx  1 ())(  2 () yfy  2 ()),(  1 () yfy  1 ())(  2 () xfx  2 ())) and n  2, (33) gives 2 (377) p1exp(1 )(  1 ( x )  f 1 ( x ))(  2 ( y )  f 2 ( y )) 2 p2exp(1  )(  1 ( y )  f 1 ( y ))(  2 ( x )  f 2 ( x )).

p1 p 2 p 1 p 2 (()1xfx  1 ())(  2 () xfx  2 ())(  2 () yfy  2 ())(()  1 yfy  1 ()).  1 t 1 Multiplying both sides by (log ) k and then integrating with respect xkk () x to x over (,)at , the inequality (377), by use of (47), becomes 2 k  (378) p1exp(1 )H I ( a , t ) (  1  f 1 )( t ) (  2 ( y )  f 2 ( y )) 2 k  p2exp(1  )(  1 ( y )  f 1 ( y ))H I ( a , t ) (  2  f 2 )( t )

k  p1 p 2 p 1 p 2 HI(,)1122 a t (  f )(   f )()(() t  2 y  f 2 ())(() y  1 y  f 1 ()). y

Integrating with respect to y over (,)at after multiplying both sides by  1 t 1 (log ) k , by use of (47), the inequality (378) leads to the relation ykk () y (370). 4.3.15 Theorem:

Let for each j 1,2; f j be an integrable function on [0, ) such that

(379) j()()()t f j t j t , on [0, ) , for some integrable functions  j and  j , then for each fixed tuple (,,,,,)k p q a of positive real numbers, ta , pq , we have the following results 2 q q p kppq k  k  (380) HatI(,)()()()() f jj t   Hat I (,)12 f  t  Hat I (,)12  f t  j1 p t  q p q (log ) k qppkk p q a  HI( a , t ) f 1 f 2()(), t  H I ( a , t )  1  2 t    pp () k 2 q q p kppq k  k  (381) HatI(,)()()()() jj f t   Hat I (,)12  f t  Hat I (,)12 f  t  j1 p t  q p q (log ) k qppkk p q a  HI( a , t )  1  2()(). t  H I ( a , t ) f 1 f 2 t    pp () k Proof: 2 (i) In view of (379), for any x(,) a t , if we set a( fjj ( x ) ( x )), for j1 n 1, the relation (31) becomes 22q q p q pq p p q p (382) (()fj x j ()) x   (() f j x   j ()) x   . jj11pp  1 t 1 Multiplying both sides by (log ) k and then integrating with respect xkk () x to x over (,)at , the inequality (382), by use of (47), becomes

22q q p q kpq p k  p q p k  HatI(,)(,)(,)( f jj )() t   Hat I ( f jj   )() t   Hat I (1) jj11pp which leads to (380) by use of (48). 2 (ii) In context of (379), for any x(,) a t , if we set a( jj ( x ) f ( x )) , for j1 n 1, the relation (31) becomes

22q q p q pq p p q p (383) (()())jx f j x   (()())  j x  f j x   . jj11pp  1 t 1 Multiplying both sides by (log ) k and then integrating with respect xkk () x to x over (,)at , the inequality (383), by use of (47), becomes

2 q k  p (384) HI(,) a t()() j f j t j1 q p2 q qppkk p q HI(,)(,) a t(  j  f j )( t )   H I a t (1) ppj1 which leads to (381) by use of (48). 4.3.16 Theorem:

Let for each j 1,2; f j be an integrable function on [0, ) such that

(385) j()()()t f j t j t , on [0, ) , for some integrable functions  j and  j , then for each fixed tuple (,,,,,,)k   p q a of positive real numbers, ta , pq , we have the following results qq kkpp (386) HI( a , t )()()()() f 1 1 t H I ( a , t ) f 2 2 t qp q p k k  k  k   HatI(,)1 f()()()() t Hat I (,)2  t Hat I (,)1  t Hat I (,)2 f t  p qp q p k k  k  k   HatI(,)1 f()()()() t Hat I (,)2 f t Hat I (,)1  t Hat I (,)2  t  p t  q (log ) k pq   p a , p kk()()   qq kkpp (387) HI( a , t )()()()() 1 f 1 t H I ( a , t ) 2 f 2 t qp q p k k  k  k   HatI(,)1 ()()()() t Hat I (,)2 f t Hat I (,)1 f t Hat I (,)2  t  p qp q p k k  k  k   HatI(,)1 ()()()() t Hat I (,)2  t Hat I (,)1 f t Hat I (,)2 f t  p t  q (log ) k pq   p a . p kk()()   Proof:

(i) In perception of (385), for any x,(,) y a t , if we set

a(() f1 x  1 ())(() x f 2 y  2 ()) y , for n 1, the relation (31) becomes qq pp (388) (()f1 x 1 ())(() x f 2 y 2 ()) y qq p p q q pp(()f x   ())(() x f y   ()) y   . pp1 1 2 2  1 t 1 Multiplying both sides by (log ) k and then integrating with respect xkk () x to x over (,)at , the inequality (388), by use of (47), gives qq k  pp (389) HI( a , t )( f 1 1 ) ( t ) ( f 2 ( y ) 2 ( y )) qq p p q q ppkkI( f   )( t ) ( f ( y )   ( y ))   I (1). ppH( a , t ) 1 1 2 2 H ( a , t ) Integrating with respect to y over (,)at after multiplying both sides by  1 t 1 (log ) k , the inequality (389), by use of (47) and (48), leads to the ykk () y relation (386). (ii) In perception of (385), for any x,(,) y a t , if we set

a(1 () x  f 1 ())( x 2 () y  f 2 ()) y , for n 1, the relation (31) becomes qq pp (390) (()1x f 1 ())( x 2 () y f 2 ()) y qq p p q q pp(() x  f ())(() x  y  f ()) y   . pp1 1 2 2  1 t 1 Multiplying both sides by (log ) k and then integrating with respect xkk () x to x over (,)at , the inequality (390), by use of (47), gives qq k  pp (391) HI( a , t )( 1 f 1 ) ( t ) ( 2 ( y ) f 2 ( y )) qq p p q q ppkkI(   f )( t ) (  ( y )  f ( y ))   I (1). ppH( a , t ) 1 1 2 2 H ( a , t ) Integrating with respect to y over (,)at after multiplying both sides by  1 t 1 (log ) k , the inequality (391), by use of (47) and (48), leads to the ykk () y relation (387).

CHAPTER 5 more extenSIONS OF FI operators

In this chapter, we introduce more fractional intergral operators including the generalized form of the operator defined in (47) and extended forms of Hadamard type fractional integrals. Moreover, we discuss the properties involving the commutative law and semigroup property of the extended integral operators. The Mellin trasforms and boundedness of some of the extended operators are also discussed.

