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

Authors: Uğur Kadak; Yusuf Gürefe
Journal: International Journal of Analysis
 (2016)

---

## Abstract

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.


---

## Body

## 1. 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., [1–8]). 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 [9–11] 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 adjectivenon-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 [12–14] for details. Also some authors have also worked on the classical sequence spaces and related topics by using non-Newtonian calculus [15–17]. 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 [20–22].The main focus of this work is to extend weighted means and convex functions based on various generator functions,that is, e
x
p and q
p
(
p
∈
R
+) generators.The rest of this paper is organized as follows: in Section2, we give some required definitions and consequences related with theα-arithmetic and q
p-arithmetic. Based on two arbitrarily selected generatorsαandβ, we give some basic definitions with respect to the∗-arithmetic. We also report the most relevant and recent literature in this section. 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 intervalI
αand 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 intervalx
,
y
⊂
0
,
∞.


## 2. Preliminary, Background, and Notation

Arithmetic is any system that satisfies the whole of the ordered field axioms whose domain is a subset ofR. There are infinitely many types of arithmetic, all of which are isomorphic, that is, structurally equivalent.Agenerator
α is a one-to-one function whose domain isRand whose range is a subset R
αofRwhere R
α
=
α
x
:
x
∈
R. Each generator generates exactly one arithmetic, and conversely each arithmetic is generated by exactly one generator. If I
x
=
x, for all x
∈
R, theidentity function’s inverse is itself. In the special cases α
=
I and α
=
exp
⁡, α generates the classical and geometric arithmetic, respectively. By α
-arithmetic, we mean the arithmetic whose domain is R and whose operations are defined as follows: for x
,
y
∈
R
α and any generator α, (1)
α
-addition
x
+
˙
y
=
α
α
-
1
x
+
α
-
1
y
,
α
-subtraction
x
-
˙
y
=
α
α
-
1
x
-
α
-
1
y
,
α
-multiplication
x
×
˙
y
=
α
α
-
1
x
×
α
-
1
y
,
α
-division
x
/
˙
y
=
α
α
-
1
x
÷
α
-
1
y
,
α
-order
x
<
˙
y
⟺
α
-
1
x
<
α
-
1
y
. As a generator, we choose exp function acting from R into the set R
e
x
p
=
(
0
,
∞
) as follows: (2)
α
:
R
⟶
R
e
x
p
x
⟼
y
=
α
x
=
e
x
. It is obvious that α-arithmetic reduces to the geometric arithmetic as follows:(3)
g
e
o
m
e
t
r
i
c
a
d
d
i
t
i
o
n
x
+
˙
y
=
e
ln
⁡
x
+
ln
⁡
y
=
x
·
y
,
g
e
o
m
e
t
r
i
c
s
u
b
t
r
a
c
t
i
o
n
x
-
˙
y
=
e
ln
⁡
x
-
ln
⁡
y
=
x
÷
y
,
g
e
o
m
e
t
r
i
c
m
u
l
t
i
p
l
i
c
a
t
i
o
n
x
×
˙
y
=
e
ln
⁡
x
ln
⁡
y
=
x
ln
⁡
y
=
y
ln
⁡
x
,
g
e
o
m
e
t
r
i
c
d
i
v
i
s
i
o
n
x
/
˙
y
=
e
ln
⁡
x
/
ln
⁡
y
=
x
1
/
ln
⁡
y
,
g
e
o
m
e
t
r
i
c
o
r
d
e
r
x
<
˙
y
⟺
ln
⁡
x
<
ln
⁡
y
.Following Grossman and Katz [10] we give the infinitely many q
p-arithmetics, of which the quadratic and harmonic arithmetic are special cases for p
=
2 and p
=
-
1, respectively. The function q
p
:
R
→
R
q
⊆
R and its inverse q
p
-
1 are defined as follows p
∈
R
∖
0:(4)
q
p
x
=
x
1
/
p
,
x
>
0
0
,
x
=
0
-
-
x
1
/
p
,
x
<
0
,
q
p
-
1
x
=
x
p
,
x
>
0
0
,
x
=
0
-
-
x
p
,
x
<
0
.It is to be noted that q
p-calculus is reduced to the classical calculus for p
=
1. Additionally it is concluded that the α-summation can be given as follows: (5)
∑
α
k
=
1
n
⁡
x
k
=
α
∑
k
=
1
n
α
-
1
x
k
=
α
α
-
1
x
1
+
⋯
+
α
-
1
x
n
∀
x
k
∈
R
+
.Definition 1 (see [15]).
LetX
=
X
,
d
α be an α-metric space. Then the basic notions can be defined as follows:(a)
A sequencex
=
(
x
k
) is a function from the set N into the set R
α. The α-real number x
k denotes the value of the function at k
∈
N and is called the kth term of the sequence.
(b)
A sequence(
x
n
) in X
=
(
X
,
d
α
) is said to be α-convergent if, for every given ε
>
˙
0
˙ (ε
∈
R
α), there exist an n
0
=
n
0
(
ε
)
∈
N and x
∈
X such that d
α
x
n
,
x
=
x
n
-
˙
x
α
<
˙
ε for all n
>
n
0 and is denoted by α
lim
n
→
∞
⁡
x
n
=
x or x
n
→
α
x, as n
→
∞.
(c)
A sequencex
n in X
=
X
,
d
α is said to be α-Cauchy if for every ε
>
˙
0
˙ there is an n
0
=
n
0
(
ε
)
∈
N such that d
α
x
n
,
x
m
<
˙
ε for all m
,
n
>
n
0.
Throughout this paper, we define thepth α-exponent x
p
α and qth α-root x
1
/
q
α of x
∈
R
+ by (6)
x
2
α
=
x
×
˙
x
=
α
α
-
1
x
×
α
-
1
x
=
α
α
-
1
x
2
,
x
3
α
=
x
2
α
×
˙
x
=
α
α
-
1
α
α
-
1
x
×
α
-
1
x
×
α
-
1
x
=
α
α
-
1
x
3
,
⋮
x
p
α
=
x
p
-
1
α
×
˙
x
=
α
α
-
1
x
p
,and x
α
=
x
1
/
2
α
=
y provided there exists an y
∈
R
α such that y
2
α
=
x.


