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

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 [q.sub.p] (p [member of] [R.sup.+]) generators.

The rest of this paper is organized as follows: in Section 2, we give some required definitions and consequences related with the a-arithmetic and [q.sub.p]-arithmetic. Based on two arbitrarily selected generators [alpha] and [beta], 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 interval [I.sub.[alpha]] 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 [x, y] [subset] (0, m).

2. 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 a is a one-to-one function whose domain is R and whose range is a subset [R.sub.[alpha]] of R where [R.sub.[alpha]] = {[alpha](x) : x [member of] R}. Each generator generates exactly one arithmetic, and conversely each arithmetic is generated by exactly one generator. If I(x) = x, for all x [member of] R, the identity function's inverse is itself. In the special cases [alpha] = I and [alpha] = exp, [alpha] generates the classical and geometric arithmetic, respectively. By [alpha]-arithmetic, we mean the arithmetic whose domain is R and whose operations are defined as follows: for x, y [member of] [R.sub.[alpha]] and any generator [alpha],

[mathematical expression not reproducible], (1)

As a generator, we choose exp function acting from R into the set [R.sub.exp] = (0,[infinity]) as follows:

[alpha] : R [right arrow] [R.sub.exp] x [??] y = [alpha](x) = [e.sup.x]. (2)

It is obvious that [alpha]-arithmetic reduces to the geometric arithmetic as follows:

[mathematical expression not reproducible]. (3)

Following Grossman and Katz [10] we give the infinitely many [q.sub.p]-arithmetics, of which the quadratic and harmonic arithmetic are special cases for p = 2 and p = -1, respectively. The function [q.sub.p] : R [right arrow] [R.sub.q] [subset or equal to] R and its inverse [q.sup.-1.sub.p] are defined as follows (p [member of] R \ {0}):

[mathematical expression not reproducible]. (4)

It is to be noted that [q.sub.p]-calculus is reduced to the classical calculus for p = 1. Additionally it is concluded that the [alpha]-summation can be given as follows:

[mathematical expression not reproducible]. (5)

Definition 1 (see [15]). Let X = (X, [d.sub.[alpha]]) be an [alpha]-metric space. Then the basic notions can be defined as follows:

(a) A sequence x = ([x.sub.k]) is a function from the set N into the set [R.sub.[alpha]]. The a-real number Xk denotes the value of the function at k [member of] N and is called the kth term of the sequence.

(b) A sequence ([x.sub.n]) in X = (X, [d.sub.[alpha]]) is said to be [alpha]-convergent if, for every given [epsilon] > 0 ([epsilon] [member of] [R.sub.[alpha]]), there exist an [n.sub.0] = [n.sub.0]([epsilon]) [member of] N and x [member of] X such that [d.sub.a]([x.sub.n], x) = [[absolute value of ([x.sub.n])].sub.a] < [epsilon] for all n > [n.sub.0] and is denoted by [mathematical expression not reproducible], as n [right arrow] [infinity].

(c) A sequence ([x.sub.n]) in X = (X, [d.sub.[alpha]]) is said to be [alpha]-Cauchy if for every [epsilon] > 0 there is an [n.sub.0] = [n.sub.0]([epsilon]) [member of] N such that [d.sub.[alpha]]([x.sub.n], [x.sub.m]) < [epsilon] for all m, n > [n.sub.0].

Throughout this paper, we define the pth [alpha]-exponent [mathematical expression not reproducible] and qth a-root [mathematical expression not reproducible] of x [member of] [R.sup.+] by

[mathematical expression not reproducible], (6)

and [mathematical expression not reproducible] provided there exists an y [member of] [R.sub.[alpha]] such that [mathematical expression not reproducible].

2.1. * -Arithmetic. Suppose that [alpha] and [beta] are two arbitrarily selected generators and ("star-") also is the ordered pair of arithmetics ([beta]-arithmetic and [alpha]-arithmetic). The sets ([mathematical expression not reproducible]) and ([mathematical expression not reproducible]) are complete ordered fields and beta(alpha)-generator generates beta(alpha)arithmetic, respectively. Definitions given for [beta]-arithmetic are also valid for a-arithmetic. Also a-arithmetic is used for arguments and [beta]-arithmetic is used for values; in particular, changes in arguments and values are measured by [alpha]-differences and [beta]-differences, respectively.

