A Generalization on Weighted Means and Convex Functions with Respect to the Non-Newtonian Calculus

Home , Weighted arithmetic mean, Weighted geometric mean

Hindawi Publishing Corporation International Journal of Analysis Volume 2016, Article ID 5416751, 9 pages http://dx.doi.org/10.1155/2016/5416751

Research Article A Generalization on Weighted Means and Convex Functions with respect to the Non-Newtonian Calculus

ULur Kadak1 and Yusuf Gürefe2

1 Department of Mathematics, Bozok University, 66100 Yozgat, Turkey 2Department of Econometrics, Us¸ak University, 64300 Us¸ak, Turkey

Correspondence should be addressed to Ugur˘ Kadak; [email protected]

Received 22 July 2016; Accepted 19 September 2016

Academic Editor: Julien Salomon

Copyright © 2016 U. Kadak and Y. Gurefe.¨ This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper is devoted to investigating some characteristic features of weighted means and convex functions in terms of the non- Newtonian calculus which is a self-contained system independent of any other system of calculus. It is shown that there are infinitely many such useful types of weighted means and convex functions depending on the choice of generating functions. Moreover, some relations between classical weighted mean and its non-Newtonian version are compared and discussed in a table. Also, some geometric interpretations of convex functions are presented with respect to the non-Newtonian slope. Finally, using multiplicative continuous convex functions we give an application.

1. Introduction some cases, for example, for wage-rate (in dollars, euro, etc.) related problems, the use of bigeometric calculus which is It is well known that the theory of convex functions and a kind of non-Newtonian calculus is advocated instead of a weighted means plays a very important role in mathematics traditional Newtonian one. and other fields. There is wide literature covering this topic Many authors have extensively developed the notion of (see, e.g., [1–8]). Nowadays the study of convex functions multiplicative calculus; see [12–14] for details. Also some has evolved into a larger theory about functions which are authorshavealsoworkedontheclassicalsequencespaces adapted to other geometries of the domain and/or obey and related topics by using non-Newtonian calculus [15–17]. otherlawsofcomparisonofmeans.Alsothestudyofconvex Furthermore, Kadak et al. [18, 19] characterized the classes of functions begins in the context of real-valued functions of a matrix transformations between certain sequence spaces over real variable. More important, they will serve as a model for the non-Newtonian complex field and generalized Runge- deep generalizations into the setting of several variables. Kutta method with respect to the non-Newtonian calculus. As an alternative to the classical calculus, Grossman and For more details, see [20–22]. Katz [9–11] introduced the non-Newtonian calculus consist- The main focus of this work is to extend weighted ing of the branches of geometric, quadratic and harmonic means and convex functions based on various generator + calculus, and so forth. All these calculi can be described functions, that is, exp and 𝑞𝑝 (𝑝 ∈ R ) generators. simultaneously within the framework of a general theory. The rest of this paper is organized as follows: in Sec- They decided to use the adjective non-Newtonian to indicate tion 2, we give some required definitions and consequences any calculi other than the classical calculus. Every property in related with the 𝛼-arithmetic and 𝑞𝑝-arithmetic. Based on classical calculus has an analogue in non-Newtonian calculus two arbitrarily selected generators 𝛼 and 𝛽,wegivesome whichisamethodologythatallowsonetohaveadifferent basic definitions with respect to the ∗-arithmetic. We also look at problems which can be investigated via calculus. In report the most relevant and recent literature in this section. 2 International Journal of Analysis

In Section 3, first the definitions of non-Newtonian means arithmetic are special cases for 𝑝=2and 𝑝=−1,respectively. −1 are given which will be used for non-Newtonian convexity. In The function 𝑞𝑝 : R → R𝑞 ⊆ R and its inverse 𝑞𝑝 are defined this section, the forms of weighted means are presented and as follows (𝑝 ∈ R \{0}): an illustrative table is given. In Section 4, the generalized non- Newtonian convex function is defined on the interval 𝐼𝛼 and 1/𝑝 {𝑥 ,𝑥>0 some types of convex function are obtained by using different { 𝑞 (𝑥) = 0, 𝑥 = 0 generators. In the final section of the paper, we assert the 𝑝 { notion of multiplicative Lipschitz condition on the closed { − (−𝑥)1/𝑝 ,𝑥<0, interval [𝑥, 𝑦] ⊂ (0,. ∞) { (4) {𝑥𝑝, 𝑥>0 { 2. Preliminary, Background, and Notation −1 { 𝑞𝑝 (𝑥) = {0, 𝑥=0 { Arithmetic is any system that satisfies the whole of the { 𝑝 ordered field axioms whose domain is a subset of R.There {− (−𝑥) , 𝑥<0. are infinitely many types of arithmetic, all of which are isomorphic, that is, structurally equivalent. It is to be noted that 𝑞𝑝-calculus is reduced to the classical A generator 𝛼 is a one-to-one function whose domain is calculus for 𝑝=1. Additionally it is concluded that the 𝛼- R andwhoserangeisasubsetR𝛼 of R where R𝛼 = {𝛼(𝑥) : summation can be given as follows: 𝑥∈R}. Each generator generates exactly one arithmetic, and conversely each arithmetic is generated by exactly one 𝑛 𝑛 ∑𝑥 =𝛼{∑𝛼−1 (𝑥 )} generator. If 𝐼(𝑥),forall =𝑥 𝑥∈R,theidentity function’s 𝛼 𝑘 𝑘 𝑘=1 inverse is itself. In the special cases 𝛼=𝐼and 𝛼=exp, 𝛼 𝑘=1 (5) generates the classical and geometric arithmetic, respectively. −1 −1 + =𝛼{𝛼 (𝑥1)+⋅⋅⋅+𝛼 (𝑥𝑛)} ∀𝑥𝑘 ∈ R . By 𝛼-arithmetic, we mean the arithmetic whose domain is R and whose operations are defined as follows: for 𝑥, 𝑦 ∈ R𝛼 Definition 1 (see [15]). Let 𝑋=(𝑋,𝑑𝛼) be an 𝛼-metric space. and any generator 𝛼, Thenthebasicnotionscanbedefinedasfollows: −1 −1 𝛼-addition 𝑥 +𝑦=𝛼{𝛼̇ (𝑥) +𝛼 (𝑦)} , (a) A sequence 𝑥=(𝑥𝑘) is a function from the set N into −1 −1 the set R𝛼.The𝛼-real number 𝑥𝑘 denotes the value of 𝛼-subtraction 𝑥 −𝑦=𝛼{𝛼̇ (𝑥) −𝛼 (𝑦)} , the function at 𝑘∈N and is called the 𝑘th term of the −1 −1 sequence. 𝛼-multiplication 𝑥 ×𝑦=𝛼{𝛼̇ (𝑥) ×𝛼 (𝑦)} , (1) (b) A sequence (𝑥𝑛) in 𝑋=(𝑋,𝑑𝛼) is said to be 𝛼- ̇ −1 −1 ̇ ̇ 𝛼-division 𝑥 /𝑦=𝛼{𝛼 (𝑥) ÷𝛼 (𝑦)} , convergent if, for every given 𝜀 > 0 (𝜀∈R𝛼), there exist an 𝑛0 =𝑛0(𝜀) ∈ N and 𝑥∈𝑋such that ̇ −1 −1 ̇ ̇ 𝛼-order 𝑥 <𝑦⇐⇒𝛼 (𝑥) <𝛼 (𝑦) . 𝑑𝛼(𝑥𝑛,𝑥)= |𝑥𝑛 −𝑥|𝛼 <𝜀for all 𝑛>𝑛0 and is denoted 𝛼 𝛼 by lim𝑛→∞𝑥𝑛 =𝑥or 𝑥𝑛 󳨀→𝑥,as𝑛→∞. As a generator, we choose exp function acting from R into the set Rexp =(0,∞)as follows: (c) A sequence (𝑥𝑛) in 𝑋=(𝑋,𝑑𝛼) is said to be 𝛼-Cauchy ̇ ̇ if for every 𝜀 > 0 there is an 𝑛0 =𝑛0(𝜀) ∈ N such that 𝛼:R 󳨀→ R ̇ exp 𝑑𝛼(𝑥𝑛,𝑥𝑚) <𝜀for all 𝑚, 𝑛0 >𝑛 . (2) 𝑥 𝑝 𝑥 󳨃󳨀→ 𝑦 =𝛼 (𝑥) =𝑒 . Throughout this paper, we define the 𝑝th 𝛼-exponent 𝑥 𝛼 (1/𝑞) + and 𝑞th 𝛼-root 𝑥 𝛼 of 𝑥∈R by It is obvious that 𝛼-arithmetic reduces to the geometric arithmetic as follows: 2 −1 −1 𝑥 𝛼 =𝑥×𝑥=𝛼{𝛼̇ (𝑥) ×𝛼 (𝑥)} {ln 𝑥+ln 𝑦} geometric addition 𝑥 +𝑦=𝑒̇ =𝑥⋅𝑦, −1 2 {ln 𝑥−ln 𝑦} =𝛼{[𝛼 (𝑥)] }, geometric subtraction 𝑥 −𝑦=𝑒̇ =𝑥÷𝑦,