### 2.1.∗-Arithmetic

Suppose thatα and β are two arbitrarily selected generators and (“star-”) also is the ordered pair of arithmetics (β-arithmetic and α-arithmetic). The sets (
R
β
,
+
¨
,
-
¨
,
×
¨
,
/
¨
,
<
¨
) and (
R
α
,
+
˙
,
-
˙
,
×
˙
,
/
˙
,
<
˙
) are complete ordered fields and b
e
t
a
(
a
l
p
h
a
)-generator generates b
e
t
a
(
a
l
p
h
a
)-arithmetic, respectively. Definitions given for β-arithmetic are also valid for α-arithmetic. Also α-arithmetic is used for arguments and β-arithmetic is used for values; in particular, changes in arguments and values are measured by α-differences and β-differences, respectively.Letx
∈
(
R
α
,
+
˙
,
-
˙
,
×
˙
,
/
˙
,
<
˙
) and y
∈
(
R
β
,
+
¨
,
-
¨
,
×
¨
,
/
¨
,
<
¨
) be 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 denoted by C
∗; that is, (7)
C
∗
≔
z
∗
=
x
,
y
∣
x
∈
R
α
,
y
∈
R
β
.Definition 2 (see [17]).
(a) The∗-limit of a function f at an element a in R
α is, if it exists, the unique number b in R
β such that (8)
lim
x
→
a
∗
⁡
f
x
=
b
⟺
∀
ε
>
¨
0
¨
,
∃
δ
>
˙
0
˙
∋
f
x
-
¨
b
β
<
¨
ε
∀
x
,
ε
∈
R
α
,
x
-
˙
a
α
<
˙
δ
,for δ
∈
R
β, and is written as ∗
lim
x
→
a
⁡
f
(
x
)
=
b.
A functionf is ∗-continuous at a point a in R
α if and only if a is an argument  of f and ∗
lim
x
→
a
⁡
f
(
x
)
=
f
(
a
). When α and β are the identity function I, the  concepts of ∗-limit and ∗-continuity are reduced to those of classical limit and  classical continuity.
(b) The isomorphism fromα-arithmetic to β-arithmetic is the unique function ι (iota) which has the following three properties:(i)
ι is one to one.
(ii)
ι is from R
α to R
β.
(iii)
For any numbersu
,
v
∈
R
α, (9)
ι
u
+
˙
v
=
ι
u
+
¨
ι
v
;
ι
u
-
˙
v
=
ι
u
-
¨
ι
v
;
ι
u
×
˙
v
=
ι
u
×
¨
ι
v
;
ι
u
/
˙
v
=
ι
u
/
¨
ι
v
.It turns out that ι
(
x
)
=
β
α
-
1
x for every x in R
α and that ι
(
n
˙
)
=
n
¨ for every α-integer n
˙. Since, for example, u
+
˙
v
=
ι
-
1
ι
u
+
¨
ι
v, it should be clear that any statement in α-arithmetic can readily be transformed into a statement in β-arithmetic.Definition 3 (see [10]).
The following statements are valid:(i)
The∗-points P
1, P
2, and P
3 are ∗-collinear provided that at least one of the following holds:(10)
d
∗
P
2
,
P
1
+
¨
d
∗
P
1
,
P
3
=
d
∗
P
2
,
P
3
,
d
∗
P
1
,
P
2
+
¨
d
∗
P
2
,
P
3
=
d
∗
P
1
,
P
3
,
d
∗
P
1
,
P
3
+
¨
d
∗
P
3
,
P
2
=
d
∗
P
1
,
P
2
.
(ii)
A∗-line is a set L of at least two distinct points such that, for any distinct points P
1 and P
2 in L, a point P
3 is in L if and only if P
1, P
2, and P
3 are ∗-collinear. When α
=
β
=
I, the ∗-lines are the straight lines in two-dimensional Euclidean space.
(iii)
The∗-slope of a ∗-line through the points (
a
1
,
b
1
) and (
a
2
,
b
2
) is given by (11)
m
∗
=
b
2
-
¨
b
1
/
¨
ι
a
2
-
˙
a
1
=
β
β
-
1
b
2
-
β
-
1
b
1
α
-
1
a
2
-
α
-
1
a
1
,
a
1
≠
a
2
,for a
1
,
a
2
∈
R
α and b
1
,
b
2
∈
R
β.If the following∗-limit in (12) exists, we denote it by f
∗
(
t
), call it the ∗-derivative of f at t, and say that f is ∗-differentiable at t (see [19]):(12)
∗
lim
x
→
t
⁡
f
x
-
¨
f
t
/
¨
ι
x
-
˙
t
=
lim
x
→
t
⁡
β
β
-
1
f
x
-
β
-
1
f
t
α
-
1
x
-
α
-
1
t
=
lim
x
→
t
⁡
β
β
-
1
f
x
-
β
-
1
f
t
x
-
t
x
-
t
α
-
1
x
-
α
-
1
t
=
β
β
-
1
∘
f
′
t
α
-
1
′
t
.


