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
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 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 [12–14] 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, exp and qp(p∈R+) generators.
The rest of this paper is organized as follows: in Section 2, we give some required definitions and consequences related with theα-arithmetic and qp-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 intervalIαand 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.
A generator α is a one-to-one function whose domain isRand whose range is a subset RαofRwhere Rα=αx:x∈R. Each generator generates exactly one arithmetic, and conversely each arithmetic is generated by exactly one generator. If Ix=x, for all x∈R, the identity 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∈Rα and any generator α, (1)α-addition x+˙y=αα-1x+α-1y,α-subtraction x-˙y=αα-1x-α-1y,α-multiplication x×˙y=αα-1x×α-1y,α-division x/˙y=αα-1x÷α-1y,α-order x<˙y⟺α-1x<α-1y. As a generator, we choose exp function acting from R into the set Rexp=(0,∞) as follows: (2)α:R⟶Rexpx⟼y=αx=ex. It is obvious that α-arithmetic reduces to the geometric arithmetic as follows:(3)geometricadditionx+˙y=elnx+lny=x·y,geometricsubtractionx-˙y=elnx-lny=x÷y,geometricmultiplicationx×˙y=elnxlny=xlny=ylnx,geometricdivisionx/˙y=elnx/lny=x1/lny,geometricorderx<˙y⟺lnx<lny.
Following Grossman and Katz [10] we give the infinitely many qp-arithmetics, of which the quadratic and harmonic arithmetic are special cases for p=2 and p=-1, respectively. The function qp:R→Rq⊆R and its inverse qp-1 are defined as follows p∈R∖0:(4)qpx=x1/p,x>00,x=0--x1/p,x<0,qp-1x=xp,x>00,x=0--xp,x<0.It is to be noted that qp-calculus is reduced to the classical calculus for p=1. Additionally it is concluded that the α-summation can be given as follows: (5)∑αk=1nxk=α∑k=1nα-1xk=αα-1x1+⋯+α-1xn∀xk∈R+.
Definition 1 (see [<xref ref-type="bibr" rid="B15">15</xref>]).
Let X=X,dα be an α-metric space. Then the basic notions can be defined as follows:
A sequence x=(xk) is a function from the set N into the set Rα. The α-real number xk denotes the value of the function at k∈N and is called the kth term of the sequence.
A sequence (xn) in X=(X,dα) is said to be α-convergent if, for every given ε>˙0˙ (ε∈Rα), there exist an n0=n0(ε)∈N and x∈X such that dαxn,x=xn-˙xα<˙ε for all n>n0 and is denoted by αlimn→∞xn=x or xn→αx, as n→∞.
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 dαxn,xm<˙ε for all m,n>n0.
Throughout this paper, we define the pth α-exponent xpα and qth α-root x1/qα of x∈R+ by (6)x2α=x×˙x=αα-1x×α-1x=αα-1x2,x3α=x2α×˙x=αα-1αα-1x×α-1x×α-1x=αα-1x3,⋮xpα=xp-1α×˙x=αα-1xp,and xα=x1/2α=y provided there exists an y∈Rα such that y2α=x.
Suppose that α and β are two arbitrarily selected generators and (“star-”) also is the ordered pair of arithmetics (β-arithmetic and α-arithmetic). The sets (Rβ,+¨,-¨,×¨,/¨,<¨) and (Rα,+˙,-˙,×˙,/˙,<˙) are complete ordered 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 arguments and β-arithmetic is used for values; in particular, changes in arguments and values are measured by α-differences and β-differences, respectively.
Let x∈(Rα,+˙,-˙,×˙,/˙,<˙) and y∈(Rβ,+¨,-¨,×¨,/¨,<¨) 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∈Rα,y∈Rβ.
Definition 2 (see [<xref ref-type="bibr" rid="B17">17</xref>]).