3𝛼 2𝛼 {ln 𝑥 ln 𝑦} ln 𝑦 𝑥 =𝑥 ×𝑥̇ geometric multiplication 𝑥 ×𝑦=𝑒̇ =𝑥 (3) =𝛼{𝛼−1 {𝛼 [𝛼−1 (𝑥) ×𝛼−1 (𝑥)]} × 𝛼−1 (𝑥)} =𝑦ln 𝑥, (6) −1 3 ̇ {ln 𝑥/ ln 𝑦} 1/ ln 𝑦 =𝛼{[𝛼 (𝑥)] }, geometric division 𝑥 /𝑦=𝑒 =𝑥 , 𝑥 <𝑦⇐⇒̇ (𝑥) < (𝑦) . . geometric order ln ln .

Following Grossman and Katz [10] we give the infinitely 𝑝 (𝑝−1) −1 𝑝 𝑥 𝛼 =𝑥 𝛼 ×𝑥=𝛼{[𝛼̇ (𝑥)] }, many 𝑞𝑝-arithmetics, of which the quadratic and harmonic International Journal of Analysis 3

𝛼 (1/2)𝛼 and √𝑥=𝑥 =𝑦provided there exists an 𝑦∈R𝛼 such 𝛼-arithmetic can readily be transformed into a statement in 2 that 𝑦 𝛼 =𝑥. 𝛽-arithmetic.

Definition 3 (see [10]). The following statements are valid: 2.1. ∗-Arithmetic. Suppose that 𝛼 and 𝛽 are two arbitrarily selected generators and (“star-”) also is the ordered pair 𝛽 𝛼 of arithmetics ( -arithmetic and -arithmetic). The sets ∗ 𝑃 𝑃 𝑃 ∗ ̈ ̈ ̇ ̇ (i) The -points 1, 2,and 3 are -collinear provided (R𝛽, +,̈ −,̈ ×,̈ /, <) and (R𝛼, +,̇ −,̇ ×,̇ /, <) are complete ordered that at least one of the following holds: fields and 𝑏𝑒𝑡𝑎(𝑎𝑙𝑝ℎ𝑎)-generator generates 𝑏𝑒𝑡𝑎(𝑎𝑙𝑝ℎ𝑎)- arithmetic, respectively. Definitions given for 𝛽-arithmetic are also valid for 𝛼-arithmetic. Also 𝛼-arithmetic is used for 𝑑∗ (𝑃 ,𝑃) +𝑑̈ ∗ (𝑃 ,𝑃)=𝑑∗ (𝑃 ,𝑃), arguments and 𝛽-arithmeticisusedforvalues;inparticu- 2 1 1 3 2 3 𝛼 lar, changes in arguments and values are measured by - 𝑑∗ (𝑃 ,𝑃) +𝑑̈ ∗ (𝑃 ,𝑃)=𝑑∗ (𝑃 ,𝑃), differences and 𝛽-differences, respectively. 1 2 2 3 1 3 (10) 𝑥∈(R , +,̇ −,̇ ×,̇ /,̇ <)̇ 𝑦∈(R , +,̈ −,̈ ×,̈ /,̈ <)̈ ∗ ∗ ∗ Let 𝛼 and 𝛽 be 𝑑 (𝑃1,𝑃3) +𝑑̈ (𝑃3,𝑃2)=𝑑 (𝑃1,𝑃2). arbitrarily chosen elements from corresponding arithmetic. Then the ordered pair (𝑥, 𝑦) is called a ∗-point and the set of all ∗-points is called the set of ∗-complex numbers and is ∗ (ii) A ∗-line is a set 𝐿 of at least two distinct points such denoted by C ;thatis, that, for any distinct points 𝑃1 and 𝑃2 in 𝐿,apoint𝑃3 is in 𝐿 if and only if 𝑃1, 𝑃2,and𝑃3 are ∗-collinear. When C∗ ﬂ {𝑧∗ =(𝑥,𝑦)|𝑥∈R ,𝑦∈R }. 𝛼 𝛽 (7) 𝛼=𝛽=𝐼,the∗-lines are the straight lines in two- dimensional Euclidean space. Definition 2 (see [17]). (a) The ∗-limit of a function 𝑓 at an element 𝑎 in R𝛼 is, if it exists, the unique number 𝑏 in R𝛽 such that (iii) The ∗-slope of a ∗-line through the points (𝑎1,𝑏1) and (𝑎2,𝑏2) is given by ∗ 𝑓 (𝑥) =𝑏⇐⇒ 𝑥→𝑎lim ̈ ̇ 󵄨 󵄨 ∀𝜀 >̈ 0, ∃𝛿 >̇ 0∋󵄨𝑓 (𝑥) −𝑏̈ 󵄨 <𝜀̈ (8) ∗ ̈ 𝛽 𝑚 =(𝑏2 −𝑏̈ 1) /𝜄(𝑎2 −𝑎̇ 1) 󵄨 󵄨 󵄨 ̇ 󵄨 ̇ −1 −1 ∀𝑥, 𝜀 ∈ R𝛼, 󵄨𝑥 −𝑎󵄨𝛼 <𝛿, 𝛽 (𝑏 )−𝛽 (𝑏 ) (11) =𝛽{ 2 1 }, (𝑎 =𝑎̸ ), −1 −1 1 2 ∗ 𝛼 (𝑎2)−𝛼 (𝑎1) for 𝛿∈R𝛽,andiswrittenas lim𝑥→𝑎𝑓(𝑥). =𝑏 Afunction𝑓 is ∗-continuous at a point 𝑎 in R𝛼 if and only ∗ if 𝑎 is an argument of 𝑓 and lim𝑥→𝑎𝑓(𝑥) = 𝑓(𝑎).When 𝑎 ,𝑎 ∈ R 𝑏 ,𝑏 ∈ R 𝛼 and 𝛽 are the identity function 𝐼,theconceptsof∗-limit for 1 2 𝛼 and 1 2 𝛽. and ∗-continuity are reduced to those of classical limit and If the following ∗-limit in (12) exists, we denote it by classical continuity. ∗ 𝛼 𝛽 𝑓 (𝑡),callitthe∗-derivative of 𝑓 at 𝑡,andsaythat𝑓 is ∗- (b) The isomorphism from -arithmetic to -arithmetic 𝑡 is the unique function 𝜄 (iota) which has the following three differentiable at (see [19]): properties:

(i) 𝜄 is one to one. ∗ (𝑓 (𝑥) −𝑓̈ (𝑡)) /𝜄(𝑥̈ −𝑡)̇ lim𝑥→𝑡 (ii) 𝜄 is from R𝛼 to R𝛽. 𝛽−1 {𝑓 (𝑥)}−𝛽−1 {𝑓 (𝑡)} 𝑢, V ∈ R = lim𝛽{ } (iii) For any numbers 𝛼, 𝑥→𝑡 𝛼−1 (𝑥) −𝛼−1 (𝑡)

𝛽−1 {𝑓 (𝑥)}−𝛽−1 {𝑓 (𝑡)} (12) 𝜄(𝑢+̇ V)=𝜄(𝑢) +𝜄̈ (V) ; = lim𝛽{ 𝑥→𝑡 𝑥−𝑡 𝜄(𝑢−̇ V)=𝜄(𝑢) −𝜄̈ (V) ; −1 󸀠 (9) 𝑥−𝑡 {(𝛽 ∘𝑓) (𝑡)} 𝜄(𝑢×̇ V)=𝜄(𝑢) ×𝜄̈ (V) ; ⋅ }=𝛽{ } . 𝛼−1 (𝑥) −𝛼−1 (𝑡) (𝛼−1)󸀠 (𝑡) { } 𝜄(𝑢/̇ V)=𝜄(𝑢) /𝜄̈ (V) .

−1 3. Non-Newtonian (Weighted) Means It turns out that 𝜄(𝑥) = 𝛽{𝛼 (𝑥)} for every 𝑥 in R𝛼 and that 𝜄(𝑛)̇ = 𝑛̈ for every 𝛼-integer 𝑛̇. Since, for example, Definition 4 (𝛼-arithmetic mean). Consider that 𝑛 positive −1 𝑢 +̇ V =𝜄 {𝜄(𝑢) +𝜄(̈ V)},itshouldbeclearthatanystatementin real numbers 𝑥1,𝑥2,...,𝑥𝑛 are given. The 𝛼-mean (average), 4 International Journal of Analysis

denoted by 𝐴𝛼,isthe𝛼-sum of 𝑥𝑛’s 𝛼-divided by 𝑛̇ for all 𝐺exp and 𝐺𝑝 are called multiplicative geometric mean and 𝑛∈N.Thatis, 𝑝-geometric mean, respectively. It would clearly have 𝐺𝑝 = 𝐴exp for 𝑝=1. 𝑛 𝑛 𝛼−1 (𝑥 ) 𝐴 = ∑𝑥 /̇ 𝑛=̇ ∑𝛼{ 𝑘 } 𝛼 𝑥 ,𝑥 ,...,𝑥 ∈ R+ 𝛼 𝛼 𝑘 𝛼 𝑛 Definition 7 ( -harmonic mean). Let 1 2 𝑛 and 𝑘=1 𝑘=1 −1 𝛼 (𝑥𝑛) =0̸ for each 𝑛∈N.The𝛼-harmonic mean 𝐻𝛼 is (13) 𝛼−1 (𝑥 )+𝛼−1 (𝑥 )+⋅⋅⋅+𝛼−1 (𝑥 ) defined by =𝛼{ 1 2 𝑛 }. 𝑛

For 𝛼=exp, we obtain that 𝑛 𝐻 = 𝑛̇ /̇ ∑ (1̇/𝑥̇ ) 𝛼 𝛼 𝑘 𝑘=1 𝑛 1/𝑛 𝐴 =(∏𝑥 ) =(𝑥 ⋅𝑥 ⋅⋅⋅𝑥 )1/𝑛 . ̇ ̇ ̇ ̇ ̇ ̇ ̇ exp 𝑘 1 2 𝑛 (14) = 𝑛̇ /(1 /𝑥1 +̇ 1 /𝑥2 +⋅⋅⋅̇ +̇ 1 /𝑥𝑛) (19) 𝑘=1 𝑛 =𝛼{ }. 𝛼=𝑞 −1 −1 −1 Similarly, for 𝑝,weget 1/𝛼 (𝑥1)+1/𝛼 (𝑥2)+⋅⋅⋅+1/𝛼 (𝑥𝑛)

𝑥𝑝 +𝑥𝑝 +⋅⋅⋅+𝑥𝑝 1/𝑝 𝐴 =( 1 2 𝑛 ) ,𝑝∈R \ {0} . (15) 𝑝 𝑛 Similarly, one obtains that

𝐴exp and 𝐴𝑝 arecalledmultiplicativearithmeticmeanand 𝑝-arithmetic mean (as usually known p-mean), respectively. One can conclude that 𝐴𝑝 reduces to arithmetic mean and 𝐻exp harmonic mean in the ordinary sense for 𝑝=1and 𝑝=−1, respectively. 𝑛 = exp { }, 1/ln (𝑥1)+1/ln (𝑥2)+⋅⋅⋅+1/ln (𝑥𝑛) Remark 5. It is clear that Definition 4 can be written by (20) using various generators. In particular if we take 𝛽-arithmetic (𝑥𝑛 >1), instead of 𝛼-arithmetic then the mean can be defined by 𝑛 1/𝑝 𝐻 ={ } . 𝑛 𝑝 1/ (𝑥 )𝑝 +1/(𝑥 )𝑝 +⋅⋅⋅+1/(𝑥 )𝑝 𝐴 = ∑𝑥 /̈ 𝑛.̈ 1 2 𝑛 𝛽 𝛽 𝑘 (16) 𝑘=1