5.1 EXTENSIONS OF THE FRACTIONAL INTEGRAL OPERATOR We have defined, in (47), an extension of Hadamard fractional integral so far called Hadamard k -fractional integral. In this section, we define, a number of Hadamard type and other fractional integral operators and discuss their properties. For any tuple (,,)ka of positive real numbers, ta and (,,)   RRR    , we define t  11  t 1 (392) k,,,  I  f( t ) (log t )  ( )  (log )k f ( ) d , H(,) a t   ktk ()a    t  11  t 1 (393) k,,,  I  f( t ) (log t )  ( )  (log )k f ( ) d , H  (,)at   kk ()a t t   and for any pair (,)k  of positive real numbers, t  0 and , we define   11 t  1 (394) k,,,  I  f( t ) (log )  ( )  (log )k f (  ) d  , (,)tt  ktk ()t      11 t  1 (395) k,,,  I  f( t ) (log )  ( )  (log )k f (  ) d  . (,)tt kk ()t t   t Likewise, if  t 1 1F 1 ; ; log( ) , kk  then for any tuple (,,,)ka of positive real numbers, and (,,)    R3 , we define t  11  t 1 (396) k,,,,   I  f( t ) (log t )  ( )  (log )k ( ) f (  ) d  , HC( a , t )  1 ktk ()a   t  11  t 1 (397) k,,,,   I  f( t ) (log t )  ( )  (log )k ( ) f (  ) d  , HC (at , )  1 kk ()a t t

93 and if    2 1F 1 ; ; log( ) , k k t then for any tuple (,,)k  of positive real numbers, t  0 and (,,)    R3 , we define   11 t  1 (398) k,,,,   I  f( t ) (log )  ( )  (log )k (  ) f (  ) d  , C( t , ) t 2 ktk ()t     11 t  1 (399) k,,,,   I  f( t ) (log )  ( )  (log )k (  ) f (  ) d  . C( t , ) t 2 kk ()t t t Also, for any tuple (,,)ka of positive real numbers, ta ,   D and any (,,)   RRR    , we define t  1   (t )1  (  ) (400) k,,,,   I  f( t ) (log t )  ( )  (log )k f ( ) d , H(,) a t   ktk ()()()a      and for any pair (,)k  of positive real numbers, t  0, and , we define   1t    (  )1  (  ) (401) k,,,,   I  f( t ) (log )  ( )  (log )k f (  ) d  . (,)tt  ktk ()()()t      Moreover, for convenience, we write (394) for 0 as   11 1 (402) k I f( t ) (log )k f ( ) d , (,)t   ktk ()t   and if any of the operators (392)-(401) is denoted by If, we define  0 I f f .

5.2 PROPERTIES OF THE EXTENDED OPERATORS We have already discussed the properties of the extended operator defined in (47). In this section, we discuss the properties of the operators defined in (392)-(401). 5.2.1 Theorem: For each fixed tuple (,,,)ka of positive real numbers, ta and fixed (,,)   RRR    , we have k,,,,,,,,, k  k  (403) HI(,)(,)(,) a t H I a t f( t ) H I a t f ( t ). Proof: Using (392), we find that kk,,,,,,        (404) HI(,)(,) a t H I a t f() t

t  11  t 1  (logt ) ( )  (log )k k,,,    I  f ( ) d ,   Ha(,)  ktk ()a    where   11 u  1 k,,,  I  f( ) (log )  ( )  (log )k f ( u ) du . H(,) a  u kk ()a u   u Applying Fubini’s theorem, it follows that kk,,,,,,        HI(,)(,) a t H I a t f() t tt 1 1ut  1 11  ( ) (logt ) (log )kk (log ) d f ( u ) du . 2 u  kkk()()  au u t     u Using (25), it becomes kk,,,,,,        (405) HI(,)(,) a t H I a t f() t t  B( ,  ) 1 u  t 1  k ( ) (logt ) (log )k f ( u ) du .  u kkk()()  a u t   u Thus, using (7) and (392), (405) becomes kk,,,,,,        HI(,)(,) a t H I a t f() t t  11ut  1  ( ) (logt ) (log )k f ( u ) du  u kk ()  a u t   u k,,,      HI(,) a t f( t ).

5.2.2 Corollary: For each fixed tuple (,,,)ka of positive real numbers, ta and fixed (,,)   RRR    , we have k,,,,,,,,,,,, k  k  k  (406) HatI(,)(,)(,)(,) Hat I f( t ) Hat I Hat I f ( t ). Proof: The result (406) follows from (403). 5.2.3 Theorem: For each fixed tuple of positive real numbers, and fixed , we have (407) k,,,,,,,,,I k  I f( t ) k  I f ( t ). HHH(,)(,)(,)a t  a t  a t Proof: Using (393) and applying Fubini’s theorem, we find that (408) kk,,,,,,  I     I  f() t HH(,)(,)a t a t tt 1 1ut  1 11  ( ) (logt ) (log )kk (log ) d f ( u ) du . 2 u  kkk()()  au t t     u Using (25), it becomes (409) kk,,,,,,  I     I  f() t HH(,)(,)a t a t

t  B( ,  ) 1 u  t 1  k ( ) (logt ) (log )k f ( u ) du .  u kkk()()  a t t   u Thus, using (7) and (393), (409) becomes kk,,,,,,  I     I  f() t HH(,)(,)a t a t t  11ut  1  ( ) (logt ) (log )k f ( u ) du  u kk ()  a t t   u  k,,,  I   f( t ). H  (,)at 5.2.4 Corollary: For each fixed tuple (,,,)ka of positive real numbers, ta , and fixed (,,)   RRR    , we have (410) k,,,,,,,,,,,,I k  I f()() t k  I k  I f t . HHHH(,)(,)(,)(,)a t  a t  a t  a t Proof: The result (410) follows from (407). 5.2.5 Theorem: For each fixed tuple (,,)k  of positive real numbers, t  0 and fixed (,,)   RRR    , we have k,,,,,,,,, k  k  (411) I(,)(,)(,)t I t  f( t ) I t  f ( t ). Proof: Using (394) and applying Fubini’s theorem, we find that kk,,,,,,        (412) I(,)(,)tt I f() t  u  1 1tu  1 11  ( ) (logu ) (log )kk (log ) d f ( u ) du . 2 t  kkk()()  tt u u    t  Using (26), it becomes kk,,,,,,        (413) I(,)(,)tt I f() t   B( ,  ) 1 t  u 1  k ( ) (logu ) (log )k f ( u ) du .  t kkk()()  t u u   t Thus, using (7) and (394), (413) becomes kk,,,,,,        I(,)(,)tt I f() t   11tu  1  ( ) (logu ) (log )k f ( u ) du  t kk ()  t u u   t k,,,      I(,)t  f( t ).

5.2.6 Corollary: For each fixed tuple of positive real numbers, and fixed , we have

k,,,,,,,,,,,, k  k  k  (414) I(,)(,)(,)(,)t I t  f( t ) I t  I t  f ( t ). Proof: The result (414) follows from (411).

5.2.7 Theorem: For each fixed tuple (,,)k  of positive real numbers, t  0 and fixed (,,)   RRR    , we have k,,,,,,,,, k  k  (415) I(,)(,)(,)t   I t  f( t )  I t  f ( t ). Proof: Using (395) and applying Fubini’s theorem, we find that kk,,,,,,        (416) I(,)(,)tt   I  f() t  u  1 1tu  1 11  ( ) (logu ) (log )kk (log ) d f ( u ) du . 2 t  kkk()()  tt t u    t  Using (26), it becomes kk,,,,,,        (417) I(,)(,)tt   I  f() t   B( ,  ) 1 t  u 1  k ( ) (logu ) (log )k f ( u ) du .  t kkk()()  t t u   t Thus, using (7) and (395), (417) becomes kk,,,,,,        I(,)(,)tt   I  f() t   11tu  1  ( ) (logu ) (log )k f ( u ) du  t kk ()  t t u   t k,,,      I(,)t f( t ).

