Unification of Generalized and -Convexity

Abstract

In the present note, we will introduce the definition of generalized convex function. We will investigate some properties of generalized convex function. Moreover, we will develop Jensen’s type, Schur type, and Hermite Hadamard type inequalities for generalized convex function.

1. Introduction

Convexity is a basic and common idea which can be followed back to Archimedes (around 250 B.C.), regarding his acclaimed gauge of the estimation of (utilizing recorded and encircled standard polygons). He saw the significant reality that the parameters of a convex figure are lesser than the parameters of some other convex figure, encompassing it. As an issue of certainties, we experience convexity constantly and from numerous points of view. The most dull precedent is our standing up position, which is verified as long as the vertical projection of our focal point of gravity lies inside the convex envelope of our feet! Additionally, convexity greatly affects our regular daily existence through its various applications in industry, business, prescription, workmanship, and so forth. So are the issues on ideal assignment of assets and balance of nonagreeable recreations.

The theory of convex functions is a piece of the general subject of convexity since a raised capacity is one whose epigraph is a convex set. In any case, it is a theory significant essentially, which contacts practically all parts of science. Likely, the principal subject who makes important the experience with this theory is the graphical examination. With this event, we learn on the second derivative text of convexity, a useful asset in perceiving convexity. At that point, an issue of finding the extremal estimations of functions of several variables raises and the utilization of Hessian as a higher dimensional speculation of the second derivatives. Going to advancement issues in infinite dimensional spaces is the following stage; however, in spite of the specialized refinement in taking care of such issues, the fundamental thoughts are really comparable with one variable case. For detailed study, we refer Jensen [1, 2]. Anyway he was not the first who is dealing with such functions. We may refer [3–5]. During the entire twentieth Century, an exceptional research movement was done and critical outcomes were acquired in mathematical economics, nonlinear optimization, convex analysis, and so on. An extraordinary job in the advancement of the subject of convex functions was played by the acclaimed book of Hardy et al. [6].

Generally, there are two fundamental properties of convex functions that made them so broadly utilized in mathematics:(1)The maximum is attained at a boundary point.(2)Any local minimum is also a global minimum. Moreover, a strictly convex function can have at most one minimum value.

The modern perspective on convex functions involves an amazing and rich communication among geometry and analysis, which makes the reader to share a feeling of fervor. In an essential paper dedicated to the Brunn–Minkowski imbalance, Gardner [7, 8], depicted this reality in lovely expressions: “[convexity] appears like an octopus, tentacles reaching far and wide, its shape and color changing as it roams one area to the next. [And] it is quite clear that research opportunities abound”. During the years various striking books are written to the theory and utilizations of convex functions. We refer here [9–14].

Convexity plays an important role in nonlinear programming and optimization theory. Although, several results have been derived under convexity assumptions, many real world problems are nonconvex in nature. So it is always appreciable to study nonconvex functions, which are close to convex function in some sense. The following is the double inequality for the convex function, where is known as Hermite Hadamard inequality,

The inequality (1) has been extended and generalized for various classes of convex function, see [15–18]. For further readings, we refer some books, see [19–21].

Definition 1 (-convex set [15]). The interval is said to be a -convex set if for all and , where or , , , and .

Definition 2 (-convex function [15]). Let be a -convex set. A function is said to be -convex function, iffor all and .
One of the novel generalization of convexity is -convexity introduced by Delavar and Dragomir [16].

Definition 3 (-convex function [18]). The function is called convex with respect to for appropriate , iffor all and .

Definition 4 (Nonnegatively homogeneous [18]). A function is said to be nonnegatively homogeneous if for all and all .
Now we are ready to introduce the definition of generalized convex function.

Definition 5 (Generalized -convex function). Let be a bifunction for appropriate and be a -convex set. We say that is a generalized convex function, iffor all and
Of course (4) is -convexity for , -convexity for and classical convex function for and simultaneously.
We observe that by taking in (4) we getfor any and for any Also, if we take in (4), we getfor any . The second condition obviously implies the first.