𝛼 𝑥 ,𝑥 ,...,𝑥 ∈ R+ Definition 6 ( -geometric mean). Let 1 2 𝑛 . 𝐻exp and 𝐻𝑝 are called multiplicative harmonic mean and 𝑝- The 𝛼-geometric mean, namely, 𝐺𝛼,is𝑛th 𝛼-root of the 𝛼- harmonic mean, respectively. Obviously the inclusion (20) is product of (𝑥𝑛)’s: reduced to ordinary harmonic mean and ordinary arithmetic mean for 𝑝=1and 𝑝=−1,respectively. (1/𝑛) 1/𝑛 𝑛 𝛼 { 𝑛 } 𝐺 =( ∏𝑥 ) =𝛼 (∏𝛼−1 (𝑥 )) 𝛼 𝛼 𝑘 { 𝑘 } 𝑘=1 { 𝑘=1 } (17) 3.1. Non-Newtonian Weighted Means. The weighted mean is similar to an arithmetic mean, where instead of each of the −1 −1 −1 1/𝑛 =𝛼{(𝛼 (𝑥1),𝛼 (𝑥2),...,𝛼 (𝑥𝑛)) }. data points contributing equally to the final average, some data points contribute more than others. Moreover the notion of weighted mean plays a role in descriptive statistics and We conclude similarly, by taking the generators 𝛼=exp also occurs in a more general form in several other areas of 𝛼=𝑞 𝛼 or 𝑝,thatthe -geometric mean can be interpreted as mathematics. follows: The following definitions can give the relationships between the non-Newtonian weighted means and ordinary 1/𝑛 𝐺exp = exp {(ln 𝑥1, ln 𝑥2,...,ln 𝑥𝑛) }, (𝑥𝑛 >1) weighted means.

1/𝑝 Definition 8 (weighted 𝛼-arithmetic mean). Formally, the 𝑝 𝑝 𝑝 1/𝑛 1/𝑛 (18) 𝐺𝑝 ={(𝑥1 ,𝑥2 ,...,𝑥𝑛 ) } =(𝑥1,𝑥2,...,𝑥𝑛) , weighted 𝛼-arithmetic mean of a nonempty set of data {𝑥1,𝑥2,...,𝑥𝑛} with nonnegative weights {𝑤1,𝑤2,...,𝑤𝑛} is (𝑝 =0).̸ the quantity International Journal of Analysis 5

𝑛 𝑛 ∑𝑛 𝑤 𝛼−1 (𝑥 ) 𝐴̃ = ∑ (𝑥 ×̇ 𝑤̇ ) /̇ ∑𝑤̇ = (𝑥 ×̇ 𝑤̇ +𝑥̇ ×̇ 𝑤̇ +̇ ⋅⋅⋅ +𝑥̇ ×̇ 𝑤̇ ) /̇ (𝑤̇ +̇ 𝑤̇ +̇ ⋅⋅⋅ +̇ 𝑤̇ ) =𝛼{ 𝑖=1 𝑖 𝑖 } . 𝛼 𝛼 𝑖 𝑖 𝛼 𝑖 1 1 2 2 𝑛 𝑛 1 2 𝑛 ∑𝑛 𝑤 (21) 𝑖=1 𝑖=1 𝑖=1 𝑖

̃ 𝑤1/(𝑤1+⋅⋅⋅+𝑤𝑛) The formulas are simplified when the weights are 𝛼- 𝐺exp = exp {[ln (𝑥1)] , 𝑛 ̇ ̇ normalized such that they 𝛼-sum up to 𝛼∑𝑖=1𝑤𝑖 = 1.For 𝑤 /(𝑤 +⋅⋅⋅+𝑤 ) 𝑤 /(𝑤 +⋅⋅⋅+𝑤 ) such normalized weights the weighted 𝛼-arithmetic mean is 2 1 𝑛 𝑛 1 𝑛 [ln (𝑥2)] ,...,[ln (𝑥𝑛)] }, ̃ 𝑛 ̇ ̇ simply 𝐴𝛼 = 𝛼∑𝑖=1 𝑥𝑖 × 𝑤𝑖.Notethatifalltheweightsare 𝛼 𝛼 equal, the weighted -arithmetic mean is the same as the - (𝑥𝑛 >1), arithmetic mean. (24) 𝛼= 𝛼=𝑞 𝛼 ̃ 𝑝𝑤1/(𝑤1+⋅⋅⋅+𝑤𝑛) 𝑝𝑤2/(𝑤1+⋅⋅⋅+𝑤𝑛) Taking exp and 𝑝,theweighted -arithmetic 𝐺𝑝 ={𝑥1 ,𝑥2 ,..., mean can be given with the weights {𝑤1,𝑤2,...,𝑤𝑛} as fol- 𝑛 𝑛 1/ ∑𝑖=1 𝑤𝑖 lows: 1/𝑝 𝑝𝑤𝑛/(𝑤1+⋅⋅⋅+𝑤𝑛) 𝑤𝑖 𝑥𝑛 } =(∏𝑥𝑖 ) . 𝑖=1

𝑛 𝑛 𝑛 1/ ∑𝑖=1 𝑤𝑖 ∑ ln (𝑥𝑖)𝑤𝑖 𝑤 ̃ ̃ ̃ 𝑖=1 𝑖 𝐺exp and 𝐺𝑝 arecalledweightedmultiplicativegeometric 𝐴exp = exp { 𝑛 }=(∏𝑥𝑖 ) , ∑ 𝑤𝑖 ̃ 𝑖=1 𝑖=1 mean and weighted 𝑞-geometric mean. Also we have 𝐺𝑝 = 𝐴̃ 𝑥 >1 (𝑥𝑛 >1), (22) exp for all 𝑛 .

1/𝑝 𝛼 {𝑤 , 𝑤 (𝑥 )𝑝 +𝑤 (𝑥 )𝑝 +⋅⋅⋅+𝑤 (𝑥 )𝑝 Definition 10 (weighted -harmonic mean). If a set 1 ̃ 1 1 2 2 𝑛 𝑛 𝑤 ,...,𝑤 } 𝐴𝑞 ={ } . 2 𝑛 of weights is associated with the data set 𝑝 𝑤 +𝑤 +⋅⋅⋅+𝑤 1 2 𝑛 {𝑥1,𝑥2,...,𝑥𝑛} then the weighted 𝛼-harmonic mean is defined by