(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 (8)limx→a∗fx=b⟺∀ε>¨0¨,∃δ>˙0˙∋fx-¨bβ<¨ε∀x,ε∈Rα,x-˙aα<˙δ,for δ∈Rβ, and is written as ∗limx→af(x)=b.
A function f 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 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 from α-arithmetic to β-arithmetic is the unique function ι (iota) which has the following three properties:
ι is one to one.
ι is from Rα to Rβ.
For any numbers u,v∈Rα, (9)ιu+˙v=ιu+¨ιv;ιu-˙v=ιu-¨ιv;ιu×˙v=ιu×¨ιv;ιu/˙v=ιu/¨ιv.
It turns out that ι(x)=βα-1x for every x in Rα and that ι(n˙)=n¨ for every α-integer n˙. Since, for example, u+˙v=ι-1ιu+¨ιv, it should be clear that any statement in α-arithmetic can readily be transformed into a statement in β-arithmetic.
Definition 3 (see [<xref ref-type="bibr" rid="B10">10</xref>]).
The following statements are valid:
The ∗-points P1, P2, and P3 are ∗-collinear provided that at least one of the following holds:(10)d∗P2,P1+¨d∗P1,P3=d∗P2,P3,d∗P1,P2+¨d∗P2,P3=d∗P1,P3,d∗P1,P3+¨d∗P3,P2=d∗P1,P2.
A ∗-line is a set L of at least two distinct points such that, for any distinct points P1 and P2 in L, a point P3 is in L if and only if P1, P2, and P3 are ∗-collinear. When α=β=I, the ∗-lines are the straight lines in two-dimensional Euclidean space.
The ∗-slope of a ∗-line through the points (a1,b1) and (a2,b2) is given by (11)m∗=b2-¨b1/¨ιa2-˙a1=ββ-1b2-β-1b1α-1a2-α-1a1,a1≠a2,for a1,a2∈Rα and b1,b2∈Rβ.
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)∗limx→tfx-¨ft/¨ιx-˙t=limx→tββ-1fx-β-1ftα-1x-α-1t=limx→tββ-1fx-β-1ftx-tx-tα-1x-α-1t=ββ-1∘f′tα-1′t.
Consider that n positive real numbers x1,x2,…,xn are given. The α-mean (average), denoted by Aα, is the α-sum of xn’s α-divided by n˙ for all n∈N. That is, (13)Aα=∑αk=1nxk/˙n˙=∑αk=1nαα-1xkn=αα-1x1+α-1x2+⋯+α-1xnn.For α=exp, we obtain that (14)Aexp=∏k=1nxk1/n=x1·x2⋯xn1/n.Similarly, for α=qp, we get (15)Ap=x1p+x2p+⋯+xnpn1/p,p∈R∖0.Aexp and Ap are called multiplicative arithmetic mean and p-arithmetic mean (as usually known p-mean), respectively. One can conclude that Ap reduces to arithmetic mean and harmonic mean in the ordinary sense for p=1 and p=-1, respectively.
Remark 5.
It is clear that Definition 4 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)Aβ=∑βk=1nxk/¨n¨.
Letx1,x2,…,xn∈R+. Theα-geometric mean, namely,Gα, isnthα-root of theα-product ofxn’s:(17)Gα=∏αk=1nxk1/nα=α∏k=1nα-1xk1/n=αα-1x1,α-1x2,…,α-1xn1/n.
We conclude similarly, by taking the generatorsα=exporα=qp, that theα-geometric mean can be interpreted as follows: (18)Gexp=explnx1,lnx2,…,lnxn1/n,xn>1Gp=x1p,x2p,…,xnp1/n1/p=x1,x2,…,xn1/n,p≠0.GexpandGpare called multiplicative geometric mean andp-geometric mean, respectively. It would clearly haveGp=Aexpforp=1.