Example 6. Consider a function defined byand define a bifunction as:The above function is generalized convex, but not convex function.

Example 7. Let , and where . Then is generalized convex function.

Proof. Take,Hence is generalized convex.

The paper is organized as follows: In the next section, we will derive some basic properties of generalized convex functions. In the last section, we will develop Hermite Hadamard, Jensen, Schur and Fejer type inequalities for generalized convex functions.

2. Basic Results

Proposition 8. (operation which preserves generalized convexity). Let be two generalized convex functions, then the following statements hold:(1)If is additive then is generalized convex.(2)If is nonnegatively homogeneous, then for any , is generalized convex.(3)If is generalized convex function, then

Proof. The proof of (1) and (2) is straightforward.
The proof of (3) is given as: For any convex interval, , we have for some which implies thatSince is arbitrary, soWhich is requried.

Proposition 9. If be a generalized convex function and is bounded above, has lower and upper bounds.

Proof. Suppose that is upper bound of on . From proposition (8), part (3)Now set .
For lower bound of consider an arbitrary point in the form in [a, b], thenNow consider and we obtain the required result.

Theorem 10. For be a -convex set. A function is generalized convex if and only if for any with

Proof. Suppose that is generalized convex function. Consider arbitrary with So there is a such that , namely . From generalized convexity of we haveImplies that,which is equivalent to above determinant being nonnegative.
Also, for and for For inverse implication, consider , where is -convex set with . Choosing any
we have and soby expanding the determinant we get,which implies,for any . So, is generalized convex function.

Theorem 11. Let is non empty collection of generalized convex functions such that(a)there exist and such that ,(b)for each exists in then the function defined by for each is generalized convex.

Proof. For any and , we havewhich is required.

3. Main Results

Theorem 12. (Schur type inequality). Let be a bifunction for appropriate and let be a function defined on interval I such that belongs to generalized convex function. Then such that and the following inequality holds:

Proof. Let be the generalized convex function and let be given. Then we can easily see thatandSetting and in (4) we have Assuming and multiplying both sides of the inequality above by ,
we obtain inequality (25).

Remark 13. In fact, if , , , and , then inequality (25) gives the schur type inequality, (see [20]).

Theorem 14. (Hermite Hadamard type inequality). Let be generalized convex function for with condition , then we obtain the inequality

Proof. Take and ,
impliesSo,By definition of generalized convex functions, we haveIntegrating above w.r.t ‘’ on [0, 1], we getwhich implies,Now,Similarly,Adding (35) and (36), we obtainCombining (34) and (37), we obtain the inequality (29).

Remark 15. If we put then (25) becomes Hermite Hadamard type inequality for -convexity, (see [16]).

Remark 16. If we put in (25) then we get Hermite hadamard type inequality for -convexity, (see [19]).

Remark 17. If we put = 1 and in (25) then we obtain classical Hermite Hadamard type inequality for convex functions.

We will use the following relations in the proof of Theorem 19 which is Jensen’s type inequality for generalized convex functions.

Theorem 18. Let be a -convex function, For and we have . Also when , for and , we have

Theorem 19. (Jensen’s Type inequality). Let with . Let generalized convex function and be nondecreasing, non negatively sub linear variable. Then we have following inequalitywhere and and .

Proof. Since be nondecreasing nonnegatively sub linear in first variable, so from Theorem 18 it follows that

Remark 20. For in (39), then we have Jensen’s type inequality for -convex function, see [17].

Remark 21. For in (39), then we have Jensen type inequality for -convex function.

Remark 22. For and in (39), then we have the Jensen type inequality for classical convex function.

Theorem 23. (Fejer type inequality). Let be nonnegative generalized convex functions such that , thenwhere,

Proof. Since and are generalized convex functions, we havefor all . Since and are nonnegative, soIntegrating both sides of above inequality over (0, 1), we obtain the inequalitySetting , we getThen, we obtain the (41) inequality.

Remark 24. If we put and in (41), then it reduces for classical convex functions.

Data Availability

All data are included within this paper.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Funding

Funding is not available for this research.