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

This paper is devoted to investigating some characteristic features of weighted means and convex functions in terms of the nonNewtonian 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.


Introduction
It is well known that the theory of convex functions and weighted means plays a very important role in mathematics and other fields.There is wide literature covering this topic (see, e.g., [1][2][3][4][5][6][7][8]). Nowadays the study of convex functions has evolved into a larger theory about functions which are adapted to other geometries of the domain and/or obey other laws of comparison of means.Also the study of convex functions begins in the context of real-valued functions of a real variable.More important, they will serve as a model for deep generalizations into the setting of several variables.
As an alternative to the classical calculus, Grossman and Katz [9][10][11] introduced the non-Newtonian calculus consisting of the branches of geometric, quadratic and harmonic calculus, and so forth.All these calculi can be described simultaneously within the framework of a general theory.They decided to use the adjective non-Newtonian to indicate any calculi other than the classical calculus.Every property in classical calculus has an analogue in non-Newtonian calculus which is a methodology that allows one to have a different look at problems which can be investigated via calculus.In some cases, for example, for wage-rate (in dollars, euro, etc.) related problems, the use of bigeometric calculus which is a kind of non-Newtonian calculus is advocated instead of a traditional Newtonian one.
Many authors have extensively developed the notion of multiplicative calculus; see [12][13][14] for details.Also some authors have also worked on the classical sequence spaces and related topics by using non-Newtonian calculus [15][16][17].Furthermore, Kadak et al. [18,19] characterized the classes of matrix transformations between certain sequence spaces over the non-Newtonian complex field and generalized Runge-Kutta method with respect to the non-Newtonian calculus.For more details, see [20][21][22].
The main focus of this work is to extend weighted means and convex functions based on various generator functions, that is, exp and   ( ∈ R + ) generators.
The rest of this paper is organized as follows: in Section 2, we give some required definitions and consequences related with the -arithmetic and   -arithmetic.Based on two arbitrarily selected generators  and , we give some basic definitions with respect to the * -arithmetic.We also report the most relevant and recent literature in this section.

International Journal of Analysis
In Section 3, first the definitions of non-Newtonian means are given which will be used for non-Newtonian convexity.In this section, the forms of weighted means are presented and an illustrative table is given.In Section 4, the generalized non-Newtonian convex function is defined on the interval   and some types of convex function are obtained by using different generators.In the final section of the paper, we assert the notion of multiplicative Lipschitz condition on the closed interval [, ] ⊂ (0, ∞).