Let [mathematical expression not reproducible] and ([mathematical expression not reproducible]) 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.sup.*]; that is,

[C.sup.*] = {[z.sup.*] = (x ,y)[absolute value of (x [member of] [R.sub.[alpha]], y [member of] [R.sub.[beta]]}. (7)

Definition 2 (see [17]). (a) The *-limit of a function f at an element a in [R.sub.[alpha]] is, if it exists, the unique number b in [R.sub.[beta]] such that

[mathematical expression not reproducible], (8)

for [delta] [member of] [R.sub.[beta]], and is written as * [lim.sub.x[right arrow]a] f(x) = b.

A function f is * -continuous at a point a in [R.sub.[alpha]] if and only if a is an argument of f and [sup.*][lim.sub.x[right arrow]a] f(x) = f(a). When a and p 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 [alpha]-arithmetic to [beta]-arithmetic is the unique function i (iota) which has the following three properties:

(i) i is one to one.

(ii) i is from [R.sub.[alpha]] to [R.sub.[beta]].

(iii) For any numbers u, v [member of] [R.sub.[alpha][alpha]],

[mathematical expression not reproducible] (9)

It turns out that i(x) = [beta]{[[alpha].sup.-1] (x)} for every x in [R.sub.[alpha]] and that i([??]) = n for every [alpha]-integer n. Since, for example, [mathematical expression not reproducible], it should be clear that any statement in [alpha]-arithmetic can readily be transformed into a statement in [beta]-arithmetic.

Definition 3 (see [10]). The following statements are valid:

(i) The * -points [P.sub.1], [P.sub.2], and [P.sub.3] are * -collinear provided that at least one of the following holds:

[mathematical expression not reproducible], (10)

(ii) A * -line is a set L of at least two distinct points such that, for any distinct points [P.sub.1] and [P.sub.2] in L, a point [P.sub.3] is in L if and only if [P.sub.1], [P.sub.2], and [P.sub.3] are * -collinear. When [alpha] = [beta] = I, the * -lines are the straight lines in two-dimensional Euclidean space.

(iii) The * -slope of a * -line through the points ([a.sub.1], [b.sub.1]) and ([a.sub.2], [b.sub.2]) is given by

[mathematical expression not reproducible], (11)

for [a.sub.1], [a.sub.2] [member of] [R.sub.[alpha]] and [b.sub.1], [b.sub.2] [member of] [R.sub.[beta]].

If the following * -limit in (12) exists, we denote it by [f.sup.*] (t), call it the * -derivative of f at t, and say that f is * -differentiable at t (see [19]):

[mathematical expression not reproducible]. (12)

3. Non-Newtonian (Weighted) Means

Definition 4 ([alpha]-arithmetic mean). Consider that n positive real numbers [x.sub.1], [x.sub.2], ..., [x.sub.n] are given. The [alpha]-mean (average), denoted by [A.sub.[alpha]], is the [alpha]-sum of [x.sub.n]'s a-divided by [??] for all n [member of] N. That is,

[mathematical expression not reproducible]. (13)

For [alpha] = exp, we obtain that

[mathematical expression not reproducible]. (14)

Similarly, for [alpha] = [q.sub.p], we get

[A.sub.p] = [([x.sup.p.sub.1] + [x.sup.p.sub.2] + ... + [x.sup.p.sub.n]/n).sup.1/p], p [member of] R \{0}. (15)

[A.sub.exp] and [A.sub.p] are called multiplicative arithmetic mean and p-arithmetic mean (as usually known p-mean), respectively. One can conclude that [A.sub.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 Definition 4 can be written by using various generators. In particular if we take [beta]-arithmetic instead of a-arithmetic then the mean can be defined by

[mathematical expression not reproducible]. (16)

Definition 6 (a-geometric mean). Let [x.sub.1], [x.sub.2], ..., [x.sub.n] [member of] [R.sup.+]. The [alpha]-geometric mean, namely, [G.sub.[alpha]], is nth a-root of the [alpha]-product of ([[x.sub.n])'s:

[mathematical expression not reproducible]. (17)

We conclude similarly, by taking the generators [alpha] = exp or [alpha] = [q.sub.p], that the a-geometric mean can be interpreted as follows:

[mathematical expression not reproducible]. (18)

[G.sub.exp] and [G.sub.p] are called multiplicative geometric mean and p-geometric mean, respectively. It would clearly have [G.sub.p] = [A.sub.exp] for p = 1.