5.2.8 Corollary: For each fixe d tuple of positive real numbers, and fixed , we have k,,,,,,,,,,,, k  k  k  (418) I(,)(,)(,)(,)t   I t  f( t )  I t   I t  f ( t ). Proof: The result (418) follows from (415). 5.2.9 Lemma: For any tuple (,,)k  of positive real numbers and non negative integer j , we have    Bj(,)() j (419) k k k . k 2()()  kk k()() kjk

Proof: The result (419) follows by use of (6), (7), (12) and (16) as:     B(,)()() j    j k k k k 1 2   k ()()  11 kk ()j k2 kkk()() k k kk  ()()j j kk             kkk()()()()   j k  k k kj k  ()() kkjj     . 1  kk kk()() kkj()() k jk k 5.2.10 Lemma: For any tuple (,,)k  of positive real numbers, tu and pair (,) R2 , one has  1 t k t (log ) 11t 11 (420) ( ) (log )kk (log ) d u , 2  1 kk()()()  k u  u  u k  k    where    t (421) 1 1F 1 ; ;(   )log . k k u Proof: Let t  11t 11 (422) Id ( ) (log )kk (log )  . 1 2  kkk()() u  u  u  t log log dt  t If we put y  u , 1y  ,  log dy and  ()y . Thus, (422) t t log log  u uu u u becomes  1 t k (log ) 1  t 11 Iu ( )y (1 y ) kk y dy ,    . 1 2  kukk()()0 Using (20), we can write it as  1 t k (log ) 1   1 t 11 j  Iu ( log )j (1 y ) kk y dy 1 2   kkk()()!j0 j u 0

98 which, by definition of beta function, becomes  t 1 (log ) k  1 t  u j . I1 2  ( log ) B ( , j ) kkk()()!j0 j u k k Using (419), it becomes  1 ttk  j (log ) ( )j ( log ) I  u k u , 1 kj()!  k j0 () k j implying, by use of (19), that  t 1 (log ) k u    t IF1 1 1 ; ; log kk ()  k k u which leads to (420). 5.2.11 Lemma: For any tuple (,,)k  of positive real numbers, ut and pair (,) R2 , one has  1 u k u (log ) 11uu 11 (423) ( ) (log )kk (log ) d t , 2  2 kk()()()  k t   t  k  k    where    u (424) 2 1F 1 ; ;(   )log . k k t Proof: Let u  11uu 11 (425) Id ( ) (log )kk (log )  . 2 2  ktkk()() t     u log log du uu If we put y  t , 1y  ,  log dy and  ()1 y . Thus, (425) u u log log  t  t t t becomes  1 u k (log ) 1  u 11 It ( )(1y ) (1 y ) kk y dy ,    . 2 2  ktkk()()0 Using (20), we can write it as  1 u k (log ) 1   1 u 11 j  It ( log )j ykk (1 y ) dy 2 2   kkk()()!j0 j t 0

99 which, by definition of beta function, becomes  u 1 (log ) k  1 u  t j . I2 2  ( log ) B ( , j ) kkk()()!j0 j t k k Using (419), it becomes  1 uuk  j (log ) ( )j ( log ) I  t k t , 2 kj()!  k j0 () k j implying, by use of (19), that  u 1 (log ) k t    u IF2 1 1 ; ; log kk ()  k k t which leads to (423). 5.2.12 Theorem: For any fixed tuple (,,,)ka of positive real numbers, ta and fixed (,,)    R3 , one has k,,,0 k ,,,0   k ,,,,  (426) HI(,)(,)(,) a t H I a t f()() t HC I a t f t . Proof: Using (392), we find, by use of Fubini’s theorem, that kk, ,  ,0  ,  ,  ,0  HI(,)(,) a t H I a t f() t tt 1 u t11  d  du  (logt ) ( )  ( )   (log )kk (log ) f ( u ) . 2 u  kkk()() au t u  u  u By use of (420) and (421), it becomes

t  11 ut1  (logt ) ( ) (log )k ( ) f ( u ) du  u 1 kk ()a u t u which leads to (426) by use of (396). 5.2.13 Theorem: For any fixed tuple of positive real numbers, and fixed , one has (427) k,,,0I k ,,,0  I  f()() t k ,,,,  I f t . HHHC(,)(,)(,)a t  a t  a t Proof: Using (393), it follows, by use of Fubini’s theorem, that kk, ,  ,0I  ,  ,  ,0 I  f() t HH(,)(,)a t a t

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tt 1 u t11  d  du  (logt ) ( )  ( )   (log )kk (log ) f ( u ) . 2 u  kkk()() au t u  u  t By use of (420) and (421), it becomes kk, ,  ,0I  ,  ,  ,0 I  f() t HH(,)(,)a t a t t  11 ut1  (logt ) ( ) (log )k ( ) f ( u ) du  u 1 kk ()a t t u which leads to (427) by use of (397). 5.2.14 Theorem: For any fixed tuple (,,)k  of positive real numbers, t  0 and fixed (,,)    R3 , one has k,,,0 k ,,,0  k ,,,,  (428) I(,)(,)(,)t I t  f()() t C I t  f t . Proof: Using Fubini’s theorem, we find, by use of (394), that kk, ,  ,0  ,  ,  ,0  I(,)(,)tt I f() t  u  1 t u11 u d du  (logu ) ( )  ( )   (log )kk (log ) f ( u ) . 2 t  kkk()() tt u  t   u By use of (423) and (424), it becomes kk, ,  ,0  ,  ,  ,0  I(,)(,)tt I f() t   11 tu1  (logu ) ( ) (log )k ( ) f ( u ) du  t 2 kk ()t u u t which leads to (428) by use of (398). 5.2.15 Theorem: For any fixed tuple (,,)k  of positive real numbers, t  0 and fixed , one has k,,,0 k ,,,0  k ,,,,  (429) *(,)*(,)*(,)It I t  f()() t C I t  f t . Proof: Using Fubini’s theorem, we find, by use of (395), that kk, ,  ,0  ,  ,  ,0  *(,)*(,)Itt I f() t  u  1 t u11 u d du  (logu ) ( )  ( )   (log )kk (log ) f ( u ) . 2 t  kkk()() tt u  t   t By use of (423) and (424), it becomes kk, ,  ,0  ,  ,  ,0  *(,)*(,)Itt I f() t   11 tu1  (logu ) ( ) (log )k ( ) f ( u ) du  t 2 kk ()t t u t which leads to (429) by use of (399).

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5.2.16 Lemma: For any tuple (,,)k  of positive real numbers and   D , one has t     ()()()()tt1    1     1 (430) (log )k (log ) kd (log ) k kB (  ,  ), tu ,  k u ()()()()  uu    u     ()()()() 1 uu  1     1 (431) (log )k (log ) kd (log ) k kB (  ,  ), ut .  k t ()()()()tt      Proof: Let t  ()()()t 11     (432) Id (log )kk (log )  . k  u ()()()  u   () ()t log log ()u () ()()   t If we put y  , 1y , d  log dy and y :0 1. ()t ()t log log ()()   u ()u ()u Thus, (432) becomes  1     ()()tt1  1  1   1   I(log )k (1  y ) k y k dy  (log ) k B ( , ) . k  ()()u0 u k k By use of (7), it leads to the relation (430).

The relation (431) is, in fact, another form of (430), which can be obtained () log ()t either by substituting y  or directly by ut and  in (430). ()u log ()t 5.2.17 Theorem: For each fixed tuple (,,,)ka of positive real numbers, ta ,   D and fixed (,,)   RRR    , we have k,,,,,,,,,,,, k  k  (433) HI(,)(,)(,) a t H I a t f( t ) H I a t f ( t ). Proof: Using (400), we find by applying Fubini’s theorem, that kk,,,,,,,,          (434) HI(,)(,) a t H I a t f() t 1t uu  ( )  ( ) (log)()()t I f u du , 2  uk kkk()()()  a t    u where t  ()()()t 11     Id (log )kk (log )  . k  ()()()  u   u

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Using (430), it becomes kk,,,,,,,,          (435) HI(,)(,) a t H I a t f() t t  B(,)()()  u   t1  u  k (logt ) ( ) (log )k f ( u ) du .  u kkk()()()()  a t    u  u Thus, using (7) and (400), (435) becomes kk,,,,,,,,          HI(,)(,) a t H I a t f() t t  1u   ( t )1  ( u )  ( ) (logt ) (log )k f ( u ) du  u kk ()()()  a t    u  u k,,,,       HI(,) a t f( t ). 5.2.18 Corollary: For each fixed tuple (,,,)ka of positive real numbers, ta ,   D and fixed (,,)   RRR    , we have k,,,,,,,,,,,,,,,, k  k  k  (436) HatI(,)(,)(,)(,) Hat I f( t ) Hat I Hat I f ( t ). Proof: The result (436) follows from (433). 5.2.19 Theorem: For each fixed tuple (,,)k  of positive real numbers, t  0, and fixed , we have k,,,,,,,,,,,, k  k  (437) I(,)(,)(,)t I t  f( t ) I t  f ( t ). Proof: Using (401) and applying Fubini’s theorem, we find that kk,,,,,,,,          (438) I(,)(,)tt I f() t 1 tu  ( )  ( ) (log)()()u I f u du , 2  tk kkk()()()  t u    u where u  ()()() 11 u   Id (log )kk (log )  . k  ()()()t     t Using (431), it becomes kk,,,,,,,,          (439) I(,)(,)tt I f() t   B(,)()()  t   u1  u  k (logu ) ( ) (log )k f ( u ) du .  t kkk()()()()  t u    t  u Thus, using (7) and (401), (439) becomes kk,,,,,,,,          I(,)(,)tt I f() t   1t   ( u )1  ( u )  (logu ) ( ) (log )k f ( u ) du  t kk ()()()  t u    t  u

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k,,,,       I(,)t  f( t ).