## 2.1.∗-Arithmetic

Suppose thatα and β are two arbitrarily selected generators and (“star-”) also is the ordered pair of arithmetics (β-arithmetic and α-arithmetic). The sets (
R
β
,
+
¨
,
-
¨
,
×
¨
,
/
¨
,
<
¨
) and (
R
α
,
+
˙
,
-
˙
,
×
˙
,
/
˙
,
<
˙
) are complete ordered fields and b
e
t
a
(
a
l
p
h
a
)-generator generates b
e
t
a
(
a
l
p
h
a
)-arithmetic, respectively. Definitions given for β-arithmetic are also valid for α-arithmetic. Also α-arithmetic is used for arguments and β-arithmetic is used for values; in particular, changes in arguments and values are measured by α-differences and β-differences, respectively.Letx
∈
(
R
α
,
+
˙
,
-
˙
,
×
˙
,
/
˙
,
<
˙
) and y
∈
(
R
β
,
+
¨
,
-
¨
,
×
¨
,
/
¨
,
<
¨
) be 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 denoted by C
∗; that is, (7)
C
∗
≔
z
∗
=
x
,
y
∣
x
∈
R
α
,
y
∈
R
β
.Definition 2 (see [17]).
(a) The∗-limit of a function f at an element a in R
α is, if it exists, the unique number b in R
β such that (8)
lim
x
→
a
∗
⁡
f
x
=
b
⟺
∀
ε
>
¨
0
¨
,
∃
δ
>
˙
0
˙
∋
f
x
-
¨
b
β
<
¨
ε
∀
x
,
ε
∈
R
α
,
x
-
˙
a
α
<
˙
δ
,for δ
∈
R
β, and is written as ∗
lim
x
→
a
⁡
f
(
x
)
=
b.
A functionf is ∗-continuous at a point a in R
α if and only if a is an argument  of f and ∗
lim
x
→
a
⁡
f
(
x
)
=
f
(
a
). When α and β are the identity function I, the  concepts of ∗-limit and ∗-continuity are reduced to those of classical limit and  classical continuity.
(b) The isomorphism fromα-arithmetic to β-arithmetic is the unique function ι (iota) which has the following three properties:(i)
ι is one to one.
(ii)
ι is from R
α to R
β.
(iii)
For any numbersu
,
v
∈
R
α, (9)
ι
u
+
˙
v
=
ι
u
+
¨
ι
v
;
ι
u
-
˙
v
=
ι
u
-
¨
ι
v
;
ι
u
×
˙
v
=
ι
u
×
¨
ι
v
;
ι
u
/
˙
v
=
ι
u
/
¨
ι
v
.It turns out that ι
(
x
)
=
β
α
-
1
x for every x in R
α and that ι
(
n
˙
)
=
n
¨ for every α-integer n
˙. Since, for example, u
+
˙
v
=
ι
-
1
ι
u
+
¨
ι
v, it should be clear that any statement in α-arithmetic can readily be transformed into a statement in β-arithmetic.Definition 3 (see [10]).
The following statements are valid:(i)
The∗-points P
1, P
2, and P
3 are ∗-collinear provided that at least one of the following holds:(10)
d
∗
P
2
,
P
1
+
¨
d
∗
P
1
,
P
3
=
d
∗
P
2
,
P
3
,
d
∗
P
1
,
P
2
+
¨
d
∗
P
2
,
P
3
=
d
∗
P
1
,
P
3
,
d
∗
P
1
,
P
3
+
¨
d
∗
P
3
,
P
2
=
d
∗
P
1
,
P
2
.
(ii)
A∗-line is a set L of at least two distinct points such that, for any distinct points P
1 and P
2 in L, a point P
3 is in L if and only if P
1, P
2, and P
3 are ∗-collinear. When α
=
β
=
I, the ∗-lines are the straight lines in two-dimensional Euclidean space.
(iii)
The∗-slope of a ∗-line through the points (
a
1
,
b
1
) and (
a
2
,
b
2
) is given by (11)
m
∗
=
b
2
-
¨
b
1
/
¨
ι
a
2
-
˙
a
1
=
β
β
-
1
b
2
-
β
-
1
b
1
α
-
1