Definition 7 ([alpha]-harmonic mean). Let [x.sub.1], [x.sub.2], ..., [x.sub.n] [member of] [R.sup.+] and [[alpha].sup.-1]([x.sub.n]) [not equal to] 0 for each n [member of] N. The [alpha]-harmonic mean [H.sub.[alpha]] is defined by

[mathematical expression not reproducible]. (19)

Similarly, one obtains that

[mathematical expression not reproducible]. (20)

[H.sub.exp] and [H.sub.p] are called multiplicative harmonic mean and pharmonic mean, respectively. Obviously the inclusion (20) is reduced to ordinary harmonic mean and ordinary arithmetic mean for p = 1 and p = -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 (weighted a-arithmetic mean). Formally, the weighted a-arithmetic mean of a nonempty set of data {[x.sub.1], [x.sub.2], ..., [x.sub.n]} with nonnegative weights {[w.sub.1], [w.sub.2], ..., [w.sub.n]} is the quantity

[mathematical expression not reproducible]. (21)

The formulas are simplified when the weights are [alpha]-normalized such that they [alpha]-sum up to [sub.[alpha]][[summation].sup.n.sub.i=1] [[??].sub.i] = 1. For such normalized weights the weighted [alpha]-arithmetic mean is simply [mathematical expression not reproducible]. Note that if all the weights are equal, the weighted a-arithmetic mean is the same as the [alpha]-arithmetic mean.

Taking [alpha] = exp and[alpha] = qp, the weighted [alpha]-arithmetic mean can be given with the weights {[w.sub.1], [w.sub.2], ..., [w.sub.n]} as follows:

[mathematical expression not reproducible]. (22)

[[??].sub.exp] and [[??].sub.p] are called multiplicative weighted arithmetic mean and weighted p-arithmetic mean, respectively. [[??].sub.exp] turns out to the ordinary weighted geometric mean. Also, one easily can see that [[??].sub.p] is reduced to ordinary weighted arithmetic mean and weighted harmonic mean for p = 1 and p = -1, respectively.

Definition 9 (weighted a-geometric mean). Given a set of positive reals {[x.sub.1], [x.sub.2], ..., [x.sub.n]} and corresponding weights {[w.sub.1], [w.sub.2], ..., [w.sub.n]}, then the weighted [alpha]-geometric mean [G.sub.[alpha]] is defined by

[mathematical expression not reproducible]. (23)

Note that if all the weights are equal, the weighted [alpha]-geometric mean is the same as the [alpha]-geometric mean. Taking [alpha] = exp and [alpha] = [q.sub.p], the weighted a-geometric mean can be written for the weights {[w.sub.1], [w.sub.2], ..., [w.sub.n]} as follows:

[mathematical expression not reproducible]. (24)

[[??].sub.exp] and [[??].sub.p] are called weighted multiplicative geometric mean and weighted q-geometric mean. Also we have [mathematical expression not reproducible] for all [x.sub.n] > 1.

Definition 10 (weighted a-harmonic mean). If a set {[w.sub.1], [w.sub.2], ..., [w.sub.n]} of weights is associated with the data set {[x.sub.1], [x.sub.2], ..., [x.sub.n]} then the weighted a-harmonic mean is defined by

[mathematical expression not reproducible]. (25)

Taking [alpha] = exp and [alpha] = [q.sub.p], the weighted [alpha]-harmonic mean with the weights {[w.sub.1], [w.sub.2], ..., [w.sub.n]} can be written as follows:

[mathematical expression not reproducible]. (26)

[[??].sub.exp] and [[??].sub.p] are called multiplicative weighted harmonic mean and weighted p-geometric mean, respectively. It is obvious that [[??].sub.p] is reduced to ordinary weighted harmonic mean and ordinary weighted arithmetic mean for p = 1 and p = -1, respectively.

In Table 1, the non-Newtonian means are obtained by using different generating functions. For [alpha] = [q.sub.p], the p-means [A.sub.p], [G.sub.p] and [H.sub.p] are reduced to ordinary arithmetic mean, geometric mean, and harmonic mean, respectively. In particular some changes are observed for each value of [A.sub.p], [G.sub.p], and [H.sub.p] means depending on the choice of p. As shown in the table, for increasing values of p, the p-arithmeticmean [A.sub.p] and its weighted form [[??].sub.p] increase; in particular p tends to [infinity], and these means converge to the value of max{[x.sub.n]}. Conversely, for increasing values of p, the p-harmonic mean [H.sub.p] and its weighted forms [[??].sub.p] decrease. In particular, these means converge to the value of min{[x.sub.n]} as p [right arrow] [infinity]. Depending on the choice of p, weighted forms [[??].sub.p] and [[??].sub.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 compare [H.sub.exp] and ordinary harmonic mean in Table 1, we also see that ordinary harmonic mean is smaller than [H.sub.exp]. On the contrary [A.sub.exp] and [G.sub.exp] are smaller than their classical forms [A.sub.p] and [G.sub.p] for p = 1. Therefore, we assert that the values of [G.sub.exp], [H.sub.exp], [[??].sub.exp], and [[??].sub.exp] should be evaluated satisfactorily.