5.2.20 Corollary: For each fixed tuple (,,)k  of positive real numbers, t  0,   D and fixed (,,)   RRR    , we have k,,,,,,,,,,,,,,,, k  k  k  (440) I(,)(,)(,)(,)t I t  f( t ) I t  I t  f ( t ). Proof: The result (440) follows from (437).

5.3 OTHER EXTENDED FRACTIONAL INTEGRALS In this section, we introduce more extension of fractional integrals on the basis of the operators defined in (400), (401) and discus their properties. For any tuple (,,)ka of positive real numbers, ta ,   R , (,,)    R3 and any pair (,)(,) CD, we define kk,,,,,,,,,    (441) ()(,)(,) HII a t H a t , and for any pair (,)k  of positive real numbers, t  0, , and any pair , we define kk,,,,,,,,,    (442) ()(,)(,) IItt. 5.3.1 Theorem: For each fixed tuple of positive real numbers, , , (,,,)     R4 and fixed pair , we have kk,,,,,,,,,,    (443) ()(,)()(,)HI a t f()() t   H I a t f t . Proof: The relation (443) follows immediately from (441) and (400) as k,,,,,       ()(,) HI a t  f() t  t  ()()()tt   1    (logt ) ( )  (log )k    (  ) f (  ) d    ktk ()()()a      k,,,,,        ()(,) HI a t f() t . 5.3.2 Theorem: For each fixed pair (,)k  of positive real numbers, t  0, , (,,,)     R4 and fixed pair , we have kk,,,,,,,,,,    (444) ()(,)()(,)Itt f()() t    I  f t . Proof: The relation (444) follows immediately from (442) and (401) as

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k,,,,,       ()(,) It   f() t    ()()()tt   1    (log ) ( )  (log )k    (  )fd (  )   t ktk ()()()t      k,,,,,        ()(,)It f() t . 5.3.3 Theorem: For each fixed tuple (,,,)ka of positive real numbers, ta ,   R , (,,)    R3 and fixed pair (,)(,) CD, we have k,,,,,,,,,,,,,,, k  k  (445) ()(,)()(,)()(,) HI a t  H I a t f()() t  H I a t f t . Proof: Using (441), we find by use of Fubini’s theorem that kk,,,,,,,,,,    ()(,)()(,) HI a t  H I a t f() t      ()()tt u   u  (log)(t )   ()()() u I f u du , 2  uk kkk()()()  a t    u where t  ()()()t 11     Id (log )kk (log )  . k  u ()()()  u   Using (430), (7) and (441), it leads to the relation (445). 5.3.4 Corollary: For each fixed tuple (,,,)ka of positive real numbers, , , and fixed pair , we have k,,,,,,,,,,,,,,,,,,,, k  k  k  (446) ()(,)()(,)()(,)()(,) HatI  Hat I f()() t   Hat I  Hat I f t . Proof: The result (446) follows from (445). 5.3.5 Theorem: For each fixed tuple (,,)k  of positive real numbers, t  0, , and fixed pair , we have k,,,,,,,,,,,,,,, k  k  (447) ()(,)()(,)()(,) It   I t  f()() t  I t  f t . Proof: Using (442), we find by use of Fubini’s theorem that kk,,,,,,,,,,    ()(,)()(,) Itt   I  f() t      ()()t t   u  (log)(u )   ()()() u I f u du , 2  tk kkk()()()  t u    u where

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u  ()()() 11 u   Id (log )kk (log )  . k  t ()()()t     Using (431), (7) and (442), it leads to the relation (447).

5.3.6 Corollary: For each fixed tuple (,,)k  of positive real numbers, t  0,   R , (,,)    R3 and fixed pair (,)(,) CD, we have k,,,,,,,,,,,,,,,,,,,, k  k  k  (448) ()(,)()(,)()(,)()(,) It   I t  f()() t    I t   I t  f t . Proof: The result (448) follows from (447).

5.4 MELLIN TRANSFORM OF THE EXTENDED FRACTIONAL INTEGRALS Now, we find the Mellin transform of some of the extended fractional integrals by using the following lemma. 5.4.1 Lemma: If  shows the Mellin convolution, one has (449) Mfgt{ ( )}( s ) MfsMgs { }( ) { }( ). Proof: It follows from (2), with use of (3) and Fubini’s theorem, that    ss11td (450) Mfgts{ ()}()  fgttdt  ()    f ()() g tdt 0 0 0   1 t s1  g()() f t dt d . 00 t If we put  x , dt dx and so the inner integral, with use of (2), becomes  t   f( ) tdts1  fxx ( )( ) s  1  dx   s  fxxdx ( ) s  1   s Mfs { }( ). 0 0 0 Thus, (450), with use of (2), becomes 1 Mfgts{ ( )}( )  g ( ) ss MfsdMfsg { }( )   { }( ) (  ) 1 d  00  M{ f }( s ) M { g }( s ). 5.4.2 Theorem: For each fixed pair (,)k  of positive real numbers, t  0; if s  0, we have   k  k (451) M{Ht I(0, ) f ( t )}( s ) ( ks ) M { f }( s ).

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Proof: If we let   1 (logt ) k  , t  1, ht1()  k ()  k 0, 0t 1 then using (3), we can rewrite (47) with a  0, as  1 t k t (log )  d t d (452) k I f() t f ()  h ()() f  h  f (). t Ht(0, ) 1 1 00kk ()    But, using (2), we have  1 1 Mhs{ }( ) httdt ( )ss11 (log ttdt ) k . 11 11kk () If we substitute yt log , te y and dt ey dy . So, we have     1 1 1 1 M{ h }( s )  yk eys y e y dy  yk eys dy . 1   kk () 0 kk () 0 du Now, if we use u ys with s  0, dy  and thus, using (6), we obtain s   1 u1 du (453) M{ h }( s ) ( ) k eu 1  kk () 0  s  s        1   ()()sk1 s k k k   uk eu du  ()()().   s k   ks k kkk()()0 k k k Hence, applying Mellin transform on (452), by use of (449) and (453), we find that k  M{Ht I(0, ) fts ()}() Mhfts { 1  ()}()  MhsMfs {}() 1 {}()   (ks )k M { f }( s ). 5.4.3 Theorem: For each fixed pair (,)k  of positive real numbers, t  0; if s  0, we have   k  k (454) M{ I(,)t  f ( t )}( s ) ( ks ) M { f }( s ). Proof: If we let   1 1 (log ) k  t ht() , 0t 1, 2  k ()  k 0, t  1