𝑛 𝑛 𝐴̃ 𝐴̃ ̃ ̇ ̇ exp and 𝑝 arecalledmultiplicativeweightedarithmetic 𝐻𝛼 = ∑𝑤̇ 𝑘 / ∑ (𝑤̇ 𝑘 /𝑥𝑘) ̃ 𝛼 𝛼 mean and weighted 𝑝-arithmetic mean, respectively. 𝐴exp 𝑘=1 𝑘=1 turns out to the ordinary weighted geometric mean. Also, ̇ ̇ ̇ ̇ ̇ ̇ ̇ ̇ ̇ ̇ ̇ ̇ ̇ ̇ ̃ =(𝑤1 + ⋅⋅⋅ + 𝑤𝑛) /(𝑤1 /𝑥1 + 𝑤2 /𝑥2 +⋅⋅⋅+ 𝑤𝑛 /𝑥𝑛) (25) one easily can see that 𝐴𝑝 is reduced to ordinary weighted arithmetic mean and weighted harmonic mean for 𝑝=1and 𝑤 +𝑤 +⋅⋅⋅+𝑤 =𝛼{ 1 2 𝑛 }. 𝑝=−1,respectively. −1 −1 −1 𝑤1/𝛼 (𝑥1)+𝑤2/𝛼 (𝑥2)+⋅⋅⋅+𝑤𝑛/𝛼 (𝑥𝑛) Definition 9 (weighted 𝛼-geometric mean). Given a set of {𝑥 ,𝑥 ,...,𝑥 } positive reals 1 2 𝑛 and corresponding weights 𝛼= 𝛼=𝑞 𝛼 ̃ Taking exp and 𝑝,theweighted -harmonic {𝑤1,𝑤2,...,𝑤𝑛}, then the weighted 𝛼-geometric mean 𝐺𝛼 is mean with the weights {𝑤1,𝑤2,...,𝑤𝑛} can be written as defined by follows: ̃ 𝐻exp 𝑛 𝑛 (𝑤𝑖/∑𝑖=1 𝑤𝑖)𝛼 (𝑤 /(𝑤 +⋅⋅⋅+𝑤 )) 𝑤 +𝑤 +⋅⋅⋅+𝑤 𝐺̃ ={ ∏𝑥 } =𝑥 1 1 𝑛 𝛼 1 2 𝑛 𝛼 𝛼 𝑖 1 = exp { }, 𝑖=1 𝑤1/ ln 𝑥1 +𝑤2/ ln 𝑥2 +⋅⋅⋅+𝑤𝑛/ ln 𝑥𝑛

(𝑤 /(𝑤 +⋅⋅⋅+𝑤 )) (𝑤 /(𝑤 +⋅⋅⋅+𝑤 )) ̇ 2 1 𝑛 𝛼 ̇ ̇ 𝑛 1 𝑛 𝛼 (𝑥 >1), ×𝑥2 ×⋅⋅⋅×𝑥𝑛 𝑛 (26) (23) 𝑤 /(𝑤 +⋅⋅⋅+𝑤 ) ̃ −1 1 1 𝑛 𝐻𝑝 =𝛼{[𝛼 (𝑥1)] , 1/𝑝 𝑤 /(𝑤 +⋅⋅⋅+𝑤 ) 𝑤 /(𝑤 +⋅⋅⋅+𝑤 ) 𝑤1 +𝑤2 +⋅⋅⋅+𝑤𝑛 −1 2 1 𝑛 −1 𝑛 1 𝑛 ={ } . [𝛼 (𝑥2)] ,...,[𝛼 (𝑥𝑛)] }. 𝑝 𝑝 𝑝 𝑤1/(𝑥1) +𝑤2/(𝑥2) +⋅⋅⋅+𝑤𝑛/(𝑥𝑛)

̃ ̃ 𝐻exp and 𝐻𝑝 arecalledmultiplicativeweightedharmonic Note that if all the weights are equal, the weighted 𝛼- mean and weighted 𝑝-geometric mean, respectively. It is ̃ geometric mean is the same as the 𝛼-geometric mean. obvious that 𝐻𝑝 is reduced to ordinary weighted harmonic Taking 𝛼=exp and 𝛼=𝑞𝑝,theweighted 𝛼-geometric mean mean and ordinary weighted arithmetic mean for 𝑝=1and canbewrittenfortheweights{𝑤1,𝑤2,...,𝑤𝑛} as follows: 𝑝=−1,respectively. 6 International Journal of Analysis

Table 1: Comparison of the non-Newtonian (weighted) means and ordinary (weighted) means. ̃ ̃ ̃ ̃ ̃ Weight Data 𝑝𝐴𝑝 𝐺𝑝 𝐻𝑝 𝐴𝑝 𝐺𝑝 𝐻𝑝 𝐺exp 𝐻exp 𝐺exp 𝐻exp

𝑤1 =2 𝑥1 =15 1.0 27.50 24.74 22.64 32.82 29.88 27.27 24.02 23.36 29.05 28.26

𝑤2 =5 𝑥2 =20 0.1 24.99 24.74 24.50 30.16 29.88 29.59 24.02 23.36 29.05 28.26

𝑤3 =7 𝑥3 =25 2.0 30.61 24.74 21.14 35.77 29.88 25.21 24.02 23.36 29.05 28.26

𝑤4 =9 𝑥4 =50 5.0 38.22 24.74 18.73 41.69 29.88 21.55 24.02 23.36 29.05 28.26

InTable1,thenon-Newtonianmeansareobtainedby be ∗-concave. On the other hand the inclusion (28) can be using different generating functions. For 𝛼=𝑞𝑝,the𝑝-means written with respect to the weighted 𝛼-arithmetic mean in 𝐴𝑝,𝐺𝑝 and 𝐻𝑝 are reduced to ordinary arithmetic mean, geo- (21) as follows: metric mean, and harmonic mean, respectively. In particular ̃ ̈ ̃ 𝑓(𝐴𝛼 {𝑥, 𝑦}) ≤ 𝐴𝛽 {𝑓 (𝑥) ,𝑓(𝑦)}. (29) some changes are observed for each value of 𝐴𝑝,𝐺𝑝,and𝐻𝑝 means depending on the choice of 𝑝. As shown in the table, Remark 13. We remark that the definition of ∗-convexity in for increasing values of 𝑝,the 𝑝-arithmetic mean 𝐴𝑝 and its (27) can be evaluated by non-Newtonian coordinate system ̃ weighted form 𝐴𝑝 increase; in particular 𝑝 tends to ∞,and involving ∗-lines (see Definition 3). For 𝛼=𝛽=𝐼,the∗- these means converge to the value of max{𝑥𝑛}.Conversely, lines are straight lines in two-dimensional Euclidean space. for increasing values of 𝑝,the𝑝-harmonic mean 𝐻𝑝 and For this reason, we say that almost all the properties of ̃ ordinary Cartesian coordinate system will be valid for non- its weighted forms 𝐻𝑝 decrease. In particular, these means Newtonian coordinate system under ∗-arithmetic. converge to the value of min{𝑥𝑛} as 𝑝→∞. Depending 𝑝 𝐴̃ 𝐻̃ on the choice of ,weightedforms 𝑝 and 𝑝,canbe Also depending on the choice of generator functions, increased or decreased without changing any weights. For the definition of ∗-convexity in (27) can be interpreted as this reason, this approach brings a new perspective to the follows. concept of classical (weighted) mean. Moreover, when we 𝐻 compare exp and ordinary harmonic mean in Table 1, we Case 1. (a) If we take 𝛼=𝛽=exp and 𝜆1 =𝜇1, 𝜆2 =𝜇2 in 𝐻 also see that ordinary harmonic mean is smaller than exp. (28), then On the contrary 𝐴exp and 𝐺exp are smaller than their classical ln 𝜆1 ln 𝜆2 ln 𝜆1 ln 𝜆2 forms 𝐴𝑝 and 𝐺𝑝 for 𝑝=1. Therefore, we assert that 𝑓(𝑥 𝑦 )≤𝑓(𝑥) 𝑓(𝑦) , 𝐺 ,𝐻 , 𝐺̃ 𝐻̃ (30) the values of exp exp exp,and exp should be evaluated (𝜆 ,𝜆 ∈ [1, 𝑒]), satisfactorily. 1 2 where 𝜆1𝜆2 =𝑒 holds and 𝑓:𝐼exp → Rexp =(0,∞)is Corollary 11. Consider 𝑛 positive real numbers 𝑥1,𝑥2,...,𝑥𝑛. ̃ ̃ ̃ called bigeometric (usually known as multiplicative) convex Then, the conditions 𝐻𝛼 <𝐺𝛼 <𝐴𝛼 and 𝐻𝛼 < 𝐺𝛼 < 𝐴𝛼 hold 𝑓 + function (cf. [2]). Equivalently, is bigeometric convex if when 𝛼=exp for all 𝑥𝑛 >1and 𝛼=𝑞𝑝 for all 𝑝∈R . and only if log 𝑓(𝑥) is an ordinary convex function. (b) For 𝛼=exp and 𝛽=𝐼we have