a
2
-
α
-
1
a
1
,
a
1
≠
a
2
,for a
1
,
a
2
∈
R
α and b
1
,
b
2
∈
R
β.If the following∗-limit in (12) exists, we denote it by f
∗
(
t
), call it the ∗-derivative of f at t, and say that f is ∗-differentiable at t (see [19]):(12)
∗
lim
x
→
t
⁡
f
x
-
¨
f
t
/
¨
ι
x
-
˙
t
=
lim
x
→
t
⁡
β
β
-
1
f
x
-
β
-
1
f
t
α
-
1
x
-
α
-
1
t
=
lim
x
→
t
⁡
β
β
-
1
f
x
-
β
-
1
f
t
x
-
t
x
-
t
α
-
1
x
-
α
-
1
t
=
β
β
-
1
∘
f
′
t
α
-
1
′
t
.


## 3. Non-Newtonian (Weighted) Means

Definition 4 (α-arithmetic mean).
Consider thatn positive real numbers x
1
,
x
2
,
…
,
x
n are given. The α-mean (average), denoted by A
α, is the α-sum of x
n’s α-divided by n
˙ for all n
∈
N. That is, (13)
A
α
=
∑
α
k
=
1
n
⁡
x
k
/
˙
n
˙
=
∑
α
k
=
1
n
⁡
α
α
-
1
x
k
n
=
α
α
-
1
x
1
+
α
-
1
x
2
+
⋯
+
α
-
1
x
n
n
.For α
=
e
x
p, we obtain that (14)
A
e
x
p
=
∏
k
=
1
n
x
k
1
/
n
=
x
1
·
x
2
⋯
x
n
1
/
n
.Similarly, for α
=
q
p, we get (15)
A
p
=
x
1
p
+
x
2
p
+
⋯
+
x
n
p
n
1
/
p
,
p
∈
R
∖
0
.
A
e
x
p and A
p are called multiplicative arithmetic mean and p-arithmetic mean (as usually knownp-mean), respectively. One can conclude that A
p reduces to arithmetic mean and harmonic mean in the ordinary sense for p
=
1 and p
=
-
1, respectively.Remark 5.
It is clear that Definition4 can be written by using various generators. In particular if we take β-arithmetic instead of α-arithmetic then the mean can be defined by (16)
A
β
=
∑
β
k
=
1
n
⁡
x
k
/
¨
n
¨
.Definition 6 (α-geometric mean).
Letx
1
,
x
2
,
…
,
x
n
∈
R
+. Theα-geometric mean, namely,G
α, isnthα-root of theα-product ofx
n’s:(17)
G
α
=
∏
α
k
=
1
n
⁡
x
k
1
/
n
α
=
α
∏
k
=
1
n
α
-
1
x
k
1
/
n
=
α
α
-
1
x
1
,
α
-
1
x
2
,
…
,
α
-
1
x
n
1
/
n
.
We conclude similarly, by taking the generatorsα
=
e
x
porα
=
q
p, that theα-geometric mean can be interpreted as follows: (18)
G
e
x
p
=
exp
⁡
ln
⁡
x
1
,
ln
⁡
x
2
,
…
,
ln
⁡
x
n
1
/
n
,
x
n
>
1
G
p
=
x
1
p
,
x
2
p
,
…
,
x
n
p
1
/
n
1
/
p
=
x
1
,
x
2
,
…
,
x
n
1
/
n
,
p
≠
0
.
G
e
x
pandG
pare called multiplicative geometric mean andp-geometric mean, respectively. It would clearly haveG
p
=
A
e
x
pforp
=
1.Definition 7 (α-harmonic mean).
Letx
1
,
x
2
,
…
,
x
n
∈
R
+ and α
-
1
(
x
n
)
≠
0 for eachn
∈
N. Theα-harmonic meanH
αis defined by (19)
H
α
=
n
˙
/
˙
∑
α
k
=
1
n
1
˙
/
˙
x
k
=
n
˙
/
˙
1
˙
/
˙
x
1
+
˙
1
˙
/
˙
x
2
+
˙
⋯
+
˙
1
˙
/
˙
x
n
=
α
n
1
/
α
-
1
x
1
+
1
/
α
-
1
x
2
+
⋯
+
1
/
α
-
1
x
n
.Similarly, one obtains that(20)
H
e
x
p
=
exp
⁡
n
1
/
l
n
x
1
+
1
/
l
n
x
2
+
⋯
+
1
/
l
n
x
n
,
x
n
>
1
,
H
p
=
n
1
/
x
1
p
+
1
/
x
2
p
+
⋯
+
1
/
x
n
p
1
/
p
.
H
e
x
p and H
p are called multiplicative harmonic mean andp-harmonic mean, respectively. Obviously the inclusion (20) is reduced to ordinary harmonic mean and ordinary arithmetic mean for p
=
1andp
=
-
1, respectively.