107 then using (3), we can rewrite (402), as  1  k (log ) d t d (455) k I f() tt f ()  h ()() f  h  f (). t (t , ) 2 2 t kk () 0   But, using (2), we have  1  111 Mhs{ }( ) httdt ( )ss11 (log ) k tdt . 22 00ktk () 1 If we substitute y  log , te  y and dt e y dy . So, we have t     1 1 1 1 M{ h }( s )  yk eys  y e  y dy  yk e ys dy . 2   kk () 0 kk () 0 du Now, if we use u ys with s  0, dy  and thus, using (6), we obtain s   1 u1 du (456) M{ h }( s ) ( ) k eu 2  kk () 0 s s        1   sk1 s k k k   uk eu du  ()().  s k  ks k kkk()()0 k k k Hence, applying Mellin transform on (455), by use of (449) and (456), we find that k  MI{(t , ) fts ()}() Mh { 2  fts ()}()  MhsMfs {}() 2 {}()    (ks )k M { f }( s ). 5.4.4 Theorem: For each fixed pair (,)k  of positive real numbers, t  0 and  R ; if s  0, we have   k, ,0,0 k (457) M{Ht I(0, ) f ( t )}( s ) ( k ( s )) M { f }( s ). Proof: If we let   1 tt (log ) k  , t  1, kt1()  k ()  k 0, 0t 1 then using (3), we can rewrite (392) with a  0, as  1 t k t (log )  d  t d  (458) k, ,0,0I  f()() t   f ()  k ()() f  k  f (). t Ht(0, ) 1 1 00tkk ()    But using (2), we have

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 1 1 Mks{ }( ) kttdt ( )ss11 t  (log ttdt ) k  . 11 11kk () If we substitute yt log , te y and dt ey dy . So, we have     1 1 1 1 M{ k }( s )  yk ey() s y e y dy  yk eys() dy . 1   kk () 0 kk () 0 du Now, if we use u  y() s   with s  0, dy  and thus, using (6), we   s obtain   1 u1 du (459) M{ k }( s ) ( ) k eu 1  kk ()0   s   s    ()()sskk1   uk eu du  () kkk()()0 k k  1  k k  ( s )kk  ( k (  s )) . k Hence, applying Mellin transform on (458), by use of (449) and (459), we find that k, ,0,0 M{Ht I(0, ) fts ()}() Mkfts { 1  ()}()  MksMfs {}() 1 {}()   (k ( s ))k M { f }( s ). 5.4.5 Theorem: For each fixed pair (,)k  of positive real numbers, t  0 and  R ; if s  0, we have   k, ,0,0 k (460) M{ I(,)t  f ( t )}( s ) ( k ( s )) M { f }( s ). Proof: If we let  1   1 t (log ) k  t kt() , 0t 1, 2  k ()  k 0, t  1 then using (3), we can rewrite (394), as  1  k t (log )  t d t d (461) k, ,0,0I  f()() t  t f ()  k ()() f  k  f (). t (t , ) 2 2 00kk ()     But using (2), we have

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 1  111 Mkskttdt{ }( ) ( )ss11 (log ) k tdt   . 22 00ktk () 1 If we substitute y  log , te  y and dt e y dy . So, we have t     1 1 1 1 M{ k }( s )  yk ey() s   y e  y dy  yk eys() dy . 2   kk () 0 kk () 0 du Now, if we use u y() s  with s  0, dy  and thus, using (6), we s   obtain   1 u1 du (462) M{ k }( s ) ( ) k eu 2  kk ()0 s   s      ()()sskk1   uk eu du  () kkk()()0 k k  1  k k  (s  )kk  ( k ( s  )) . k Hence, applying Mellin transform on (461), by use of (449) and (462), we find that k, ,0,0 M{ I(t , ) ftsMkftsMksMfs ()}() { 2  ()}()  {}() 2 {}()   (k ( s ))k M { f }( s ).

5.5 BOUNDEDNESS OF THE EXTENDED FRACTIONAL INTEGRAL OPERATORS We have already discussed boundedness of Hadamard k -fractional integral in chapter 2. Now, we discuss the boundedness of some of the extended operators defined in this chapter. 5.5.1 Definition: For any interval [,]ab and real number c ; the weighted Lp -space p Xc (,) a b is defined to be the set of all Lebesgue measurable functions f on (,)ab for which b  1 c p  t f( t ) dt  , 1  p   ,  t (463)  a ess suptc f ( t )  , p   .  a t b

The norms f p , where 1p  , and f  are defined as X c X c

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1 b p 1 c p (464) fp   t f() t dt X c  a t and (465) f ess sup tc f ( t ) X  c a t b respectively. 5.5.2 Theorem: For each fixed tuple (,,,)k a b of positive real numbers, ba , ta , 2 k, ,0,0 (,)cR  ,   c and fixed 1p  ; the operator HI(,) a t is bounded in p Xc (,) a b and (466) k, ,0,0I f() t K f , H(,) a tp b X p Xc c where  1 b k (467) Kb  (log ) k () a for   c, while   () cbk (468) Kcb ( ,( )log ) kk () k a for   c. Proof: Case-1: Firstly, we consider the case 1p   . Using (392), we find, by (464), that 1 b p k, ,0,0  c k ,  ,0,0  p dt HI(,)(,) a t f()() tp   t H I a t f t Xc  a t 1 p bt  p 1 t1 d dt  tfc ( ) (log )k ( ) k () t   t k aa 1 p bt1  p 1 c td1  tp ( ) (log )k f ( ) dt . kt ()   k aa t Substituting x  , it becomes  1 t p p b 1 a  c 1 k, ,0,0 1 p   k t dx HI(,) a t f( t )p   t x (log x ) f ( ) dt X c k ()  x x k a 1 

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1 t p p b 1 a  1 c 1 t  tp x 1(log x )k f ( ) dx dt . kx ()  k a 1  1 c p p p Since f Xc (,) a b , it follows from (463) that t f()(,) t L a b , so using (29), we find that b 1 p a b  1 p 1 1 c t k, ,0,0   1 k p HIft(,) a t ( )p  x (log xtf ) ( ) dtdx Xc kx ()  k 1 ax b 1 a  b p p 1 1 t  x 11(log x )k  tpc  f ( ) dt dx . kx ()  k 1 ax t Putting y  , we find that x 1 bbp ax 1 k, ,0,0 1 c   1 pc  1 p I f( t ) x (log x )k y f ( y ) dy dx H(,) a t X p  c k ()  k 1 a  1 bbp ax 1 11cc 1 p  x(log x )k y f ( y ) dy dx ky ()  k 1 a    Kfb p , Xc where b a  1 1 (469) K xc  1(log x )k dx . b  kk () 1  Now if   c, (467) is obvious from (469), that is, KKbb of (467). While if   c, putting log xt and then z t() c , we find (468) from (469) with use  of (15) , that is, KKbb of (468). Case-2: Now, we consider the case p . Using (392), we find that t  1 td1 tc k, ,0,0 I  f() t ()(log)   ck  c f () . H(,) a t  ktk ()a   t Substituting u  , it gives  ck, ,0,0  (470) tH I(,) a t f() t K t f  , Xc

112 where t a  1 1 K uc  1(log u ) k du t  kk () 1  which has already been discussed with b for t . So, if   c, KKKt t b of  (467) as (,)x is an increasing function. Likewise if   c, KKKt t b of (468). Thus, (470) leads to ck, ,0,0 tH I(,) a t f() t K b f  , X c implying by (465) that (466) holds for this case (i.e. p ) too. This completes the proof. 5.5.3 Corollary: For each fixed tuple (,,,)k a b of positive real numbers, ba , ta , 1   R ,   and fixed 1p  ; the operator k, ,0,0I is bounded in Lp (,) a b p H(,) a t and k, ,0,0 (471) I f() t K f p , H(,) a tLp b L where  1 b k (472) Kb  (log ) k () a 1 for   , while p  1  ()  k p  1 b (473) Kb ( ,( )log ) kk () k p a 1 for   . p Proof: 1 Replacing c by , the result (471) follows from (466). p 5.5.4 Corollary: For each fixed tuple of positive real numbers, , , k  c  0 and fixed ; the Hadamard k -fractional intergral operator HI(,) a t p is bounded in Xc (,) a b and k  (474) I f() t K f p , H(,) a t p X Xc c where

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1 b  (475) K  (log ) k k () a for c  0, while   ()cbk  (476) Kc ( , log ) kk () k a for c  0. Proof: kk,0,0,0  Since HII(,)(,) a t H a t , the result (474) follow from (466) by using   0.