Let x1,x2,…,xn∈R+ and α-1(xn)≠0 for eachn∈N. Theα-harmonic meanHαis defined by (19)Hα=n˙/˙∑αk=1n1˙/˙xk=n˙/˙1˙/˙x1+˙1˙/˙x2+˙⋯+˙1˙/˙xn=αn1/α-1x1+1/α-1x2+⋯+1/α-1xn.Similarly, one obtains that(20)Hexp=expn1/lnx1+1/lnx2+⋯+1/lnxn,xn>1,Hp=n1/x1p+1/x2p+⋯+1/xnp1/p.Hexp and Hp 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.
Formally, the weightedα-arithmetic mean of a nonempty set of data{x1,x2,…,xn}with nonnegative weights{w1,w2,…,wn}is the quantity(21)A~α=∑αi=1nxi×˙w˙i/˙∑αi=1nw˙i=x1×˙w˙1+˙x2×˙w˙2+˙⋯+˙xn×˙w˙n/˙w˙1+˙w˙2+˙⋯+˙w˙n=α∑i=1nwiα-1xi∑i=1nwi.
The formulas are simplified when the weights areα-normalized such that theyα-sum up to∑i=1nαw˙i=1˙. For such normalized weights the weightedα-arithmetic mean is simply A~α=∑i=1nxi×˙w˙iα. Note that if all the weights are equal, the weightedα-arithmetic mean is the same as theα-arithmetic mean.
Takingα=exp andα=qp, the weightedα-arithmetic mean can be given with the weights{w1,w2,…,wn}as follows: (22)A~exp=exp∑i=1nlnxiwi∑i=1nwi=∏i=1nxiwi1/∑i=1nwi,xn>1,A~qp=w1x1p+w2x2p+⋯+wnxnpw1+w2+⋯+wn1/p.A~exp and A~p are called multiplicative weighted arithmetic mean and weightedp-arithmetic mean, respectively.A~expturns 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.
Given a set of positive reals{x1,x2,…,xn}and corresponding weights{w1,w2,…,wn}, then the weightedα-geometric mean G~α is defined by (23)G~α=∏αi=1nxiwi/∑i=1nwiα=x1w1/w1+⋯+wnα×˙x2w2/w1+⋯+wnα×˙⋯×˙xnwn/w1+⋯+wnα=αα-1x1w1/w1+⋯+wn,α-1x2w2/w1+⋯+wn,…,α-1xnwn/w1+⋯+wn.
Note that if all the weights are equal, the weightedα-geometric mean is the same as theα-geometric mean. Takingα=expandα=qp, the weightedα-geometric mean can be written for the weights{w1,w2,…,wn}as follows: (24)G~exp=explnx1w1/w1+⋯+wn,lnx2w2/w1+⋯+wn,…,lnxnwn/w1+⋯+wn,xn>1,G~p=x1pw1/w1+⋯+wn,x2pw2/w1+⋯+wn,…,xnpwn/w1+⋯+wn1/p=∏i=1nxiwi1/∑i=1nwi.G~exp and G~p are called weighted multiplicative geometric mean and weightedq-geometric mean. Also we haveG~p=A~exp for allxn>1.
If a setw1,w2,…,wnof weights is associated with the data setx1,x2,…,xnthen the weightedα-harmonic mean is defined by (25)H~α=∑αk=1nw˙k/˙∑αk=1nw˙k/˙xk=w˙1+˙⋯+˙w˙n/˙w˙1/˙x1+˙w˙2/˙x2+˙⋯+˙w˙n/˙xn=αw1+w2+⋯+wnw1/α-1x1+w2/α-1x2+⋯+wn/α-1xn.