### 3.1. 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 (weightedα-arithmetic mean).
Formally, the weightedα-arithmetic mean of a nonempty set of data{
x
1
,
x
2
,
…
,
x
n
}with nonnegative weights{
w
1
,
w
2
,
…
,
w
n
}is the quantity(21)
A
~
α
=
∑
α
i
=
1
n
⁡
x
i
×
˙
w
˙
i
/
˙
∑
α
i
=
1
n
⁡
w
˙
i
=
x
1
×
˙
w
˙
1
+
˙
x
2
×
˙
w
˙
2
+
˙
⋯
+
˙
x
n
×
˙
w
˙
n
/
˙
w
˙
1
+
˙
w
˙
2
+
˙
⋯
+
˙
w
˙
n
=
α
∑
i
=
1
n
w
i
α
-
1
x
i
∑
i
=
1
n
w
i
.
The formulas are simplified when the weights areα-normalized such that theyα-sum up to∑
i
=
1
n
‍
α
w
˙
i
=
1
˙. For such normalized weights the weightedα-arithmetic mean is simply A
~
α
=
∑
i
=
1
n
x
i
×
˙
w
˙
i
α. Note that if all the weights are equal, the weightedα-arithmetic mean is the same as theα-arithmetic mean.
Takingα
=
e
x
p andα
=
q
p, the weightedα-arithmetic mean can be given with the weights{
w
1
,
w
2
,
…
,
w
n
}as follows: (22)
A
~
e
x
p
=
exp
⁡
∑
i
=
1
n
ln
⁡
x
i
w
i
∑
i
=
1
n
w
i
=
∏
i
=
1
n
x
i
w
i
1
/
∑
i
=
1
n
w
i
,
x
n
>
1
,
A
~
q
p
=
w
1
x
1
p
+
w
2
x
2
p
+
⋯
+
w
n
x
n
p
w
1
+
w
2
+
⋯
+
w
n
1
/
p
.
A
~
e
x
p and A
~
p are called multiplicative weighted arithmetic mean and weightedp-arithmetic mean, respectively.A
~
e
x
pturns out to the ordinary weighted geometric mean. Also, one easily can see that A
~
p is reduced to ordinary weighted arithmetic mean and weighted harmonic mean forp
=
1andp
=
-
1, respectively.Definition 9 (weightedα-geometric mean).
Given a set of positive reals{
x
1
,
x
2
,
…
,
x
n
}and corresponding weights{
w
1
,
w
2
,
…
,
w
n
}, then the weightedα-geometric mean G
~
α is defined by (23)
G
~
α
=
∏
α
i
=
1
n
⁡
x
i
w
i
/
∑
i
=
1
n
w
i
α
=
x
1
w
1
/
w
1
+
⋯
+
w
n
α
×
˙
x
2
w
2
/
w
1
+
⋯
+
w
n
α
×
˙
⋯
×
˙
x
n
w
n
/
w
1
+
⋯
+
w
n
α
=
α
α
-
1
x
1
w
1
/
w
1
+
⋯
+
w
n
,
α
-
1
x
2
w
2
/
w
1
+
⋯
+
w
n
,
…
,
α
-
1
x
n
w
n
/
w
1
+
⋯
+
w
n
.
Note that if all the weights are equal, the weightedα-geometric mean is the same as theα-geometric mean. Takingα
=
e
x
pandα
=
q
p, the weightedα-geometric mean can be written for the weights{
w
1
,
w
2
,
…
,
w
n
}as follows: (24)
G
~
e
x
p
=
e
x
p
ln
⁡
x
1
w
1
/
w
1
+
⋯
+
w
n
,
l
n
x
2
w
2
/
w
1
+
⋯
+
w
n
,
…
,
l
n
x
n
w
n
/
w
1
+
⋯
+
w
n
,
x
n
>
1
,
G
~
p
=
x
1
p
w
1
/
w
1
+
⋯
+
w
n
,
x
2
p
w
2
/
w
1
+
⋯
+
w
n
,
…
,
x
n
p
w
n
/
w
1
+
⋯
+
w
n
1
/
p
=
∏
i
=
1
n
x
i
w
i
1
/
∑
i
=
1
n
w
i
.
G
~
e
x
p and G
~
p are called weighted multiplicative geometric mean and weightedq-geometric mean. Also we haveG
~
p
=
A
~
e
x
p for allx
n
>
1.Definition 10 (weightedα-harmonic mean).
If a setw
1
,
w
2
,
…
,
w
nof weights is associated with the data setx
1
,
x
2
,
…
,
x
nthen the weightedα-harmonic mean is defined by (25)
H
~
α
=
∑
α
k
=
1
n
⁡
w
˙
k
/
˙
∑
α
k
=
1
n
⁡
w
˙
k
/
˙
x
k
=
w
˙
1
+
˙
⋯
+
˙
w
˙
n
/
˙
w
˙
1
/
˙
x
1
+
˙
w
˙
2
/
˙
x
2
+
˙
⋯
+
˙
w
˙
n
/
˙
x
n
=
α
w
1
+
w
2
+
⋯
+
w
n
w
1
/
α
-
1
x
1
+
w
2
/
α
-
1
x
2
+
⋯
+
w
n
/
α
-
1
x
n
.
Takingα
=
e
x
pandα
=
q
p, the weightedα-harmonic mean with the weights{
w
1
,
w
2
,
…
,
w
n
}can be written as follows: (26)
H
~
e
x
p
=
e
x
p
w
1
+
w
2
+
⋯
+
w
n
w
1
/
ln
⁡
x
1
+
w
2
/
ln
⁡
x
2
+
⋯
+
w
n
/
ln
⁡
x
n
,
x
n
>
1
,
H
~
p
=
w
1
+
w
2
+
⋯
+
w
n
w
1
/
x
1
p
+
w
2
/
x
2
p
+
⋯
+
w
n
/
x
n
p
1
/
p
.
H
~
e
x
p and H
~
p are called multiplicative weighted harmonic mean and weightedp-geometric mean, respectively. It is obvious that H
~
p is reduced to ordinary weighted harmonic mean and ordinary weighted arithmetic mean forp
=
1andp
=
-
1, respectively.In Table1, the non-Newtonian means are obtained by using different generating functions. Forα
=
q
p, thep-meansA
p
,
G
pandH
pare reduced to ordinary arithmetic mean, geometric mean, and harmonic mean, respectively. In particular some changes are observed for each value ofA
p
,
G
p, andH
pmeans depending on the choice ofp. As shown in the table, for increasing values ofp, thep-arithmetic meanA
pand its weighted formA
~
p increase; in particularptends to∞, and these means converge to the value of max
⁡
{
x
n
}. Conversely, for increasing values ofp, thep-harmonic mean H
p and its weighted forms H
~
p decrease. In particular, these means converge to the value of min
⁡
{
x
n
}asp
→
∞. Depending on the choice ofp, weighted formsA
~
p and H
~
p, 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 compareH
e
x
pand ordinary harmonic mean in Table 1, we also see that ordinary harmonic mean is smaller thanH
e
x
p. On the contraryA
e
x
pandG
e
x
pare smaller than their classical formsA
pandG
pforp
=
1. Therefore, we assert that the values of G
e
x
p
,
H
e
x
p
,
G
~
e
x
p, and H
~
e
x
p should be evaluated satisfactorily.Table 1
Comparison of the non-Newtonian (weighted) means and ordinary (weighted) means.
Weight
Data
p
A
p
G
p
H
p
A
~
p
G
~
p
H
~
p
G
e
x
p
H
e
x
p
G
~
e
x
p
H
~
e
x
p
w
1
=
2
x
1
=
15
1.0
27.50
24.74
22.64
32.82
29.88
27.27
24.02
23.36
29.05
28.26
w
2
=
5
x
2
=
20
0.1
24.99
24.74
24.50
30.16
29.88
29.59
24.02
23.36
29.05
28.26
w
3
=
7
x
3
=
25
2.0
30.61
24.74
21.14
35.77
29.88
25.21
24.02
23.36
29.05
28.26
w
4
=
9
x
4
=
50
5.0
38.22
24.74
18.73
41.69
29.88
21.55
24.02
23.36
29.05
28.26Corollary 11.
Considernpositive real numbersx
1
,
x
2
,
…
,
x
n. Then, the conditionsH
α
<
G
α
<
A
α and H
~
α
<
G
~
α
<
A
~
α hold whenα
=
e
x
pfor allx
n
>
1andα
=
q
pfor allp
∈
R
+.