Corollary 11. Consider n positive real numbers [x.sub.1], [x.sub.2], ..., [x.sub.n]. Then, the conditions [H.sub.[alpha]] < [G.sub.[alpha]] < [A.sub.[alpha]] and [mathematical expression not reproducible] hold when [alpha] = exp for all [x.sub.n] > 1 and [alpha] = [q.sub.p] for all p [member of] [R.sup.+].

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). Let [I.sub.[alpha]] be an interval in [R.sub.[alpha]]. Then f: [I.sub.[alpha]] [right arrow] [R.sub.[beta]] is said to be * -convex if

[mathematical expression not reproducible] (27)

holds, where [mathematical expression not reproducible] and [mathematical expression not reproducible] for all [[lambda].sub.1], [[lambda].sub.2] [member of] [[mathematical expression not reproducible]] and [mathematical expression not reproducible]. Therefore, by combining this with the generators [alpha] and [beta], we deduce that

[mathematical expression not reproducible]. (28)

If (28) is strict for all x [not equal to] y, then f is said to be strictly * -convex. If the inequality in (28) is reversed, then f is said to be * -concave. On the other hand the inclusion (28) can be written with respect to the weighted [alpha]-arithmetic mean in (21) as follows:

[mathematical expression not reproducible]. (29)

Remark 13. We remark that the definition of * -convexity in (27) can be evaluated by non-Newtonian coordinate system involving * -lines (see Definition 3). For [alpha] = [beta] = 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 take [alpha] = [beta] = exp and [[lambda].sub.1] = [[mu].sub.1] [[lambda].sub.2] = [[mu].sub.2] in (28), then

[mathematical expression not reproducible], (30)

where [[lambda].sub.1][[lambda].sub.2] = e holds and f : [I.sub.exp] [right arrow] [R.sub.exp] = (0, [infinity]) is called bigeometric (usually known as multiplicative) convex function (cf. [2]). Equivalently, f is bigeometric convex if and only if log f(x) is an ordinary convex function.

(b) For [alpha] = exp and [beta] = I we have

[mathematical expression not reproducible], (31)

where [[lambda].sub.1][[lambda].sub.2] = e and [[mu].sub.1] + [[mu].sub.2] = 1. In this case the function f: [I.sub.exp] [right arrow] R is called geometric convex function. Every geometric convex (usually known as log-convex) function is also convex (cf. [2]).

[mathematical expression not reproducible], (32)

where [[mu].sub.1][[mu].sub.2] = e and [[lambda].sub.1] + [[lambda].sub.2] = 1, and f: I [right arrow] [R.sub.exp] is called anageometric convex function.

Case 2. (a) If [alpha] = [beta] = [q.sub.p] in (28) then

[mathematical expression not reproducible], (33)

where [[lambda].sub.1], [A.sub.2] [member of] [0,1], [[lambda].sup.p.sub.1] + [[lambda].sup.p.sub.2] = 1, and [mathematical expression not reproducible] is called QQ-convex function.

(b) For [alpha] = [q.sub.p] and [beta] = I, we write that

[mathematical expression not reproducible], (34)

where [[lambda].sup.p.sub.1] + [[lambda].sup.p.sub.2] = 1, [[mu].sub.1] + [[mu].sub.2] = 1, and [mathematical expression not reproducible] is called QI-convex function.

(c) For [alpha] = I and [beta] = [q.sub.p], we obtain that

[mathematical expression not reproducible], (35)

where [[mu].sup.p.sub.1] + [u.sup.p.sub.2] = 1, [[lambda].sub.1] + [[lambda].sub.2] = 1, and f: I [right arrow] R is called IQ-convex function.

