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

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A Generalization on Weighted Means and Convex Functions with Respect to the Non-Newtonian Calculus 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 ( ∈ 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 -arithmetic and -arithmetic. Based on classical calculus has an analogue in non-Newtonian calculus two arbitrarily selected generators and ,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 =2and =−1,respectively. −1 are given which will be used for non-Newtonian convexity. In The function : R → R ⊆ R and its inverse are defined this section, the forms of weighted means are presented and as follows ( ∈ R \{0}): an illustrative table is given. In Section 4, the generalized non- Newtonian convex function is defined on the interval and 1/ { ,>0 some types of convex function are obtained by using different { () = 0, = 0 generators. In the final section of the paper, we assert the { notion of multiplicative Lipschitz condition on the closed { − (−)1/ ,<0, interval [, ] ⊂ (0,. ∞) { (4) {, >0 { 2. Preliminary, Background, and Notation −1 { () = {0, =0 { Arithmetic is any system that satisfies the whole of the { ordered field axioms whose domain is a subset of R.There {− (−) , <0. are infinitely many types of arithmetic, all of which are isomorphic, that is, structurally equivalent. It is to be noted that -calculus is reduced to the classical A generator is a one-to-one function whose domain is calculus for =1. Additionally it is concluded that the - R andwhoserangeisasubsetR of R where R = {() : summation can be given as follows: ∈R}. Each generator generates exactly one arithmetic, and conversely each arithmetic is generated by exactly one ∑ ={∑−1 ( )} generator. If (),forall = ∈R,theidentity function’s =1 inverse is itself. In the special cases =and =exp, =1 (5) generates the classical and geometric arithmetic, respectively. −1 −1 + ={ (1)+⋅⋅⋅+ ()} ∀ ∈ R . By -arithmetic, we mean the arithmetic whose domain is R and whose operations are defined as follows: for , ∈ R Definition 1 (see [15]). Let =(,) be an -metric space. and any generator , Thenthebasicnotionscanbedefinedasfollows: −1 −1 -addition +={̇ () + ()} , (a) A sequence =() is a function from the set N into −1 −1 the set R.The-real number denotes the value of -subtraction −={̇ () − ()} , the function at ∈N and is called the th term of the −1 −1 sequence. -multiplication ×={̇ () × ()} , (1) (b) A sequence () in =(,) is said to be - ̇ −1 −1 ̇ ̇ -division /={ () ÷ ()} , convergent if, for every given > 0 (∈R), there exist an 0 =0() ∈ N and ∈such that ̇ −1 −1 ̇ ̇ -order <⇐⇒ () < () . (,)= | −| <for all >0 and is denoted by lim→∞ =or →,as→∞. As a generator, we choose exp function acting from R into the set Rexp =(0,∞)as follows: (c) A sequence () in =(,) is said to be -Cauchy ̇ ̇ if for every > 0 there is an 0 =0() ∈ N such that :R → R ̇ exp (,) <for all , 0 > . (2) → = () = . Throughout this paper, we define the th -exponent (1/) + and th -root of ∈R by It is obvious that -arithmetic reduces to the geometric arith- metic as follows: 2 −1 −1 =×={̇ () × ()} {ln +ln } geometric addition +=̇ =⋅, −1 2 {ln −ln } ={[ ()] }, geometric subtraction −=̇ =÷, 3 2 {ln ln } ln = ×̇ geometric multiplication ×=̇ = (3) ={−1 { [−1 () ×−1 ()]} × −1 ()} =ln , (6) −1 3 ̇ {ln / ln } 1/ ln ={[ ()] }, geometric division /= = , <⇐⇒̇ () < () . geometric order ln ln . Following Grossman and Katz [10] we give the infinitely (−1) −1 = ×={[̇ ()] }, many -arithmetics, of which the quadratic and harmonic International Journal of Analysis 3 (1/2) and √= =provided there exists an ∈R such -arithmetic can readily be transformed into a statement in 2 that =. -arithmetic. Definition 3 (see [10]). The following statements are valid: 2.1. ∗-Arithmetic. Suppose that and are two arbitrarily selected generators and (“star-”) also is the ordered pair of arithmetics ( -arithmetic and -arithmetic). The sets ∗ ∗ ̈ ̈ ̇ ̇ (i) The -points 1, 2,and 3 are -collinear provided (R, +,̈ −,̈ ×,̈ /, <) and (R, +,̇ −,̇ ×,̇ /, <) are complete ordered that at least one of the following holds: fields and (ℎ)-generator generates (ℎ)- arithmetic, respectively. Definitions given for -arithmetic are also valid for -arithmetic. Also -arithmetic is used for ∗ ( ,) +̈ ∗ ( ,)=∗ ( ,), arguments and -arithmeticisusedforvalues;inparticu- 2 1 1 3 2 3 lar, changes in arguments and values are measured by - ∗ ( ,) +̈ ∗ ( ,)=∗ ( ,), differences and -differences, respectively. 1 2 2 3 1 3 (10) ∈(R , +,̇ −,̇ ×,̇ /,̇ <)̇ ∈(R , +,̈ −,̈ ×,̈ /,̈ <)̈ ∗ ∗ ∗ Let and be (1,3) +̈ (3,2)= (1,2). arbitrarily chosen elements from corresponding arithmetic. Then the ordered pair (, ) 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 of at least two distinct points such denoted by C ;thatis, that, for any distinct points 1 and 2 in ,apoint3 is in if and only if 1, 2,and3 are ∗-collinear. When C∗ fl {∗ =(,)|∈R ,∈R }. (7) ==,the∗-lines are the straight lines in two- dimensional Euclidean space. Definition 2 (see [17]). (a) The ∗-limit of a function at an element in R is, if it exists, the unique number in R such that (iii) The ∗-slope of a ∗-line through the points (1,1) and (2,2) is given by ∗ () =⇐⇒ →lim ̈ ̇ ∀ >̈ 0, ∃ >̇ 0∋ () −̈ <̈ (8) ∗ ̈ =(2 −̈ 1) /(2 −̇ 1) ̇ ̇ −1 −1 ∀, ∈ R, − <, ( )− ( ) (11) ={ 2 1 }, ( ≠ ), −1 −1 1 2 ∗ (2)− (1) for ∈R,andiswrittenas lim→(). = Afunction is ∗-continuous at a point in R if and only ∗ if is an argument of and lim→() = ().When , ∈ R , ∈ R and are the identity function ,theconceptsof∗-limit for 1 2 and 1 2 . and ∗-continuity are reduced to those of classical limit and If the following ∗-limit in (12) exists, we denote it by classical continuity. ∗ (),callitthe∗-derivative of at ,andsaythat is ∗- (b) The isomorphism from -arithmetic to -arithmetic is the unique function (iota) which has the following three differentiable at (see [19]): properties: (i) is one to one. ∗ ( () −̈ ()) /(̈ −)̇ lim→ (ii) is from R to R. −1 { ()}−−1 { ()} , V ∈ R = lim{ } (iii) For any numbers , → −1 () −−1 () −1 { ()}−−1 { ()} (12) (+̇ V)=() +̈ (V) ; = lim{ → − (−̇ V)=() −̈ (V) ; −1 (9) − {( ∘) ()} (×̇ V)=() ×̈ (V) ; ⋅ }={ } . −1 () −−1 () (−1) () { } (/̇ V)=() /̈ (V) . −1 3. Non-Newtonian (Weighted) Means It turns out that () = { ()} for every in R and that ()̇ = ̈ for every -integer ̇. Since, for example, Definition 4 (-arithmetic mean).
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