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CHAPTER 6 the extended FD operators

We have already introduced a number of extended fractinoal intergral operators in chapter 1 and chapter 5. In this chapter, we introduce extended fractional derivatives related to the fractional integral operators. The extended derivative oparators are based on k -gamma function and the operators t D and

Dt defined in chapter 2. We discuss the properties of the extended derivatives, particularly, their composition with the related fractional integral operators. We also explore, how fractional derivative becomes reciprocal of the related fractional integral.

6.1 FRACTIONAL DERIVATIVE RELATED TO HADAMARD k-FRACTIONAL INTEGRAL In this section, we introduce extended fractional derivative related to Hadamard k -fractional integral and discus the composition of the operators. 6.1.1 Definition: For any tuple (,,)ka of positive real numbers, ta , we define Hadamard k fractional derivative, of order  , of a function fx() as t  11t n1 (477) knD f() t ()(log) Dk f () d , H(,) a t1n t  kk () nk a    where n 1. k 6.1.2 Theorem: For each fixed tuple (,,,)ka of positive real numbers, ta and  , we have k k  k   (478) HD(,)(,)(,) a t H I a t f()() t H I a t f t . Proof: Using (47) and (477), we find, by Fubini’s theorem, that 11t (479) kD k I f()()()() t D n I f u du H(,)(,) a t H a t2n t k kkk()()  nk  a u where t  1 t n 11  (480) Id (log )kk (log ) . k  u  u Using (25), (480) becomes

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 t n1 (481) I kB( nk  , ) (log ) k . kk u Using (7) and (481), (479) becomes t  11t n1 kD k I f() t ()(log) D n k f () u du . H(,)(,) a t H a t1n t  kk ()   nka u u

Applying the operator t D once, it becomes kk HD(,)(,) a t H I a t f() t  (n  1) t  1 t n2  k (D )n1 (log )k f ( u ) du . 1n t  kk ()   nka u u Repeated application of the operator leads to the relation

n  k ()t  n 1 t 1 kkD I f( t ) k (log )k f ( u ) du . H(,)(,) a t H a t  kk ()   nka u u Using (18), it leads to the relation t  11t 1 kkD I f( t ) (log )k f ( u ) du H(,)(,) a t H a t  kk ()a u u which leads to the relation (478) by use of (47). 6.1.3 Corollary: For each fixed tuple (,,)ka of positive real numbers, ta , we have kk (482) HD(,)(,) a t H I a t f()() t f t . Proof: Replacing  by  , the result (482) follows from (478).

6.2 FRACTIONAL DERIVATIVES RELATED TO OTHER EXTENDED FRACTIONAL INTEGRALS In this section, we introduce extended fractional derivatives related to some of the fractional integrals introduced in chapter 5 and discus the compositions of the operators. 6.2.1 Definition: For any tuple (,,)ka of positive real numbers, ta and  k,,,    (,,)   RRR   , we define HD(,) a t f() t as k,,,,,, n   n    k nk  (483) HDftktt(,)(,) a t() (log)(  )( t D )(log)( tt  ) H Ift a t (),  where n 1. k

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6.2.2 Theorem: For each fixed tuple (,,,)ka of positive real numbers, ta ,  , and (,,)   RRR    , we have k,,,,,,,,, k  k  (484) HD(,)(,)(,) a t H I a t f()() t H I a t f t . Proof: Replacing  by nk , (483) becomes k,,,   nk n   n   HDftk(,) a t() (log)( tt  )( t D )(log)( tt  ) ft () which implies, by use of (392), that k,,,,,,   nk k     HD(,)(,) a t H I a t f() t n t  kt1 1 (logt ) ( t  )  ( D )n (log  )  (    )  (log )k f (  ) d  . t  kk ()a  

Applying the operator t D once, it becomes k,,,,,,   nk k     HD(,)(,) a t H I a t f() t

n  k ( 1) t  1 t 2 k (logt ) ( t  )  ( D )n 1 (log  )   (    )  (log )k f (  ) d  . t  kk ()a   Repeated application of the operator leads to the relation k,,,,,,   nk k     HD(,)(,) a t H I a t f() t

n  kn() t  n 1 t n 1 k (logt ) ( t  )  (log  )  (    )  (log )k f (  ) d  . kk ()a   Using (17), it becomes k,,,,,,   nk k     (485) HD(,)(,) a t H I a t f() t t  nk 11  t 1  (logt ) ( ) (log )k f ( ) d   kk ()  nka  t    k,,,     nk  HI(,) a t f() t . If nk1   nk , using (403) and (485), we find that k,,,,,,,,,,,,,,, k  k  nk k  nk  k  HatD(,)(,)(,)(,)(,) Hat Ift()()() Hat D Hat Ift Hat Ift k,,,,,,   nk k    nk    HD(,)(,) a t H I a t f() t k,,,      HI(,) a t f( t ). Thus (484) is proved. 6.2.3 Corollary: For each fixed tuple (,,)ka of positive real numbers, , and , we have kk,,,,,,        (486) HD(,)(,) a t H I a t f()() t f t .

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Proof: Replacing  by  , the result (486) follows from (484). 6.2.4 Definition: For any tuple (,,)ka of positive real numbers, ta and (,,)   RRR    , we define k,,,  D  f() t as H * (,)at (487) k,,,,,,Dftktt() n (log)(   )()(log)(  D n tt    )  k Ift nk  (), HH**(,)(,)a t t a t  where n 1. k 6.2.5 Theorem: For each fixed tuple (,,,)ka of positive real numbers, ta ,  , and , we have (488) k,,,,,,,,,D k  I f()() t k  I f t . HHH***(,)(,)(,)a t a t a t Proof: Replacing  by nk , (487) becomes k,,,  Dftk nk() n (log)( tt   )(  D )(log)( n tt   )  ft () H * (,)a t t which implies, by use of (393), that k,,,,,,  D nk k    I  f() t HH**(,)(,)a t a t n t  kt1 1 (logt ) ( t  )  ( D )n (log  )  (    )  (log )k f (  ) d  . t  ktk ()a

Applying the operator Dt once, it becomes k,,,,,,  D nk k    I  f() t HH**(,)(,)a t a t

n  k ( 1) t  1 t 2 k (logt ) ( t  )  ( D )n 1 (log  )   (    )  (log )k f (  ) d  . t  ktk ()a Repeated application of the operator leads to the relation k,,,,,,  D nk k    I  f() t HH**(,)(,)a t a t

n  kn() t  n 1 t n 1 k (logt ) ( t  )  (log  )  (    )  (log )k f (  ) d  . ktk ()a Using (17), it becomes (489) k,,,,,,  D nk k    I  f() t HH**(,)(,)a t a t t  nk 11  t 1  (logt ) ( ) (log )k f ( ) d   kk ()  nka t t     k,,,  I   nk f() t . H * (,)at If nk1   nk , using (407) and (489), we find that