The * -convexity of a function f : [I.sub.[alpha]] [right arrow] [R.sub.[beta]] means geometrically that the * -points of the graph of f are under the chord joining the endpoints (a, f(a)) and (b, f(b)) on non-Newtonian coordinate system for every a, b [member of] [I.sub.[alpha]]. By taking into account the definition of * -slope in Definition 3 we have

[mathematical expression not reproducible] (36)

which implies

[mathematical expression not reproducible] (37)

for all x [member of] [[mathematical expression not reproducible]].

On the other hand (37) means that if P, Q, and R are any three * -points on the graph of f with Q between P and R, then Q is on or below chord PR. In terms of * -slope, it is equivalent to

[m.sup.*] (PQ) [??] [m.sup.*] (PR) [??] [m.sup.*] (QR) (38)

with strict inequalities when f is strictly * -convex.

Now to avoid the repetition of the similar statements, we give some necessary theorems and lemmas.

Lemma 14 (Jensen's inequality). A [beta]-real-valued function f defined on an interval [I.sub.[alpha]] is * -convex if and only if

[mathematical expression not reproducible]) (39)

holds, where [sub.[alpha]][[summation].sup.n.sub.k=1] [[lambda].sub.k] = [??] and [[summation].sup.n.sub.k=1] [[mu].sub.k] = [??] for all [mathematical expression not reproducible] and [mathematical expression not reproducible].

Proof. The proof is straightforward, hence omitted.

Theorem 15. Let f : [I.sub.a] [right arrow] [R.sub.[beta]] be a * -continuous function. Then f is * -convex if and only if f is midpoint *-convex, that is,

[x.sub.1], [x.sub.2] [member of] [I.sub.[alpha]] implies f([A.sub.[alpha]] {[x.sub.1], [x.sub.2]}) [??] [A.sub.[beta]] {f ([x.sub.1]), f ([x.sub.2])}. (40)

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

Theorem 16 (cf. [2]). Let f: [I.sub.exp] [right arrow] [R.sub.exp] be a * -differentiable function (see [19]) on a subinterval [I.sub.exp] [subset or equal to] (0, [infinity]). Then the following assertions are equivalent:

(i) f is bigeometric convex (concave).

(ii) The function [f.sup.*] (x) is increasing (decreasing).

Corollary 17. A positive [beta]-real-valued function f defined on an interval [I.sub.exp] is bigeometric convex if and only if

[mathematical expression not reproducible] 41)

holds, where [[PI].sup.n.sub.k=1] [[lambda].sub.k] = e for all [x.sub.1], [x.sub.2], ..., [x.sub.n] [member of] [I.sub.exp] and [[lambda].sub.1], [[lambda].sub.2], ..., [[lambda].sub.n] [member of] [1,e]. Besides, we have

f ([[??].sub.exp] {[x.sub.1], [x.sub.2], ..., [X.sub.n]}) [less than or equal to] [[??].sub.exp] {f([x.sub.1]), f([x.sub.2]), ..., f([x.sub.n])}.

Corollary 18. A [beta]-real-valued function f defined on an interval [mathematical expression not reproducible] is QQ-convex if and only if

[mathematical expression not reproducible]. (43)

holds, where [[summation].sup.n.sub.k=1] [[lambda].sup.p.sub.k] = 1 for all [mathematical expression not reproducible] and [[lambda].sub.1], [[lambda].sub.2], ..., [[lambda].sub.n] [member of] [0,1]. Thus, we have

f([[??].sub.p] {[x.sub.1], [x.sub.2], ..., [x.sub.n]})

[less than or equal to] [[??].sub.p] {f([x.sub.1]), f([x.sub.2]), ..., f([x.sub.n])}, (p [member of] [R.sup.+]) .

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.

Let f be a bigeometric (multiplicative) convex function and finite on a closed interval [x, y] [susbet] [R.sup.+]. It is obvious that f is bounded from above by M = max{f(x), f(y)}, since, for any z = [x.sup.[lambda] [y.sup.1-[lambda]] in the interval, f(z) [less than or equal to] f[(x).sup.[lambda]] f[(y).sup.1-[lambda]] for [lambda] [member of] [1, ej. It is also bounded from below as we see by writing an arbitrary point in the form t[square root of (xy)] for t [member of] [R.sup.+]. Then

[f.sup.2] ([square root of (xy)]) [less than or equal to] f(t[square root of (xy)]) f([square root of (xy)]/t). (45)

Using M as the upper bound f([square root of (xy)]/t) we obtain

f(t[square root of (xy)])[greater than or equal to] 1/M [f.sup.2] ([square root of (xy)]) = m. (46)

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, yj of the interior of the domain, there is a constant K > 0 so that, for any two points a, b [member of] [x, y] [subset [R.sup.+],

f(a)/f(b) [less than or equal to] [(a/b).sup.K]. (47)

A function that satisfies (47) for some K and all a and b in an interval is said to satisfy bigeometric Lipschitz condition on the interval.