Preliminary, Background, and Notation
Arithmetic is any system that satisfies the whole of the ordered field axioms whose domain is a subset of R.There are infinitely many types of arithmetic, all of which are isomorphic, that is, structurally equivalent.
A generator  is a one-to-one function whose domain is R and whose range is a subset R  of R where R  = {() :  ∈ R}.Each generator generates exactly one arithmetic, and conversely each arithmetic is generated by exactly one generator.If () = , for all  ∈ R, the identity function's inverse is itself.In the special cases  =  and  = exp,  generates the classical and geometric arithmetic, respectively.By -arithmetic, we mean the arithmetic whose domain is R and whose operations are defined as follows: for ,  ∈ R  and any generator , As a generator, we choose exp function acting from R into the set R exp = (0, ∞) as follows: It is obvious that -arithmetic reduces to the geometric arithmetic as follows: geometric addition  +  =  {ln +ln } =  ⋅ , geometric subtraction  −  =  {ln −ln } =  ÷ , geometric multiplication  ×  =  {ln  ln } =  ln  =  ln  , geometric division  /  =  {ln / ln } =  1/ ln  , geometric order  <  ⇐⇒ ln () < ln () .
(3) Following Grossman and Katz [10] we give the infinitely many   -arithmetics, of which the quadratic and harmonic arithmetic are special cases for  = 2 and  = −1, respectively.The function   : R → R  ⊆ R and its inverse  −1  are defined as follows ( ∈ R \ {0}): It is to be noted that   -calculus is reduced to the classical calculus for  = 1.Additionally it is concluded that the summation can be given as follows: ( Definition 1 (see [15]).Let  = (,   ) be an -metric space.
Then the basic notions can be defined as follows: (c) A sequence (  ) in  = (,   ) is said to be -Cauchy if for every  > 0 there is an  0 =  0 () ∈ N such that   (  ,   ) <  for all ,  >  0 .
Throughout this paper, we define the th -exponent    and th -root  (1/)  of  ∈ R + by and  √ =  (1/2)  =  provided there exists an  ∈ R  such that  2  = .
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 (R  , + , − , × , / , < ) and (R  , + , − , × , / , < ) are complete ordered fields and (ℎ)-generator generates (ℎ)arithmetic, respectively.Definitions given for -arithmetic are also valid for -arithmetic.Also -arithmetic is used for arguments and -arithmetic is used for values; in particular, changes in arguments and values are measured by differences and -differences, respectively.
A function  is * -continuous at a point  in R  if and only if  is an argument of  and * lim → () = ().When  and  are the identity function , the concepts of * -limit and * -continuity are reduced to those of classical limit and classical continuity.
(b) The isomorphism from -arithmetic to -arithmetic is the unique function  (iota) which has the following three properties: It turns out that () = { −1 ()} for every  in R  and that ( ṅ ) = n for every -integer ṅ .Since, for example,  + V =  −1 {() + (V)}, it should be clear that any statement in -arithmetic can readily be transformed into a statement in -arithmetic.Definition 3 (see [10]).The following statements are valid: (i) The * -points  1 ,  2 , and  3 are * -collinear provided that at least one of the following holds: for If the following * -limit in (12) exists, we denote it by  * (), call it the * -derivative of  at , and say that  is *differentiable at  (see [19]

Non-Newtonian (Weighted) Means
For  = exp, we obtain that Similarly, for  =   , we get exp and   are called multiplicative arithmetic mean and -arithmetic mean (as usually known p-mean), respectively.One can conclude that   reduces to arithmetic mean and harmonic mean in the ordinary sense for  = 1 and  = −1, respectively.
Remark 5.It is clear that Definition 4 can be written by using various generators.In particular if we take -arithmetic instead of -arithmetic then the mean can be defined by Definition 6 (-geometric mean).Let  1 ,  2 , . . .,   ∈ R + .The -geometric mean, namely,   , is th -root of the product of (  )'s: We conclude similarly, by taking the generators  = exp or  =   , that the -geometric mean can be interpreted as follows: exp and   are called multiplicative geometric mean and -geometric mean, respectively.It would clearly have   =  exp for  = 1.