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k,,,,,,,,,,,,,,,D k  Ift()()() k  D nk k  Ift nk  k  Ift HHHHH*****(,)(,)(,)(,)(,)a t a t a t a t a t  k,,,,,,  D nk k    I nk   f() t HH**(,)(,)a t a t  k,,,  I   f( t ). H * (,)at Thus (488) is proved. 6.2.6 Corollary: For each fixed tuple (,,)ka of positive real numbers, ta , and (,,)   RRR    , we have (490) kk,,,,,,  D     I  f()() t f t . HH(,)(,)a t a t Proof: Replacing  by  , the result (490) follows from (488). 6.2.7 Definition: For any pair (,)k  of positive real numbers, t  0 and k,,,    , we define D(,)t  f() t as k,,,,,, n   n    k nk  (491) Dftktt(,)(,)t() (log)(  )(  t D )(log)( tt  ) Ift t () ,  where n 1. k 6.2.8 Theorem: For each fixed tuple (,,)k  of positive real numbers, t  0,  , and , we have k,,,,,,,,, k  k  (492) D(,)(,)(,)t I t  f()() t I t  f t . Proof: Replacing  by nk , (491) becomes k,,,   nk n   n   Dftktt(,)tt () (log)(  )(  D )(log)( tt  ) ft () which implies, by use of (394), that k,,,,,,   nk k     D(,)(,)tt I f() t n   k 1  1 (logt ) ( t  )  (  D )n (log  )  (    )  (log )k f (  ) d  . t  ktk ()t

Applying the operator () t D once, it becomes k,,,,,,   nk k     D(,)(,)tt I f() t  k n ( 1) k n 1 (log)(t t  )(  t D ) () I , kk () where   1  2 I (log ) (   ) (log )k f (  ) d  . t  t

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Repeated application of the operator leads to the relation k,,,,,,   nk k     D(,)(,)tt I f() t

n  kn()   n 1  n 1 k (logt ) ( t  )  (log  )  (    )  (log )k f (  ) d  . ktk ()t Using (17), it becomes k,,,,,,   nk k     (493) D(,)(,)tt I f() t   nk 11 t  1  (log ) ( ) (log )k fd (  )   t kk ()  nkt     t k,,,     nk  I(,)t  f() t . If nk1   nk , using (411) and (493), we find that k,,,,,,,,,,,,,,, k  k  nk k  nk  k  D(,)(,)(,)(,)(,)t Ift t ()()() D t  Ift t  Ift t  k,,,,,,   nk k    nk    D(,)(,)tt I f() t k,,,      I(,)t  f( t ). 6.2.9 Corollary: For each fixed pair (,)k  of positive real numbers, t  0, and (,,)   RRR    , we have kk,,,,,,        (494) D(,)(,)tt I f()() t f t . Proof: Replacing  by  , the result (494) follows from (492). 6.2.10 Definition: For any pair (,)k  of positive real numbers, t  0 and k,,,    , we define D(,)t f() t as k,,,,,, n   n    k nk  (495) Dftktt(,)(,)t () (log)(  )(  D t )(log)( tt  )  Ift t  (),  where n 1. k 6.2.11 Theorem: For each fixed tuple (,,)k  of positive real numbers, t  0,  , and , we have k,,,,,,,,, k  k  (496) D(,)(,)(,)t   I t  f()() t  I t  f t . Proof: Replacing  by nk , (495) becomes k,,,   nk n   n   Dftktt(,)tt() (log)(  )(  D )(log)( tt  ) ft () which implies, by use of (395), that

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k,,,,,,   nk k     D(,)(,)tt   I  f() t n   k 1  1 (logt ) ( t  )  (  D )n (log  )  (    )  (log )k f (  ) d  . t  ktk ()t

Applying the operator ()Dt once, it becomes k,,,,,,   nk k     D(,)(,)tt   I  f() t  k n ( 1) k n 1 (log)(t t  )(  Dt ) () I , kk () where   1  2 I (log ) (   ) (log )k f (  ) d  . t  t Repeated application of the operator leads to the relation k,,,,,,   nk k     D(,)(,)tt   I  f() t

n  kn()   n 1  n 1 k (logt ) ( t  )  (log  )  (    )  (log )k f (  ) d  . ktk ()t Using (17), it becomes k,,,,,,   nk k     (497) D(,)(,)tt   I  f() t   nk 11 t  1  (log ) ( ) (log )k fd (  )   t kk ()  nkt     t k,,,     nk  I(,)t f() t . If nk1   nk , using (415) and (497), we find that k,,,,,,,,,,,,,,, k  k  nk k  nk  k  D(,)(,)(,)(,)(,)t   Ift t ()()()  D t   Ift t   Ift t  k,,,,,,   nk k    nk    D(,)(,)tt   I  f() t k,,,      I(,)t f( t ). 6.2.12 Corollary: For each fixed pair (,)k  of positive real numbers, t  0, and (,,)   RRR    , we have kk,,,,,,        (498) D(,)(,)tt   I  f()() t f t . Proof: Replacing  by  , the result (498) follows from (496).

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CHAPTER 7 Application of the extended FC operators

In this chapter, we find solutions of a number of intergral equations using some of the extended fractional integral operators defined and discussed in the previous chapters. We also illustrate, with an example, the use of some of the extended fractional calculus operators to find solutions of fractional differential equations.

7.1 SOLUTIONS OF INTEGRAL EQUATIONS USING HADAMARD k-FRACTIONAL INTEGRAL OPERATOR In this section, we present some fractional integral equations and find the solutions of the integral equations using Hadamard k -fractional integral k  operator HtI(0, ) . We want to find the function f for which the equation  1 t k t (log ) 1  (499)  F( ;  ;log)() f  d   g () t ,   0, t  0,  11kk 0 k () t is satisfied. To do so, firstly, we prove the following results. 7.1.1 Lemma: For any tuple (,,)k  of positive real numbers and  R , we have  11 1 (1 y ) kk y k (500) F( ;  ; zy ) dy F (  ; (   ); z ).  1 1k k 1 1 k k 0 k()()()  k   k    Proof: Let  11 1 (1 yy ) kk (501) I F(;;) zy dy .  11kk 0 kk()() Using (21), we can write it as 1  r 1 ( ) 11 r  z I kr(1 y ) kk y dy . k()()()! k r0 k  r 0 r It follows, by use of (5), that  r kz()kr I Bk (,) kr . k()()()! k r0 k  r r Now using (7) and (10), we find that kz () r I   kr k(   )r0 ( k (    )) r r !

122 which, with use of (21), leads to (500). 7.1.2 Corollary: For any tuple (,,)k  of positive real numbers, tu and  R , we have

 11 t kk t (log ) (log ) 1 u (502)  u Fd( ;  ;log )   11kk u kk()()     t 1 k(log ) k u u 11F (kk ; (   );log ) . k () t Proof:  log u Substituting y  u and z  log , the result (502) follows from (500) t log t u immediately. 7.1.3 Lemma: For any pair (,)k  of positive real numbers and ; if  1 t k t (log ) d (503) F(,)()  f t  F (;;log)()   f  , t  0, k k 11 k k 0 k ()t then for each   0, one has k  (504) HI(0, t ) F(, k k  )() f t F (,( k  k   ))() f t . Proof: Using (503), it follows by (47) that k  HI(0, t ) F(,)() k k f t  11 t kk t (log ) (log ) u du d   u F( ; ;log ) f ( u ) 11kk 00kukk()()    which implies, by Fubini’s theorem, that k  HI(0, t ) F(,)() k k f t tt  1 t11 u d du  f( u ) (log )kk (log ) F ( ; ;log ) . 11kk kkk()() 0 u  u   u Using (502), it becomes k  HI(0, t ) F(,)() k k f t t  1 t1 u du (log )k F ( ; (   );log ) f ( u ) .  11kk k ()0 u t u

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Hence, by (503), it leads to (504). Now, we are well in position to find the solution of the integral equation (499). We find the solution, by using (504) in the following theorem. 7.1.4 Theorem: The solution of the integral equation (499) is given by 1 (505) f()() t k I t k I  k I  g t . kt H(0,) t H (0,) t H (0,) t Proof: Using (503), we can write (499) as

F(,)()()kk f t g t , implying that kk HI(0, t ) F(,)()() k k f t H I (0, t ) g t . Using (504), it becomes k  F(k , k (   )) f ( t ) H I(0, t ) g ( t ) which, by (504), gives kk (506) HI(0, t ) F(,)()() k k f t H I (0, t ) g t . But k (507) F( , ) f ( t ) k I ( tf ( t )). k kt H(0, t ) Thus, (506) becomes k kI k I ( tf ( t )) k I  g ( t ) H(0,) tt H (0,) t H (0,) t which immediately leads to (505). 7.1.5 Theorem: The solution of the integral equation  1 t k t (log ) 1 t (508)  F( ;  ;log)() f  d   g () t ,   0, t  0,  11kk 0 k ()   is given by 1 (509) f()() t k I  t k I   k I    t 1 g t . k H(0,) t H (0,) t H (0,) t Proof: t Using (22) for z  log , (508) can be written as   1 t k t (log ) 1 t   F( (  );  ;log ) f (  ) d  g ( t )  11kk 0 k ()   t

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1 1 which is (499) with ft(), gt() and  instead of ft(), gt() and  t t respectively. Hence, by (505), we obtain (509).