4. Non-Newtonian Convexity ln 𝜆1 ln 𝜆2 𝑓(𝑥 𝑦 )≤𝜇1𝑓 (𝑥) +𝜇2𝑓(𝑦), ∗ (31) In this section, the notion of non-Newtonian convex ( - (𝜆 ,𝜆 ∈ [1, 𝑒] ;𝜇,𝜇 ∈ [0, 1]), convex) functions will be given by using different genera- 1 2 1 2 tors. Furthermore the relationships between ∗-convexity and where 𝜆1𝜆2 =𝑒and 𝜇1 +𝜇2 =1.Inthiscasethefunction 𝑓: non-Newtonian weighted mean will be determined. 𝐼exp → R is called geometric convex function. Every geometric convex (usually known as log-convex) function is ∗ 𝐼 Definition 12 (generalized -convex function). Let 𝛼 be an also convex (cf. [2]). R 𝑓:𝐼 → R ∗ interval in 𝛼.Then 𝛼 𝛽 is said to be -convex if (c) Taking 𝛼=𝐼and 𝛽=exp, one obtains ln 𝜇 ln 𝜇 ̇ ̇ ̇ ̈ ̈ ̈ ̈ 𝑓(𝜆 𝑥+𝜆 𝑦) ≤ 𝑓 𝑥 1 𝑓(𝑦) 2 , 𝑓 (𝜆1 ×𝑥+𝜆2 ×𝑦) ≤𝜇1 ×𝑓(𝑥) +𝜇2 ×𝑓(𝑦) (27) 1 2 ( ) (32) ̇ ̈ (𝜇1,𝜇2 ∈ [1, 𝑒] ;𝜆1,𝜆2 ∈ [0, 1]), holds, where 𝜆1 +𝜆̇ 2 = 1 and 𝜇1 +𝜇̈ 2 = 1 for all 𝜆1,𝜆2 ∈ [0,̇ 1]̇ 𝜇 ,𝜇 ∈[0,̈ 1]̈ and 1 2 . Therefore, by combining this with where 𝜇1𝜇2 =𝑒and 𝜆1 +𝜆2 =1,and𝑓:𝐼→Rexp is called the generators 𝛼 and 𝛽,wededucethat anageometric convex function. −1 −1 −1 −1 𝑓(𝛼{𝛼 (𝜆1)𝛼 (𝑥) +𝛼 (𝜆2)𝛼 (𝑦)}) Case 2. (a) If 𝛼=𝛽=𝑞𝑝 in (28) then (28) ≤𝛽{𝛽̈ −1 (𝜇 )𝛽−1𝑓 𝑥 +𝛽−1 (𝜇 )𝛽−1𝑓(𝑦)}. 𝑝 𝑝 1/𝑝 1 ( ) 2 𝑓(((𝜆1𝑥) +(𝜆2𝑦) ) ) (33) 𝑥 =𝑦̸ 𝑓 ∗ 1/𝑝 If (28) is strict for all ,then is said to be strictly - ≤((𝜆 𝑓 (𝑥))𝑝 +(𝜆 𝑓(𝑦))𝑝) (𝑝 ∈ R+), convex. If the inequality in (28) is reversed, then 𝑓 is said to 1 2 International Journal of Analysis 7

𝜆 ,𝜆 ∈ [0, 1] 𝜆𝑝 +𝜆𝑝 =1, 𝑓:𝐼 → R Theorem 15. 𝑓:𝐼 → R ∗ where 1 2 , 1 2 and 𝑞𝑝 𝑞𝑝 is Let 𝛼 𝛽 be a -continuous function. called 𝑄𝑄-convex function. Then 𝑓 is ∗-convex if and only if 𝑓 is midpoint ∗-convex, (b) For 𝛼=𝑞𝑝 and 𝛽=𝐼,wewritethat that is, 𝑥 ,𝑥 ∈𝐼 𝑝 𝑝 1/𝑝 1 2 𝛼 implies 𝑓(((𝜆1𝑥) +(𝜆2𝑦) ) )≤𝜇1𝑓 (𝑥) +𝜇2𝑓(𝑦), (40) ̈ (34) 𝑓(𝐴𝛼 {𝑥1,𝑥2}) ≤𝐴𝛽 {𝑓 1(𝑥 ),𝑓(𝑥2)} . (𝜆1,𝜆2,𝜇1,𝜇2 ∈ [0, 1]), Proof. The proof can be easily obtained using the inequality 𝜆𝑝 +𝜆𝑝 =1𝜇 +𝜇 =1, 𝑓:𝐼 → R (39) in Lemma 14. where 1 2 , 1 2 and 𝑞𝑝 is called 𝑄𝐼-convex function. Theorem 16 (cf. [2]). Let 𝑓:𝐼exp → Rexp be a ∗-differentiable (c) For 𝛼=𝐼and 𝛽=𝑞𝑝,weobtainthat function (see [19]) on a subinterval 𝐼exp ⊆(0,∞).Thenthe following assertions are equivalent: 𝑝 𝑝 1/𝑝 𝑓(𝜆1𝑥+𝜆2𝑦) ≤ ((𝜇1𝑓 (𝑥)) +(𝜇2𝑓(𝑦)) ) , (35) (i) 𝑓 is bigeometric convex (concave). (𝜆 ,𝜆 ,𝜇 ,𝜇 ∈ [0, 1]), ∗ 1 2 1 2 (ii) The function 𝑓 (𝑥) is increasing (decreasing). 𝑝 𝑝 Corollary 17. 𝛽 𝑓 where 𝜇1 +𝜇2 =1, 𝜆1 +𝜆2 =1,and 𝑓:𝐼→R𝑞 is called 𝐼𝑄- Apositive -real-valued function defined on 𝑝 𝐼 convex function. an interval exp is bigeometric convex if and only if The ∗-convexity of a function 𝑓:𝐼𝛼 → R𝛽 means geo- ln 𝜆1 ln 𝜆2 ln 𝜆𝑛 metrically that the ∗-points of the graph of 𝑓 are under the 𝑓(𝑥1 ,𝑥2 ,...,𝑥𝑛 ) chord joining the endpoints (𝑎, 𝑓(𝑎)) and (𝑏, 𝑓(𝑏)) on non- (41) ln 𝜆 ln 𝜆 ln 𝜆 𝑎, 𝑏 ∈𝐼 1 2 𝑛 Newtonian coordinate system for every 𝛼.Bytaking ≤𝑓(𝑥1) ,𝑓(𝑥2) ,...,𝑓(𝑥𝑛) into account the definition of ∗-slope in Definition 3 we have 𝑛 holds, where ∏𝑘=1𝜆𝑘 =𝑒for all 𝑥1,𝑥2,...,𝑥𝑛 ∈𝐼exp and ̈ ̈ ̇ (𝑓 (𝑥) −𝑓(𝑎)) /𝜄(𝑥−𝑎) 𝜆1,𝜆2,...,𝜆𝑛 ∈[1,𝑒]. Besides, we have (36) ≤(𝑓̈ (𝑏) −𝑓̈ (𝑎)) /𝜄(𝑏̈ −𝑎)̇ ̃ 𝑓(𝐴exp {𝑥1,𝑥2,...,𝑥𝑛}) (42) ̃ which implies ≤ 𝐴exp {𝑓 1(𝑥 ),𝑓(𝑥2),...,𝑓(𝑥𝑛)} . ̈ ̈ 𝑓 (𝑥) ≤𝑓(𝑎) +((𝑓̈ (𝑏) −𝑓̈ (𝑎)) /𝜄(𝑏−𝑎))̇ Corollary 18. A 𝛽-real-valued function 𝑓 defined on an inter- (37) val 𝐼𝑞 is 𝑄𝑄-convex if and only if ×𝜄(𝑥̈ −𝑎)̇ 𝑝