Takingα=expandα=qp, the weightedα-harmonic mean with the weights{w1,w2,…,wn}can be written as follows: (26)H~exp=expw1+w2+⋯+wnw1/lnx1+w2/lnx2+⋯+wn/lnxn,xn>1,H~p=w1+w2+⋯+wnw1/x1p+w2/x2p+⋯+wn/xnp1/p.H~exp 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 Table 1, the non-Newtonian means are obtained by using different generating functions. Forα=qp, thep-meansAp,GpandHpare reduced to ordinary arithmetic mean, geometric mean, and harmonic mean, respectively. In particular some changes are observed for each value ofAp,Gp, andHpmeans depending on the choice ofp. As shown in the table, for increasing values ofp, thep-arithmetic meanApand its weighted formA~p increase; in particularptends to∞, and these means converge to the value of max{xn}. Conversely, for increasing values ofp, thep-harmonic mean Hp and its weighted forms H~p decrease. In particular, these means converge to the value of min{xn}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 compareHexpand ordinary harmonic mean in Table 1, we also see that ordinary harmonic mean is smaller thanHexp. On the contraryAexpandGexpare smaller than their classical formsApandGpforp=1. Therefore, we assert that the values of Gexp,Hexp,G~exp, and H~exp should be evaluated satisfactorily.
Comparison of the non-Newtonian (weighted) means and ordinary (weighted) means.
Weight
Data
p
Ap
Gp
Hp
A~p
G~p
H~p
Gexp
Hexp
G~exp
H~exp
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
Corollary 11.
Considernpositive real numbersx1,x2,…,xn. Then, the conditionsHα<Gα<Aα and H~α<G~α<A~α hold whenα=expfor allxn>1andα=qpfor 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.
LetIαbe an interval inRα. Thenf:Iα→Rβis said to be∗-convex if(27)fλ1×˙x+˙λ2×˙y≤¨μ1×¨fx+¨μ2×¨fyholds, whereλ1+˙λ2=1˙andμ1+¨μ2=1¨for allλ1,λ2∈[0˙,1˙]andμ1,μ2∈[0¨,1¨]. Therefore, by combining this with the generatorsαandβ, we deduce that (28)fαα-1λ1α-1x+α-1λ2α-1y≤¨ββ-1μ1β-1fx+β-1μ2β-1fy.
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)fA~αx,y≤¨A~βfx,fy.
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=expandλ1=μ1,λ2=μ2in (28), then (30)fxlnλ1ylnλ2≤fxlnλ1fylnλ2,λ1,λ2∈1,e,whereλ1λ2=eholds andf:Iexp→Rexp=(0,∞)is called bigeometric (usually known as multiplicative) convex function (cf. [2]). Equivalently,fis bigeometric convex if and only if logf(x)is an ordinary convex function.
(b) Forα=expandβ=Iwe have (31)fxlnλ1ylnλ2≤μ1fx+μ2fy,λ1,λ2∈1,e;μ1,μ2∈0,1,whereλ1λ2=eandμ1+μ2=1. In this case the functionf:Iexp→R is called geometric convex function. Every geometric convex (usually known as log-convex) function is also convex (cf. [2]).
(c) Takingα=Iandβ=exp, one obtains (32)fλ1x+λ2y≤fxlnμ1fylnμ2,μ1,μ2∈1,e;λ1,λ2∈0,1,whereμ1μ2=eandλ1+λ2=1, andf:I→Rexp is called anageometric convex function.
Case 2.
(a) Ifα=β=qpin (28) then (33)fλ1xp+λ2yp1/p≤λ1fxp+λ2fyp1/pp∈R+,whereλ1,λ2∈[0,1], λ1p+λ2p=1, and f:Iqp→Rqp is calledQQ-convex function.
(b) Forα=qpandβ=I, we write that (34)fλ1xp+λ2yp1/p≤μ1fx+μ2fy,λ1,λ2,μ1,μ2∈0,1,whereλ1p+λ2p=1, μ1+μ2=1, and f:Iqp→R is calledQI-convex function.
(c) Forα=Iandβ=qp, we obtain that (35)fλ1x+λ2y≤μ1fxp+μ2fyp1/p,λ1,λ2,μ1,μ2∈0,1,whereμ1p+μ2p=1, λ1+λ2=1, and f:I→Rqp is calledIQ-convex function.