Non-Newtonian Weighted Means.
The weighted mean is similar to an arithmetic mean, where instead of each of the 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 also occurs in a more general form in several other areas of mathematics.
The following definitions can give the relationships between the non-Newtonian weighted means and ordinary weighted means.Definition 8 (weighted -arithmetic mean).Formally, the weighted -arithmetic mean of a nonempty set of data { 1 ,  2 , . . .,   } with nonnegative weights { 1 ,  2 , . . .,   } is the quantity The formulas are simplified when the weights are normalized such that they -sum up to  ∑  =1 ẇ = 1 .For such normalized weights the weighted -arithmetic mean is simply Ã =  ∑  =1   × ẇ .Note that if all the weights are equal, the weighted -arithmetic mean is the same as the arithmetic mean.
Taking  = exp and  =   , the weighted -arithmetic mean can be given with the weights { 1 ,  2 , . . .,   } as follows: Ãexp and Ã are called multiplicative weighted arithmetic mean and weighted -arithmetic mean, respectively.Ãexp turns out to the ordinary weighted geometric mean.Also, one easily can see that Ã is reduced to ordinary weighted arithmetic mean and weighted harmonic mean for  = 1 and  = −1, respectively.
Definition 9 (weighted -geometric mean).Given a set of positive reals { 1 ,  2 , . . .,   } and corresponding weights { 1 ,  2 , . . .,   }, then the weighted -geometric mean G is defined by Note that if all the weights are equal, the weighted geometric mean is the same as the -geometric mean.Taking  = exp and  =   , the weighted -geometric mean can be written for the weights { 1 ,  2 , . . .,   } as follows: , Gexp and G are called weighted multiplicative geometric mean and weighted -geometric mean.Also we have G = Ãexp for all   > 1.
Definition 10 (weighted -harmonic mean).If a set { 1 ,  2 , . . .,   } of weights is associated with the data set { 1 ,  2 , . . .,   } then the weighted -harmonic mean is defined by Taking  = exp and  =   , the weighted -harmonic mean with the weights { 1 ,  2 , . . .,   } can be written as follows: Hexp and H are called multiplicative weighted harmonic mean and weighted -geometric mean, respectively.It is obvious that H is reduced to ordinary weighted harmonic mean and ordinary weighted arithmetic mean for  = 1 and  = −1, respectively.In Table 1, the non-Newtonian means are obtained by using different generating functions.For  =   , the -means   ,   and   are reduced to ordinary arithmetic mean, geometric mean, and harmonic mean, respectively.In particular some changes are observed for each value of   ,   , and   means depending on the choice of .As shown in the table, for increasing values of , the -arithmetic mean   and its weighted form Ã increase; in particular  tends to ∞, and these means converge to the value of max{  }.Conversely, for increasing values of , the -harmonic mean   and its weighted forms H decrease.In particular, these means converge to the value of min{  as  → ∞.Depending on the choice of , weighted forms Ã and H , can be increased or decreased without changing any weights.For this reason, this approach brings a new perspective to the concept of classical (weighted) mean.Moreover, when we compare  exp and ordinary harmonic mean in Table 1, we also see that ordinary harmonic mean is smaller than  exp .On the contrary  exp and  exp are smaller than their classical forms   and   for  = 1.Therefore, we assert that the values of  exp ,  exp , Gexp , and Hexp should be evaluated satisfactorily.

Non-Newtonian Convexity
In this section, the notion of non-Newtonian convex ( *convex) functions will be given by using different generators.Furthermore the relationships between * -convexity and non-Newtonian weighted mean will be determined.
For this reason, we say that almost all the properties of ordinary Cartesian coordinate system will be valid for non-Newtonian coordinate system under * -arithmetic.
Also depending on the choice of generator functions, the definition of * -convexity in (27) can be interpreted as follows.
On the other hand (37) means that if , , and  are any three * -points on the graph of  with  between  and , then  is on or below chord .In terms of * -slope, it is equivalent to with strict inequalities when  is strictly * -convex.Now to avoid the repetition of the similar statements, we give some necessary theorems and lemmas.(40)

Lemma 14 (Jensen's inequality
Proof.The proof can be easily obtained using the inequality (39) in Lemma 14.

An Application of Multiplicative Continuity
In this section based on the definition of bigeometric convex function and multiplicative continuity, we get an analogue of ordinary Lipschitz condition on any closed interval.
Let  be a bigeometric (multiplicative) convex function and finite on a closed interval [, ] ⊂ R + .It is obvious that is bounded from above by  = max{(), ()}, since, for any  =    1− in the interval, () ≤ ()  () 1− for (a) A sequence  = (  ) is a function from the set N into the set R  .The -real number   denotes the value of the function at  ∈ N and is called the th term of the sequence.(b) A sequence (  ) in  = (,  ) is said to be convergent if, for every given  > 0 ( ∈ R  ), there exist an  0 =  0 () ∈ N and  ∈  such that   (  , ) = |  − |  <  for all  >  0 and is denoted by  lim →∞   =  or     → , as  → ∞.

Table 1 :
Comparison of the non-Newtonian (weighted) means and ordinary (weighted) means.