## 3.1. 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 (weightedα-arithmetic mean).
Formally, the weightedα-arithmetic mean of a nonempty set of data{
x
1
,
x
2
,
…
,
x
n
}with nonnegative weights{
w
1
,
w
2
,
…
,
w
n
}is the quantity(21)
A
~
α
=
∑
α
i
=
1
n
⁡
x
i
×
˙
w
˙
i
/
˙
∑
α
i
=
1
n
⁡
w
˙
i
=
x
1
×
˙
w
˙
1
+
˙
x
2
×
˙
w
˙
2
+
˙
⋯
+
˙
x
n
×
˙
w
˙
n
/
˙
w
˙
1
+
˙
w
˙
2
+
˙
⋯
+
˙
w
˙
n
=
α
∑
i
=
1
n
w
i
α
-
1
x
i
∑
i
=
1
n
w
i
.
The formulas are simplified when the weights areα-normalized such that theyα-sum up to∑
i
=
1
n
‍
α
w
˙
i
=
1
˙. For such normalized weights the weightedα-arithmetic mean is simply A
~
α
=
∑
i
=
1
n
x
i
×
˙
w
˙
i
α. Note that if all the weights are equal, the weightedα-arithmetic mean is the same as theα-arithmetic mean.
Takingα
=
e
x
p andα
=
q
p, the weightedα-arithmetic mean can be given with the weights{
w
1
,
w
2
,
…
,
w
n
}as follows: (22)
A
~
e
x
p
=
exp
⁡
∑
i
=
1
n
ln
⁡
x
i
w
i
∑
i
=
1
n
w
i
=
∏
i
=
1
n
x
i
w
i
1
/
∑
i
=
1
n
w
i
,
x
n
>
1
,
A
~
q
p
=
w
1
x
1
p
+
w
2
x
2
p
+
⋯
+
w
n
x
n
p
w
1
+
w
2
+
⋯
+
w
n
1
/
p
.
A
~
e
x
p and A
~
p are called multiplicative weighted arithmetic mean and weightedp-arithmetic mean, respectively.A
~
e
x
pturns out to the ordinary weighted geometric mean. Also, one easily can see that A
~
p is reduced to ordinary weighted arithmetic mean and weighted harmonic mean forp
=
1andp
=
-
1, respectively.Definition 9 (weightedα-geometric mean).
Given a set of positive reals{
x
1
,
x
2
,
…
,
x
n
}and corresponding weights{
w
1
,
w
2
,
…
,
w
n
}, then the weightedα-geometric mean G
~
α is defined by (23)
G
~
α
=
∏
α
i
=
1
n
⁡
x
i
w
i
/
∑
i
=
1
n
w
i
α
=
x
1
w
1
/
w
1
+
⋯
+
w
n
α
×
˙
x
2
w
2
/
w
1
+
⋯
+
w
n
α
×
˙
⋯
×
˙
x
n
w
n
/
w
1
+
⋯
+
w
n
α
=
α
α
-
1
x
1
w
1
/
w
1
+
⋯
+
w
n
,
α
-
1
x
2
w
2
/
w
1
+
⋯
+
w
n
,
…
,
α
-
1
x
n
w
n
/
w
1
+
⋯
+
w
n
.
Note that if all the weights are equal, the weightedα-geometric mean is the same as theα-geometric mean. Takingα
=
e
x
pandα
=
q
p, the weightedα-geometric mean can be written for the weights{
w
1
,
w
2
,
…
,
w
n
}as follows: (24)
G
~
e
x
p
=
e
x
p
ln
⁡
x
1
w
1
/
w
1
+
⋯
+
w
n
,
l
n
x
2
w
2
/
w
1
+
⋯
+
w
n
,
…
,
l
n
x
n
w
n
/
w
1
+
⋯
+
w
n
,
x
n
>
1
,
G
~
p
=
x
1
p
w
1
/
w
1
+
⋯
+
w
n
,
x
2
p
w
2
/
w
1
+
⋯
+
w
n
,
…
,
x
n
p
w
n
/
w
1
+
⋯
+
w
n
1
/
p
=
∏
i
=
1
n
x
i
w
i
1
/
∑
i
=
1
n
w
i
.
G
~
e
x
p and G
~
p are called weighted multiplicative geometric mean and weightedq-geometric mean. Also we haveG
~
p
=
A
~
e
x
p for allx
n
>
1.Definition 10 (weightedα-harmonic mean).
If a setw
1
,
w
2
,
…
,
w
nof weights is associated with the data setx
1
,
x
2
,
…
,
x
nthen the weightedα-harmonic mean is defined by (25)
H
~
α
=
∑
α
k
=
1
n
⁡
w
˙
k
/
˙
∑
α
k
=
1
n
⁡
w
˙
k
/
˙
x
k
=
w
˙
1
+
˙
⋯
+
˙
w
˙
n
/
˙
w
˙
1
/
˙
x
1
+
˙
w
˙
2
/
˙
x
2
+
˙
⋯
+
˙
w
˙
n
/
˙
x
n
=
α
w
1
+
w
2
+
⋯
+
w
n
w
1
/
α
-
1
x
1
+
w
2
/
α
-
1
x
2
+
⋯
+
w
n
/
α
-
1
x
n
.
Takingα
=
e
x
pandα
=
q
p, the weightedα-harmonic mean with the weights{
w
1
,
w
2
,
…
,
w
n
}can be written as follows: (26)
H
~
e
x
p
=
e
x
p
w
1
+
w
2
+
⋯
+
w
n
w
1
/
ln
⁡
x
1
+
w
2
/
ln
⁡
x
2
+
⋯
+
w
n
/
ln
⁡
x
n
,
x
n
>
1
,
H
~
p
=
w
1
+
w
2
+
⋯
+
w
n
w
1
/
x
1
p
+
w
2
/
x
2
p
+
⋯
+
w
n
/
x
n
p
1
/
p
.
H
~
e
x
p and H
~
p are called multiplicative weighted harmonic mean and weightedp-geometric mean, respectively. It is obvious that H
~
p is reduced to ordinary weighted harmonic mean and ordinary weighted arithmetic mean forp
=
1andp
=
-
1, respectively.In Table1, the non-Newtonian means are obtained by using different generating functions. Forα
=
q
p, thep-meansA
p
,
G
pandH
pare reduced to ordinary arithmetic mean, geometric mean, and harmonic mean, respectively. In particular some changes are observed for each value ofA
p
,
G
p, andH
pmeans depending on the choice ofp. As shown in the table, for increasing values ofp, thep-arithmetic meanA
pand its weighted formA
~
p increase; in particularptends to∞, and these means converge to the value of max
⁡
{
x
n
}. Conversely, for increasing values ofp, thep-harmonic mean H
p and its weighted forms H
~
p decrease. In particular, these means converge to the value of min
⁡
{
x
n
}asp
→
∞. Depending on the choice ofp, weighted formsA
~
p and H
~
p, 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 compareH
e
x
pand ordinary harmonic mean in Table 1, we also see that ordinary harmonic mean is smaller thanH
e
x
p. On the contraryA
e
x
pandG
e
x
pare smaller than their classical formsA
pandG
pforp
=
1. Therefore, we assert that the values of G
e
x
p
,
H
e
x
p
,
G
~
e
x
p, and H
~
e
x
p should be evaluated satisfactorily.Table 1
Comparison of the non-Newtonian (weighted) means and ordinary (weighted) means.
Weight
Data
p
A
p
G
p
H
p
A
~
p
G
~
p
H
~
p
G
e
x
p
H
e
x
p
G
~
e
x
p
H
~
e
x
p
w
1
=
2
x
1
=
15
1.0
27.50
24.74
22.64
32.82
29.88
27.27
24.02
23.36
29.05
28.26
w
2
=
5
x
2
=
20
0.1
24.99
24.74
24.50
30.16
29.88
29.59
24.02
23.36
29.05
28.26
w
3
=
7
x
3
=
25
2.0
30.61
24.74
21.14
35.77
29.88
25.21
24.02
23.36
29.05
28.26
w
4
=
9
x
4
=
50
5.0
38.22
24.74
18.73
41.69
29.88
21.55
24.02
23.36
29.05
28.26Corollary 11.
Considernpositive real numbersx
1
,
x
2
,
…
,
x
n. Then, the conditionsH
α
<
G
α
<
A
α and H
~
α
<
G
~
α
<
A
~
α hold whenα
=
e
x
pfor allx
n
>
1andα
=
q
pfor allp
∈
R
+.