The∗-convexity of a functionf:Iα→Rβmeans 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∈Iα. By taking into account the definition of∗-slope in Definition 3 we have(36)fx-¨fa/¨ιx-˙a≤¨fb-¨fa/¨ιb-˙awhich implies(37)fx≤¨fa+¨fb-¨fa/¨ιb-˙a×¨ιx-˙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 chordPR. In terms of∗-slope, it is equivalent to(38)m∗PQ≤¨m∗PR≤¨m∗QRwith 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 intervalIαis∗-convex if and only if(39)f∑αk=1nλk×˙xk≤¨∑βk=1nμk×¨fxkholds, where ∑k=1nλk=1˙α and ∑k=1nμk=1¨β for all λn∈[0˙,1˙] and μn∈[0¨,1¨].
Proof.
The proof is straightforward, hence omitted.
Theorem 15.
Letf:Iα→Rβbe a∗-continuous function. Thenfis∗-convex if and only iffis midpoint∗-convex, that is, (40)x1,x2∈IαimpliesfAαx1,x2≤¨Aβfx1,fx2.
Proof.
The proof can be easily obtained using the inequality (39) in Lemma 14.
Letf:Iexp→Rexp be a ∗-differentiable function (see [19]) on a subintervalIexp⊆(0,∞). Then the following assertions are equivalent:
fis bigeometric convex (concave).
The functionf∗(x)is increasing (decreasing).
Corollary 17.
A positiveβ-real-valued functionfdefined on an intervalIexpis bigeometric convex if and only if (41)fx1lnλ1,x2lnλ2,…,xnlnλn≤fx1lnλ1,fx2lnλ2,…,fxnlnλn holds, where∏k=1nλk=e for allx1,x2,…,xn∈Iexpandλ1,λ2,…,λn∈[1,e]. Besides, we have (42)fA~expx1,x2,…,xn≤A~expfx1,fx2,…,fxn.
Corollary 18.
Aβ-real-valued functionfdefined on an intervalIqpisQQ-convex if and only if (43)fλ1x1p+λ2x2p+⋯+λnxnp1/p≤λ1fx1p+λ2fx2p+⋯+λnfxnp1/p holds, where∑k=1nλkp=1for allx1,x2,…,xn∈Iqpandλ1,λ2,…,λn∈[0,1]. Thus, we have (44)fA~px1,x2,…,xn≤A~pfx1,fx2,…,fxn,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=xλy1-λin the interval,f(z)≤f(x)λf(y)1-λforλ∈[1,e]. It is also bounded from below as we see by writing an arbitrary point in the formtxyfort∈R+. Then (45)f2xy≤ftxyfxyt. UsingMas the upper boundf(xy/t)we obtain (46)ftxy≥1Mf2xy=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)fafb≤abK.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 interiorI0of I; that is,fis continuous onI0.
Proof.
Takeε>1so that[x/ε,yε]∈I, and letmandMbe the lower and upper bounds forfon[x/ε,yε]. Ifrandsare distinct points of[x,y]withs>r, set (48)z=sε,λ=sr1/lnεs/r,λ∈1,e. Thenz∈[x/ε,yε]ands=zlnλr1-lnλ, and we obtain (49)fs≤fzlnλfr1-lnλ=fzfrlnλfr which yields (50)lnfsfr≤lnλlnfzfr<lnsr1/lnεlnMm,fsfr≤srlnM/m/lnε, 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 inI0.
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,Ap, Gp, and Hp means are reduced to ordinary arithmetic, geometric, and harmonic mean, respectively. As shown in Table 1, for increasing values ofp,Ap and A~p means increase, especiallyp→∞; these means converge to the value of max{xn}. Conversely for increasing values ofp,Hp and H~p means decrease, especiallyp→∞; these means converge to the value of min{xn}. 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.
Competing Interests
The authors declare that they have no competing interests.
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