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.
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, and ) 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 generatorsand, 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 intervaland 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.
2. Preliminary, Background, and Notation
Arithmetic is any system that satisfies the whole of the ordered field axioms whose domain is a subset of. 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 isand whose range is a subset ofwhere . Each generator generates exactly one arithmetic, and conversely each arithmetic is generated by exactly one generator. If , for all , the identity function’s inverse is itself. In the special cases and , generates the classical and geometric arithmetic, respectively. By -arithmetic, we mean the arithmetic whose domain is and whose operations are defined as follows: for and any generator , As a generator, we choose exp function acting from into the set as follows: It is obvious that -arithmetic reduces to the geometric arithmetic as follows:
Following Grossman and Katz [10] we give the infinitely many -arithmetics, of which the quadratic and harmonic arithmetic are special cases for and , respectively. The function and its inverse are defined as follows :It is to be noted that -calculus is reduced to the classical calculus for . 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:(a)A sequence is a function from the set into the set . The -real number denotes the value of the function at and is called the th term of the sequence.(b)A sequence in is said to be -convergent if, for every given (), there exist an and such that for all and is denoted by or , as .(c)A sequence in is said to be -Cauchy if for every there is an such that for all .Throughout this paper, we define the th -exponent and th -root of by and provided there exists an such that .
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 and are complete ordered fields and -generator generates -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.
Let and be arbitrarily chosen elements from corresponding arithmetic. Then the ordered pair is called a -point and the set of all -points is called the set of -complex numbers and is denoted by ; that is,
Definition 2 (see [17]). (a) The -limit of a function at an element in is, if it exists, the unique number in such that for , and is written as .
A function is -continuous at a point in if and only if is an argument of and . 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:(i) is one to one.(ii) is from to .(iii)For any numbers , It turns out that for every in and that for every -integer . Since, for example, , 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 , , and are -collinear provided that at least one of the following holds:(ii)A -line is a set of at least two distinct points such that, for any distinct points and in , a point is in if and only if , , and are -collinear. When , the -lines are the straight lines in two-dimensional Euclidean space.(iii)The -slope of a -line through the points and is given by for and .
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]):
3. Non-Newtonian (Weighted) Means
Definition 4 (-arithmetic mean). Consider that positive real numbers are given. The -mean (average), denoted by , is the -sum of ’s -divided by for all . That is, For , we obtain that Similarly, for , we get 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 and , 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. The-geometric mean, namely,, isth-root of the-product of’s:We conclude similarly, by taking the generatorsor, that the-geometric mean can be interpreted as follows: andare called multiplicative geometric mean and-geometric mean, respectively. It would clearly havefor.
Definition 7 (-harmonic mean). Let and for each. The-harmonic meanis defined by Similarly, one obtains that and are called multiplicative harmonic mean and-harmonic mean, respectively. Obviously the inclusion (20) is reduced to ordinary harmonic mean and ordinary arithmetic mean for and, 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-arithmetic mean). Formally, the weighted-arithmetic mean of a nonempty set of datawith nonnegative weightsis the quantityThe formulas are simplified when the weights are-normalized such that they-sum up to. For such normalized weights the weighted-arithmetic mean is simply . Note that if all the weights are equal, the weighted-arithmetic mean is the same as the-arithmetic mean.
Taking and, the weighted-arithmetic mean can be given with the weightsas follows: and are called multiplicative weighted arithmetic mean and weighted-arithmetic mean, respectively.turns out to the ordinary weighted geometric mean. Also, one easily can see that is reduced to ordinary weighted arithmetic mean and weighted harmonic mean forand, respectively.
Definition 9 (weighted-geometric mean). Given a set of positive realsand corresponding weights, then the weighted-geometric mean is defined by Note that if all the weights are equal, the weighted-geometric mean is the same as the-geometric mean. Takingand, the weighted-geometric mean can be written for the weightsas follows: and are called weighted multiplicative geometric mean and weighted-geometric mean. Also we have for all.
Definition 10 (weighted-harmonic mean). If a setof weights is associated with the data setthen the weighted-harmonic mean is defined by Takingand, the weighted-harmonic mean with the weightscan be written as follows: and are called multiplicative weighted harmonic mean and weighted-geometric mean, respectively. It is obvious that is reduced to ordinary weighted harmonic mean and ordinary weighted arithmetic mean forand, respectively.