𝑝 𝑝 𝑝 1/𝑝 ̇ 𝑓[((𝜆 𝑥 ) +(𝜆 𝑥 ) +⋅⋅⋅+(𝜆 𝑥 ) ) ] for all 𝑥∈[𝑎,̇ 𝑏]. 1 1 2 2 𝑛 𝑛 𝑃, 𝑄 𝑅 On the other hand (37) means that if ,and are any 𝑝 𝑝 three ∗-points on the graph of 𝑓 with 𝑄 between 𝑃 and 𝑅, ≤((𝜆1𝑓(𝑥1)) +(𝜆2𝑓(𝑥2)) +⋅⋅⋅ (43) then 𝑄 is on or below chord 𝑃𝑅.Intermsof∗-slope, it is 𝑝 1/𝑝 equivalent to +(𝜆𝑛𝑓(𝑥𝑛)) )

∗ ̈ ∗ ̈ ∗ 𝑝 𝑚 (𝑃𝑄) ≤𝑚 (𝑃𝑅) ≤𝑚 (𝑄𝑅) (38) ∑𝑛 𝜆 =1 𝑥 ,𝑥 ,...,𝑥 ∈𝐼 holds, where 𝑘=1 𝑘 for all 1 2 𝑛 𝑞𝑝 and 𝜆1,𝜆2,...,𝜆𝑛 ∈ [0, 1].Thus,wehave with strict inequalities when 𝑓 is strictly ∗-convex. Now to avoid the repetition of the similar statements, we 𝑓(𝐴̃ {𝑥 ,𝑥 ,...,𝑥 }) give some necessary theorems and lemmas. 𝑝 1 2 𝑛 (44) ≤ 𝐴̃ {𝑓 (𝑥 ),𝑓(𝑥 ),...,𝑓(𝑥 )} , (𝑝 ∈ R+). Lemma 14 (Jensen’s inequality). A 𝛽-real-valued function 𝑓 𝑝 1 2 𝑛 defined on an interval 𝐼𝛼 is ∗-convex if and only if 5. An Application of Multiplicative Continuity 𝑛 𝑛 𝑓( ∑𝜆 ×𝑥̇ ) ≤̈ ∑𝜇 ×𝑓(𝑥̈ ) 𝛼 𝑘 𝑘 𝛽 𝑘 𝑘 (39) In this section based on the definition of bigeometric convex 𝑘=1 𝑘=1 function and multiplicative continuity, we get an analogue of ordinary Lipschitz condition on any closed interval. 𝑛 ̇ 𝑛 ̈ ̇ ̇ holds, where 𝛼∑𝑘=1 𝜆𝑘 = 1 and 𝛽∑𝑘=1 𝜇𝑘 = 1 for all 𝜆𝑛 ∈[0, 1] Let 𝑓 be a bigeometric (multiplicative) convex function ̈ ̈ [𝑥, 𝑦] ⊂ R+ and 𝜇𝑛 ∈[0, 1]. and finite on a closed interval .Itisobviousthat 𝑓is bounded from above by 𝑀=max{𝑓(𝑥), 𝑓(𝑦)},since, 𝜆 1−𝜆 𝜆 1−𝜆 Proof. The proof is straightforward, hence omitted. for any 𝑧=𝑥𝑦 in the interval, 𝑓(𝑧) ≤ 𝑓(𝑥) 𝑓(𝑦) for 8 International Journal of Analysis

𝜆∈[1,𝑒].Itisalsoboundedfrombelowasweseebywriting 6. Concluding Remarks + an arbitrary point in the form 𝑡√𝑥𝑦 for 𝑡∈R .Then Although all arithmetics are isomorphic, only by distinguish- √𝑥𝑦 ing among them do we obtain suitable tools for construct- 𝑓2 (√𝑥𝑦) ≤𝑓(𝑡√𝑥𝑦) 𝑓 ( ) . 𝑡 (45) ing all the non-Newtonian calculi. But the usefulness of arithmetic is not limited to the construction of calculi; we Using 𝑀 as the upper bound 𝑓(√𝑥𝑦/𝑡) we obtain believe there is a more fundamental reason for considering alternative arithmetics; they may also be helpful in developing 1 and understanding new systems of measurement that could 𝑓(𝑡√𝑥𝑦) ≥ 𝑓2 (√𝑥𝑦) = 𝑚. 𝑀 (46) yield simpler physical laws. In this paper, it was shown that, due to the choice of 𝐴 𝐺 𝐻 Thus a bigeometric convex function may not be continuous at generator function, 𝑝, 𝑝 ,and 𝑝 means are reduced to the boundary points of its domain. We will prove that, for any ordinary arithmetic, geometric, and harmonic mean, respec- 𝑝 𝐴 closed subinterval [𝑥, 𝑦] of the interior of the domain, there tively.AsshowninTable1,forincreasingvaluesof , 𝑝 ̃ is a constant 𝐾>0so that, for any two points 𝑎, 𝑏 ∈ [𝑥, 𝑦]⊂ and 𝐴𝑝 means increase, especially 𝑝→∞;thesemeans + R , converge to the value of max{𝑥𝑛}. Conversely for increasing 𝑝 𝐻 𝐻̃ 𝑝→∞ 𝐾 values of , 𝑝 and 𝑝 means decrease, especially ; 𝑓 (𝑎) 𝑎 {𝑥 } ≤( ) . (47) these means converge to the value of min 𝑛 . Additionally 𝑓 (𝑏) 𝑏 we give some new definitions regarding convex functions which are plotted on the non-Newtonian coordinate system. A function that satisfies (47) for some 𝐾 and all 𝑎 and 𝑏 in Obviously, for different generator functions, one can obtain an interval is said to satisfy bigeometric Lipschitz condition some new geometrical interpretations of convex functions. on the interval. Our future works will include the most famous Hermite ∗ + Hadamard inequality for the class of -convex functions. Theorem 19. Suppose that 𝑓:𝐼→R is multiplicative convex. Then, 𝑓 satisfies the multiplicative Lipschitz condition on + 0 any closed interval [𝑥, 𝑦] ⊂ R contained in the interior 𝐼 of Competing Interests 𝐼 𝑓 𝐼0 ;thatis, is continuous on . The authors declare that they have no competing interests. Proof. Take 𝜀>1so that [𝑥/𝜀, 𝑦𝜀] ∈𝐼,andlet𝑚 and 𝑀 be thelowerandupperboundsfor𝑓 on [𝑥/𝜀, 𝑦𝜀].If𝑟 and 𝑠 are References [𝑥, 𝑦] 𝑠>𝑟 distinct points of with ,set [1] C. Niculescu and L.-E. Persson, Convex Functions and Their Applications: A Contemporary Approach, Springer, Berlin, Ger- 𝑧=𝑠𝜀, many, 2006. 𝑠 1/ ln(𝜀𝑠/𝑟) (48) [2] C. P. Niculescu, “Convexity according to the geometric mean,” 𝜆=( ) , (𝜆∈(1, 𝑒)) . Mathematical Inequalities & Applications,vol.3,no.2,pp.155– 𝑟 167, 2000.