## 4. 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.Definition 12 (generalized∗-convex function).
LetI
αbe an interval inR
α. Thenf
:
I
α
→
R
βis said to be∗-convex if(27)
f
λ
1
×
˙
x
+
˙
λ
2
×
˙
y
≤
¨
μ
1
×
¨
f
x
+
¨
μ
2
×
¨
f
yholds, whereλ
1
+
˙
λ
2
=
1
˙andμ
1
+
¨
μ
2
=
1
¨for allλ
1
,
λ
2
∈
[
0
˙
,
1
˙
]andμ
1
,
μ
2
∈
[
0
¨
,
1
¨
]. Therefore, by combining this with the generatorsαandβ, we deduce that (28)
f
α
α
-
1
λ
1
α
-
1
x
+
α
-
1
λ
2
α
-
1
y
≤
¨
β
β
-
1
μ
1
β
-
1
f
x
+
β
-
1
μ
2
β
-
1
f
y
.If (28) is strict for allx
≠
y, thenfis said to be strictly∗-convex. If the inequality in (28) is reversed, thenfis said to be∗-concave. On the other hand the inclusion (28) can be written with respect to the weightedα-arithmetic mean in (21) as follows: (29)
f
A
~
α
x
,
y
≤
¨
A
~
β
f
x
,
f
y
.Remark 13.
We remark that the definition of∗-convexity in (27) can be evaluated by non-Newtonian coordinate system involving∗-lines (see Definition 3). Forα
=
β
=
I, the∗-lines are straight lines in two-dimensional Euclidean space. 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.Case 1.
(a) If we takeα
=
β
=
e
x
pandλ
1
=
μ
1,λ
2
=
μ
2in (28), then (30)
f
x
ln
⁡
λ
1
y
ln
⁡
λ
2
≤
f
x
ln
⁡
λ
1
f
y
ln
⁡
λ
2
,
λ
1
,
λ
2
∈
1
,
e
,whereλ
1
λ
2
=
eholds andf
:
I
e
x
p
→
R
e
x
p
=
(
0
,
∞
)is called bigeometric (usually known as multiplicative) convex function (cf. [2]). Equivalently,fis bigeometric convex if and only if log
⁡
f
(
x
)is an ordinary convex function.
(b) Forα
=
e
x
pandβ
=
Iwe have (31)
f
x
ln
⁡
λ
1
y
ln
⁡
λ
2
≤
μ
1
f
x
+
μ
2
f
y
,
λ
1
,
λ
2
∈
1
,
e
;
μ
1
,
μ
2
∈
0,1
,whereλ
1
λ
2
=
eandμ
1
+
μ
2
=
1. In this case the functionf
:
I
e
x
p
→
R is called geometric convex function. Every geometric convex (usually known as log-convex) function is also convex (cf. [2]).
(c) Takingα
=
Iandβ
=
e
x
p, one obtains (32)
f
λ
1
x
+
λ
2
y
≤
f
x
ln
⁡
μ
1
f
y
ln
⁡
μ
2
,
μ
1
,
μ
2
∈
1
,
e
;
λ
1
,
λ
2
∈
0,1
,whereμ
1
μ
2
=
eandλ
1
+
λ
2
=
1, andf
:
I
→
R
e
x
p is called anageometric convex function.Case 2.
(a) Ifα
=
β
=
q
pin (28) then (33)
f
λ
1
x
p
+
λ
2
y
p
1
/
p
≤
λ
1
f
x
p
+
λ
2
f
y
p
1
/
p
p
∈
R
+
,whereλ
1
,
λ
2
∈
[
0,1
], λ
1
p
+
λ
2
p
=
1, and f
:
I
q
p
→
R
q
p is calledQ
Q-convex function.
(b) Forα
=
q
pandβ
=
I, we write that (34)
f
λ
1
x
p
+
λ
2
y
p
1
/
p
≤
μ
1
f
x
+
μ
2
f
y
,
λ
1
,
λ
2
,
μ
1
,
μ
2
∈
0,1
,whereλ
1
p
+
λ
2
p
=
1, μ
1
+
μ
2
=
1, and f
:
I
q
p
→
R is calledQ
I-convex function.
(c) Forα
=
Iandβ
=
q
p, we obtain that (35)
f
λ
1
x
+
λ
2
y
≤
μ
1
f
x
p
+
μ
2
f
y
p
1
/
p
,
λ
1
,
λ
2
,
μ
1
,
μ
2
∈
0,1
,whereμ
1
p
+
μ
2
p
=
1, λ
1
+
λ
2
=
1, and f
:
I
→
R
q
p is calledI
Q-convex function.
The∗-convexity of a functionf
:
I
α
→
R
βmeans geometrically that the∗-points of the graph offare under the chord joining the endpoints(
a
,
f
(
a
)
)and(
b
,
f
(
b
)
)on non-Newtonian coordinate system for everya
,
b
∈
I
α. By taking into account the definition of∗-slope in Definition 3 we have(36)
f
x
-
¨
f
a
/
¨
ι
x
-
˙
a
≤
¨
f
b
-
¨
f
a
/
¨
ι
b
-
˙
awhich implies(37)
f
x
≤
¨
f
a
+
¨
f
b
-
¨
f
a
/
¨
ι
b
-
˙
a
×
¨
ι
x
-
˙
a
for allx
∈
[
a
˙
,
b
˙
].
On the other hand (37) means that ifP
,
Q, andRare any three∗-points on the graph offwithQbetweenPandR, thenQis on or below chordP
R. In terms of∗-slope, it is equivalent to(38)
m
∗
P
Q
≤
¨
m
∗
P
R
≤
¨
m
∗
Q
Rwith strict inequalities whenfis strictly∗-convex.
Now to avoid the repetition of the similar statements, we give some necessary theorems and lemmas.Lemma 14 (Jensen’s inequality).
Aβ-real-valued functionfdefined on an intervalI
αis∗-convex if and only if(39)
f
∑
α
k
=
1
n
⁡
λ
k
×
˙
x
k
≤
¨
∑
β
k
=
1
n
⁡
μ
k
×
¨
f
x
kholds, where ∑
k
=
1
n
λ
k
=
1
˙
α and ∑
k
=
1
n
μ
k
=
1
¨
β for all λ
n
∈
[
0
˙
,
1
˙
] and μ
n
∈
[
0
¨
,
1
¨
].Proof.
The proof is straightforward, hence omitted.Theorem 15.
Letf
:
I
α
→
R
βbe a∗-continuous function. Thenfis∗-convex if and only iffis midpoint∗-convex, that is, (40)
x
1
,
x
2
∈
I
α
i
m
p
l
i
e
s
f
A
α
x
1
,
x
2
≤
¨
A
β
f
x
1
,
f
x
2
.Proof.
The proof can be easily obtained using the inequality (39) in Lemma 14.Theorem 16 (cf. [2]).
Letf
:
I
e
x
p
→
R
e
x
p be a ∗-differentiable function (see [19]) on a subintervalI
e
x
p
⊆
(
0
,
∞
). Then the following assertions are equivalent:(i)
fis bigeometric convex (concave).
(ii)
The functionf
∗
(
x
)is increasing (decreasing).Corollary 17.
A positiveβ-real-valued functionfdefined on an intervalI
e
x
pis bigeometric convex if and only if (41)
f
x
1
ln
⁡
λ
1
,
x
2
ln
⁡
λ
2
,
…
,
x
n
ln
⁡
λ
n
≤
f
x
1
ln
⁡
λ
1
,
f
x
2
ln
⁡
λ
2
,
…
,
f
x
n
ln
⁡
λ
n holds, where∏
k
=
1
n
λ
k
=
e for allx
1
,
x
2
,
…
,
x
n
∈
I
e
x
pandλ
1
,
λ
2
,
…
,
λ
n
∈
[
1
,
e
]. Besides, we have (42)
f
A
~
e
x
p
x
1
,
x
2
,
…
,
x
n
≤
A
~
e
x
p
f
x
1
,
f
x
2
,
…
,
f
x
n
.Corollary 18.
Aβ-real-valued functionfdefined on an intervalI
q
pisQ
Q-convex if and only if (43)
f
λ
1
x
1
p
+
λ
2
x
2
p
+
⋯
+
λ
n
x
n
p
1
/
p
≤
λ
1
f
x
1
p
+
λ
2
f
x
2
p
+
⋯
+
λ
n
f
x
n
p
1
/
p holds, where∑
k
=
1
n
λ
k
p
=
1for allx
1
,
x
2
,
…
,
x
n
∈
I
q
pandλ
1
,
λ
2
,
…
,
λ
n
∈
[
0,1
]. Thus, we have (44)
f
A
~
p
x
1
,
x
2
,
…
,
x
n
≤
A
~
p
f
x
1
,
f
x
2
,
…
,
f
x
n
,
p
∈
R
+
.