In Table 1, the non-Newtonian means are obtained by using different generating functions. For, the-meansandare reduced to ordinary arithmetic mean, geometric mean, and harmonic mean, respectively. In particular some changes are observed for each value of, andmeans depending on the choice of. As shown in the table, for increasing values of, the-arithmetic meanand its weighted form increase; in particulartends to, and these means converge to the value of . Conversely, for increasing values of, the-harmonic mean and its weighted forms decrease. In particular, these means converge to the value of as. Depending on the choice of, weighted forms and , 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 compareand ordinary harmonic mean in Table 1, we also see that ordinary harmonic mean is smaller than. On the contraryandare smaller than their classical formsandfor. Therefore, we assert that the values of , and should be evaluated satisfactorily.
Corollary 11. Considerpositive real numbers. Then, the conditions and hold whenfor allandfor all.
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). Letbe an interval in. Thenis said to be-convex ifholds, whereandfor alland. Therefore, by combining this with the generatorsand, we deduce that
If (28) is strict for all, thenis said to be strictly-convex. If the inequality in (28) is reversed, thenis said to be-concave. On the other hand the inclusion (28) can be written with respect to the weighted-arithmetic mean in (21) as follows:
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, 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 takeand,in (28), then whereholds andis called bigeometric (usually known as multiplicative) convex function (cf. [2]). Equivalently,is bigeometric convex if and only if is an ordinary convex function.
(b) Forandwe have whereand. In this case the function is called geometric convex function. Every geometric convex (usually known as log-convex) function is also convex (cf. [2]).
(c) Takingand, one obtains whereand, and is called anageometric convex function.
Case 2. (a) Ifin (28) then where, , and is called-convex function.
(b) Forand, we write that where, , and is called-convex function.
(c) Forand, we obtain that where, , and is called-convex function.
The-convexity of a functionmeans geometrically that the-points of the graph ofare under the chord joining the endpointsandon non-Newtonian coordinate system for every. By taking into account the definition of-slope in Definition 3 we havewhich implies for all.
On the other hand (37) means that if, andare any three-points on the graph ofwithbetweenand, thenis on or below chord. In terms of-slope, it is equivalent towith strict inequalities whenis strictly-convex.
Now to avoid the repetition of the similar statements, we give some necessary theorems and lemmas.
Lemma 14 (Jensen’s inequality). A-real-valued functiondefined on an intervalis-convex if and only ifholds, where and for all and .
Proof. The proof is straightforward, hence omitted.
Theorem 15. Letbe a-continuous function. Thenis-convex if and only ifis midpoint-convex, that is,
Proof. The proof can be easily obtained using the inequality (39) in Lemma 14.
Theorem 16 (cf. [2]). Let be a -differentiable function (see [19]) on a subinterval. Then the following assertions are equivalent:(i)is bigeometric convex (concave).(ii)The functionis increasing (decreasing).
Corollary 17. A positive-real-valued functiondefined on an intervalis bigeometric convex if and only if holds, where for alland. Besides, we have
Corollary 18. A-real-valued functiondefined on an intervalis-convex if and only if holds, wherefor alland. Thus, we have
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.
Letbe a bigeometric (multiplicative) convex function and finite on a closed interval. It is obvious thatis bounded from above by, since, for anyin the interval,for. It is also bounded from below as we see by writing an arbitrary point in the formfor. Then Usingas the upper boundwe obtain Thus a bigeometric convex function may not be continuous at the boundary points of its domain. We will prove that, for any closed subintervalof the interior of the domain, there is a constantso that, for any two points,A function that satisfies (47) for someand allandin an interval is said to satisfy bigeometric Lipschitz condition on the interval.
Theorem 19. Suppose thatis multiplicative convex. Then,satisfies the multiplicative Lipschitz condition on any closed intervalcontained in the interiorof ; that is,is continuous on.
Proof. Takeso that, and letandbe the lower and upper bounds foron. Ifandare distinct points ofwith, set Thenand, and we obtain which yields where. Since the pointsare arbitrary, we getthat satisfies a multiplicative Lipschitz condition. The remaining part can be obtained in the similar way by takingand. Finally,is continuous, sinceis arbitrary in.
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,, , and means are reduced to ordinary arithmetic, geometric, and harmonic mean, respectively. As shown in Table 1, for increasing values of, and means increase, especially; these means converge to the value of . Conversely for increasing values of, and means decrease, especially; these means converge to the value of . 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.