7.2 SOLUTIONS OF INTEGRAL EQUATIONS USING ANOTHER EXTENDED FI OPERATOR In this section, we present some fractional integral equations and find the solutions of the integral equations using another extended fractional integral k  operator I(,)t  . We seek for the function f for which the equation  1  k  (log ) 1 t (510) t F( ;  ;log ) f (  ) d   h ( t ),   0, t  0,  11kk t k ()   is satisfied. 7.2.1 Lemma: For any tuple (,,)k  of positive real numbers, ut and  R , one has  11  kku u (log ) (log ) 1  (511) t  Fd( ;  ;log )   11kk t kk()()   u  u 1 k(log ) k t t 11F (kk ; (   );log ) . k () u Proof: u log t The result (511) follows from (500) by substituting y   and z  log . u log u t 7.2.2 Lemma: For any pair (,)k  of positive real numbers and ; if  1  k  (log ) td (512) G(,)()  f t t F (;;log)()   f  , t  0, k k 11 k k t k ()   then for each   0, we have k  (513) I(,)t G(, k k  )() f t G (,( k  k   ))() f t . Proof: Using (512), it follows by (402) that k  I(,)t G(,)() k k f t

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 11  kku (log ) (log ) du d  t  F( ; ;log ) f ( u ) 11kk t kkk()()  u u  which implies, by Fubini’s theorem, that k  I(,)t G(,)() k k f t  u  1 11u  d  du  f( u ) (log )kk (log ) F ( ; ;log ) . 11kk kkk()() tt t  u  u Using (511), it becomes

  1 u1 t du (log )k F ( ; (   );log ) f ( u ) .  11kk k ()t t u u Hence, by (512), it leads to (513). Now, we are well in position to find the solution of the integral equation (510). We find the solution, by using (513) in the following theorem. 7.2.3 Theorem: The solution of the integral equation (510) is given by t 1 (514) f()() t k I k I  k I  h t . kt(,)(,)(,)t t  t  Proof: Using (512), we can write (510) as

G(,)()()kk f t h t , implying that kk I(,)(,)t G(,)()() k k f t I t h t . Using (513), it becomes k  G(k , k (   )) f ( t ) I(,) t  h ( t ) which, by (513), gives kk (515) I(,)(,)t G(,)()() k k f t I t h t . But 1 (516) G( , ) f ( t ) ktk I ( f ( t )) . k k(,) t  t Thus, (515) becomes 1 kI kt k I ( f ( t )) k I  h ( t ) (,)(,)(,)t t t t  which immediately leads to (514). 7.2.4 Theorem: The solution of the integral equation

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 1  k  (log ) 1  (517) t F( ;  ;log ) f (  ) d   h ( t ),   0, t  0,  11kk t k () t is given by 11 (518) f()() t k I  k I   k I    th t . kt(,)(,)(,)t t  t  Proof:  Using (22) for z  log , (517) can be written as t  1  k  (log ) 1  t t F((  );  ;log)() f  d  h () t  11kk t k () t  which is (510) with tf() t , th() t and  instead of ft(), ht() and  respectively. Hence, by (514), we obtain (518).

7.3 SOLUTIONS OF FRACTIONAL DIFFERENTIAL EQUATIONS VIA EXTENDED FC OPERATORS There are many real life problems which can not be described with the classical integer order differential equations, while fractional differential equations provide the models to represent the situations properly. So, fractional differential equations has become an important filed of investigation due to its extensive applications in various areas of physics, statistics, economics and engineering. As, fractional differential equations is not focus of this work, we just introduce a simple form of fractional differential equation and find its solution using the extended fractional calculus operators defined and discussed in this work. 7.3.1 Example: Let for some k  0, kk 2 and h:[1, e ] R be a continuous function. Then the unique solution of the problem k   HtD(1, ) f( t ) h ( t ), 1  t  e , (519)  k   f(1) 0, f ( e ) H I(1, ) f ( ),  0, is given by kk    I h()()  I h e 1 k  H(1, ) H (1, e ) k (520) f( t )Ht I(1, ) h ( t )2 (log t ) .  () 1 1 k (log ) k k (2 ) Solution: Applying Hadamard k -fractional integral on the fractional differential equation of (519), we find that there exist c12, c R such that

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 12 k  kk (521) f( t )Ht I(1, ) h ( t )  c 1 (log t )  c 2 (log t ) . Since kk 2 and f (1) 0, we find that

(522) c2  0. Thus, (521) becomes  1 k  k (523) f( t )Ht I(1, ) h ( t ) c 1 (log t ) . k  Since f()() e H I(1, ) f  , it follows from (523) that  1 k k  k  k  k HI(1, e ) h() e c 1  H I (1, ) f ()  H I (1,  ) H I (1,  ) h ()(log)   c 1  .  Using (50) and (68), it becomes 2 1 kk   k () k HI(1, e ) h( e ) c 1  H I (1, ) h ( )  c 1 (log ) , k (2 ) implying that kk   HI(1, ) h()()  H I (1, e ) h e (524) c1  2 .  () 1 1 k (log ) k k (2 ) Using (524), (523) leads to (520).

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CONCLUSION We have introduced numerous extension of Hadamard and Hadamard type fractional integrals and fractional derivatives including the extended k  integral HI(,) a t called Hadamard k -fractional integral. The extensions are based on -gamma function. We have discussed a number of properties involving the commutative law, semigroup property and boundedness of the extended integral operator. We have also proved many fractional integral inequalities for the extended integral operator . The work contains the fractional integral inequalities for the Chebyshev functional, Gruss type inequalities, the fractional integral inequalities of bounded and monotonic functions, and the like, using the integral operator . To generate the fractional integral inequalities, we have used many base inequalities involving AM-GM inequalities, Holder’s inequalities and rearrangement inequalities.

Moreover, we have discussed the Mellin transforms and boundedness of some of the extended integral operators. We have also discussed the properties of the extended derivatives, involving their composition with the related fractional integral operators. Finally, we have introduced some integral equations and have found their solutions using some of the extended fractional integral operators. The use of some of the extended fractional calculus operators in finding solutions of fractional differential equations is also presented.

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Recently published papers of the author 1. Farid, G., & Habibullah, G. M. (2014). An Extension of Hermite Polynomials. Int. J. Contemp. Math. Sciences, 9(10), 455-459. 2. Farid, G., & Habibullah, G. M. (2014). Extensions of Legendre Polynomials. International Journal of Contemporary Mathematical Sciences, 9(11), 545-551. 3. Farid, G., Habibullah, G. M., & Shahzeen, S. (2014). A Generalization of Hermite Polynomials. International Journal of Contemporary Mathematical Sciences, 9(13), 615-623. 4. Farid, G., Habibullah, G. M., & Shahzeen, S. (2014). Some Inequalities Related to AM-GM, Hölder and Rearrangement Inequalities. International Journal of Mathematical Analysis, 8(60), 2959-2969. 5. Farid, G. & Habibullah, G. M. (2015). An Extension of Hadamard Fractional Integral. International Journal of Mathematical Analysis, 9(10), 471-482.

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