## 5. 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.Letfbe a bigeometric (multiplicative) convex function and finite on a closed interval[
x
,
y
]
⊂
R
+. It is obvious thatfis bounded from above byM
=
max
⁡
{
f
(
x
)
,
f
(
y
)
}, since, for anyz
=
x
λ
y
1
-
λin the interval,f
(
z
)
≤
f
(
x
)
λ
f
(
y
)
1
-
λforλ
∈
[
1
,
e
]. It is also bounded from below as we see by writing an arbitrary point in the formt
x
yfort
∈
R
+. Then (45)
f
2
x
y
≤
f
t
x
y
f
x
y
t
. UsingMas the upper boundf
(
x
y
/
t
)we obtain (46)
f
t
x
y
≥
1
M
f
2
x
y
=
m
. Thus a bigeometric convex function may not be continuous at the boundary points of its domain. We will prove that, for any closed subinterval[
x
,
y
]of the interior of the domain, there is a constantK
>
0so that, for any two pointsa
,
b
∈
[
x
,
y
]
⊂
R
+,(47)
f
a
f
b
≤
a
b
K
.A function that satisfies (47) for someKand allaandbin an interval is said to satisfy bigeometric Lipschitz condition on the interval.Theorem 19.
Suppose thatf
:
I
→
R
+is multiplicative convex. Then,fsatisfies the multiplicative Lipschitz condition on any closed interval[
x
,
y
]
⊂
R
+contained in the interiorI
0of I;that is,fis continuous onI
0.Proof.
Takeε
>
1so that[
x
/
ε
,
y
ε
]
∈
I, and letmandMbe the lower and upper bounds forfon[
x
/
ε
,
y
ε
]. Ifrandsare distinct points of[
x
,
y
]withs
>
r, set (48)
z
=
s
ε
,
λ
=
s
r
1
/
ln
⁡
ε
s
/
r
,
λ
∈
1
,
e
. Thenz
∈
[
x
/
ε
,
y
ε
]ands
=
z
ln
⁡
λ
r
1
-
ln
⁡
λ, and we obtain (49)
f
s
≤
f
z
ln
⁡
λ
f
r
1
-
ln
⁡
λ
=
f
z
f
r
ln
⁡
λ
f
r which yields (50)
ln
⁡
f
s
f
r
≤
ln
⁡
λ
ln
⁡
f
z
f
r
<
ln
⁡
s
r
1
/
ln
⁡
ε
ln
⁡
M
m
,
f
s
f
r
≤
s
r
ln
⁡
M
/
m
/
ln
⁡
ε
, whereK
=
ln
⁡
(
M
/
m
)
/
ln
⁡
(
ε
)
>
0. Since the pointsr
,
s
∈
[
x
,
y
]are arbitrary, we getfthat satisfies a multiplicative Lipschitz condition. The remaining part can be obtained in the similar way by takings
<
randz
=
s
/
ε. Finally,fis continuous, since[
x
,
y
]is arbitrary inI
0.


## 6. Concluding Remarks

Although all arithmetics are isomorphic, only by distinguishing among them do we obtain suitable tools for constructing all the non-Newtonian calculi. But the usefulness of arithmetic is not limited to the construction of calculi; we believe there is a more fundamental reason for considering alternative arithmetics; they may also be helpful in developing and understanding new systems of measurement that could yield simpler physical laws.In this paper, it was shown that, due to the choice of generator function,A
p, G
p, and H
p means are reduced to ordinary arithmetic, geometric, and harmonic mean, respectively. As shown in Table 1, for increasing values ofp,A
p and A
~
p means increase, especiallyp
→
∞; these means converge to the value of max
⁡
{
x
n
}. Conversely for increasing values ofp,H
p and H
~
p means decrease, especiallyp
→
∞; these means converge to the value of min
⁡
{
x
n
}. Additionally we give some new definitions regarding convex functions which are plotted on the non-Newtonian coordinate system. Obviously, for different generator functions, one can obtain some new geometrical interpretations of convex functions. Our future works will include the most famous Hermite Hadamard inequality for the class of∗-convex functions.



---

*Source: 5416751-2016-11-22.xml*