ln 𝜆 1−ln 𝜆 [3] R. Webster, Convexity,OxfordUniversityPress,NewYork,NY, Then 𝑧 ∈ [𝑥/𝜀, 𝑦𝜀] and 𝑠=𝑧 𝑟 ,andweobtain USA, 1995. [4] J. Banas and A. Ben Amar, “Measures of noncompactness in ln 𝜆 ln 𝜆 1−ln 𝜆 𝑓 (𝑧) locally convex spaces and fixed point theory for the sum of 𝑓 (𝑠) ≤𝑓(𝑧) 𝑓 (𝑟) =( ) 𝑓 (𝑟) (49) 𝑓 (𝑟) two operators on unbounded convex sets,” Commentationes Mathematicae Universitatis Carolinae,vol.54,no.1,pp.21–40, 2013. which yields [5] J. Matkowski, “Generalized weighted and quasi-arithmetic 𝑓 (𝑠) 𝑓 (𝑧) means,” Aequationes Mathematicae,vol.79,no.3,pp.203–212, ( )≤ 𝜆 ( ) 2010. ln 𝑓 (𝑟) ln ln 𝑓 (𝑟) [6] D. Głazowska and J. Matkowski, “An invariance of geometric 𝑠 1/ ln(𝜀) 𝑀 mean with respect to Lagrangian means,” Journal of Mathemat- < ln ( ) ln ( ), (50) ical Analysis and Applications,vol.331,no.2,pp.1187–1199,2007. 𝑟 𝑚 [7]J.Matkowski,“Generalizedweightedquasi-arithmeticmeans 𝑓 (𝑠) 𝑠 ln(𝑀/𝑚)/ ln(𝜀) and the Kolmogorov-Nagumo theorem,” Colloquium Mathe- ≤( ) , maticum,vol.133,no.1,pp.35–49,2013. 𝑓 (𝑟) 𝑟 [8] N. Merentes and K. Nikodem, “Remarks on strongly convex functions,” Aequationes Mathematicae,vol.80,no.1-2,pp.193– 𝐾= (𝑀/𝑚)/ (𝜀) > 0 𝑟, 𝑠 ∈ [𝑥, 𝑦] where ln ln .Sincethepoints 199, 2010. 𝑓 are arbitrary, we get that satisfies a multiplicative Lipschitz [9] M. Grossman, Bigeometric Calculus, Archimedes Foundation condition. The remaining part can be obtained in the similar Box 240, Rockport, Mass, USA, 1983. way by taking 𝑠<𝑟and 𝑧 = 𝑠/𝜀.Finally,𝑓 is continuous, [10] M. Grossman and R. Katz, Non-Newtonian Calculus,Newton 0 since [𝑥, 𝑦] is arbitrary in 𝐼 . Institute, 1978. International Journal of Analysis 9

[11] M. Grossman, The First Nonlinear System of Differential and Integral Calculus,1979. [12] A. E. Bashirov, E. M. Kurpınar, and A. Ozyapıcı,¨ “Multiplicative calculus and its applications,” JournalofMathematicalAnalysis and Applications,vol.337,no.1,pp.36–48,2008. [13] D. Aniszewska, “Multiplicative Runge-Kutta methods,” Nonlin- ear Dynamics,vol.50,no.1-2,pp.265–272,2007. [14] E. Mısırlı and Y. Gurefe,¨ “Multiplicative adams-bashforth- moulton methods,” Numerical Algorithms,vol.57,no.4,pp. 425–439, 2011. [15] A. F. C¸akmak and F.Bas¸ar, “Some new results on sequence spaces with respect to non-Newtonian calculus,” Journal of Inequalities and Applications,vol.2012,article228,2012. [16] A. F. C¸akmak andF.Bas¸ar, “Certain spaces of functions over the field of non-Newtonian complex numbers,” Abstract and Applied Analysis, vol. 2014, Article ID 236124, 12 pages, 2014. [17] S. Tekin and F. Bas¸ar, “Certain sequence spaces over the non- Newtonian complex field,” Abstract and Applied Analysis,vol. 2013, Article ID 739319, 11 pages, 2013. [18] U. Kadak and H. Efe, “Matrix transformations between certain sequence spaces over the non-Newtonian complex field,” The Scientific World Journal, vol. 2014, Article ID 705818, 12 pages, 2014. [19] U. Kadak and M. Ozluk, “Generalized Runge-Kutta method with respect to the non-Newtonian calculus,” Abstract and Applied Analysis, vol. 2015, Article ID 594685, 10 pages, 2015. [20] U. Kadak, “Determination of the Kothe-Toeplitz¨ duals over the non-Newtonian complex field,” The Scientific World Journal,vol. 2014,ArticleID438924,10pages,2014. [21] U. Kadak, “Non-Newtonian fuzzy numbers and related applications,” Iranian Journal of Fuzzy Systems,vol.12,no.5,pp.117–137, 2015. [22]Y.Gurefe,U.Kadak,E.Misirli,andA.Kurdi,“Anewlookat the classical sequence spaces by using multiplicative calculus,” University Politehnica of Bucharest Scientific Bulletin, Series A: Applied Mathematics and Physics,vol.78,no.2,pp.9–20,2016. Advances in Advances in Journal of Journal of Operations Research Decision Sciences Applied Mathematics Algebra Probability and Statistics Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014

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