Theorem 19. Suppose that f : I [right arrow] [R.sup.+] is multiplicative convex. Then, f satisfies the multiplicative Lipschitz condition on any closed interval [x, y] [susbet] [R.sup.+] contained in the interior [I.sup.0] of I; that is, f is continuous on [I.sup.0].

Proof. Take e > 1 so that [x/[epsilon], y[epsilon]] [member of] I, and let m and M be the lower and upper bounds for f on [x/[epsilon], y[epsilon]]. If r and s are distinct points of [x, y] with s > r, set

z = s[epsilon],

[lambda] = [(s/r).sup.1/ln([epsilon]s/r)], ([lambda] [member of] (1, e))(48)

Then z [member of] [x/[epsilon], y[epsilon]] and s = [z.sup.ln[lambda]] [r.sup.1-ln[lambda]], and we obtain

f(s) [less than or equal to] f [(z).sup.ln[lambda]] f [(r).sup.1-ln[lambda]] = f (r) (49)

which yields

[mathematical expression not reproducible], (50)

where K = ln(M/m)/ln([epsilon]) > 6. Since the points r, s [member of] [x, y] are arbitrary, we get f that satisfies a multiplicative Lipschitz condition. The remaining part can be obtained in the similar way by taking s < r and z = s/[epsilon]. Finally, f is continuous, since [x, y] is arbitrary in [I.sub.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.sub.p], [G.sub.p], and [H.sub.p] means are reduced to ordinary arithmetic, geometric, and harmonic mean, respectively. As shown in Table 1, for increasing values of p, [A.sub.p] and Gp means increase, especially p [right arrow] [infinity]; these means converge to the value of max{xn}. Conversely for increasing values of p, [H.sub.p] and [[??].sub.p] means decrease, especially p [right arrow] [infinity]; 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.

http://dx.doi.org/10.1155/2016/5416751

Competing Interests

The authors declare that they have no competing interests.

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Ugur Kadak (1) and Yusuf Gurefe (2)

(1) Department of Mathematics, Bozok University, 66100 Yozgat, Turkey

(2) Department of Econometrics, Usak University, 64300 Usak, Turkey

Correspondence should be addressed to Ugur Kadak; ugurkadak@gmail.com

Received 22 July 2016; Accepted 19 September 2016

Academic Editor: Julien Salomon
TABLE 1: Comparison of the non-Newtonian (weighted) means and ordinary
(weighted) means.

Weight               Data         P    [A.sub.p]   [G.sub.p]

[w.sub.1] = 2   [x.sub.1] = 15   1.0     27.50       24.74
[w.sub.2] = 5   [x.sub.2] = 20   0.1     24.99       24.74
[w.sub.3] = 7   [x.sub.3] = 25   2.0     30.61       24.74
[w.sub.4] = 9   [x.sub.4] = 50   5.0     38.22       24.74

Weight          [H.sub.p]   [[??].sub.p]   [[??].sub.p]

[w.sub.1] = 2     22.64        32.82          29.88
[w.sub.2] = 5     24.50        30.16          29.88
[w.sub.3] = 7     21.14        35.77          29.88
[w.sub.4] = 9     18.73        41.69          29.88

Weight          [[??].sub.p]   [G.sub.exp]   [H.sub.exp]

[w.sub.1] = 2      27.27          24.02         23.36
[w.sub.2] = 5      29.59          24.02         23.36
[w.sub.3] = 7      25.21          24.02         23.36
[w.sub.4] = 9      21.55          24.02         23.36

Weight          [[??].sub.exp]   [[??].sub.exp]

[w.sub.1] = 2       29.05            28.26
[w.sub.2] = 5       29.05            28.26
[w.sub.3] = 7       29.05            28.26
[w.sub.4] = 9       29.05            28.26
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Title Annotation:Research Article
Author:Kadak, Ugur; Gurefe, Yusuf
Publication:International Journal of Analysis
Article Type:Report
Date:Jan 1, 2016
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