On Generalized Strongly -Convex Functions of Higher Order

Abstract

The aim of this paper is to introduce the definition of a generalized strongly -convex function for higher order. We will develop some basic results related to generalized strongly -convex function of higher order. Moreover, we will develop Hermite–Hadamard-, Fejér-, and Schur-type inequalities for this generalization.

1. Introduction

Some geometric properties of convex sets and functions have been studied before 1960 by great mathematicians Hermann, Minkowski, and Werner Fenchel. The classical convexity is defined as follows: a function is said to be convex function iffor all and .

The notion of convexity is crucial to the solution of many real-world problems. Fortunately, many problems encountered in constrained control and estimation are convex. Since the convexity of sets and functions has been the main object of studies of recent years, in many new problems encountered in applied mathematics, the notion of classical convexity is not enough to reach favorite results [1–4]. Recently, several extensions have been considered for classical convexity such that some of these new concepts are based on extension of the domain of convex function or set to a generalized form [5–7]. Some new generalized concepts in this point of view are pseudoconvex function, quasiconvex functions, invex functions, preinvex functions, B-convex functions, strongly convex functions, and generalized strongly convex functions. There are several fundamental books devoted to different aspects of convex analysis optimization [8–13]. Among them, we mention convex analysis by Rickafaller [11], convex analysis and minimization algorithm by Hiriart-Urruty and Lemerechel [8], convex analysis and nonlinear optimization by Browan and Lewin [9], and introducing lectures on convex optimization by Nesterov [10].

Several inequalities, including some famous inequalities are of Schur-type, Hermite–Hadamard-type, and Fejér-type inequalities, are satisfied by convex functions. The following inequality is known as Hermite–Hadamard inequality [14–20].

Let be a convex function and let with , then the following holds:

The present paper is organized as follows: in the first section, we will give some basic definitions and some basic properties related to our work. Next, we will derive Hermite–Hadamard-type, Fejér-type and Schur-type inequalities for our definition.

2. Basic Definitions

Let us recall few definitions related to our work [21–25].

A convex function is called strictly convex if the above inequality holds strictly whenever and are distinct points and .

Definition 1. (see [21]). A function is said to be quasiconvex if for any and , we have

Definition 2. The interval is said to be -convex set iffor all and .
In [26], Zhang et al. introduced -convexity as follows.

Definition 3. Let be convex set. A function f is said to be -convex function if the following inequality holds:wherever and
It can be easily seen that, for , -convexity reduced to classical convexity of functions defined on .
In [27], Polyak introduced the strongly convex function as follows.

Definition 4. Suppose is a convex set. A function is said to be strongly convex function with modulus iffor all and

Definition 5. (see [28]). A function is said to be strongly -convex function, iffor all and

Definition 6. (see [29]). A function is said to be generalized convex function with respect to for appropriate , iffor all and .

Definition 7. (see [26]). A function is said to be generalized -convex function, iffor all and
The generalized strongly -convex function [30] is defined as follows.

Definition 8. A function is said to be a generalized strongly -convex function, ifholds for
In [31], Lin and Fukushima gave the concept of higher order strongly convex functions and also used it many mathematical programs. Mishra and Sharma [32] derived the Hermite–Hadamard-type inequalities of higher order strongly convex function.

Definition 9. (see [31]). A function on the convex and closed set is said to be strongly convex function of higher order iffor all with , where and is any positive real number. If , then higher order strongly convex function becomes strongly convex functions with
Now, in the view of above definitions, we are in the position to introduce new generalization of convexity as follows:

Definition 10. A function f is said to be generalized strongly convex of higher order iffor , with and is any positive real number
Some generalizations of strongly -convex function of higher order are given in [11] for bifunctions.

Definition 11. A function f is said to be a generalized strongly -convex of higher order iffor , with

Remark 1. (1)If we take and in (13), then we obtain (10).(2)If we take , then we get (12).(3)Inserting and , we obtain (11).

3. Basic Results

This section is to introduce some basic results related to a generalized strongly -convex function of higher order.

Definition 12. A function is said to be nonnegatively homogeneous if for all and

Definition 13. (see [33]). A function is said to be additive if for all .

Proposition 1. Let be two generalized strongly -convex functions of higher order, then the following statements hold:(i)If is additive, then is a generalized strongly -convex function of higher order.(ii)If is nonnegatively homogeneous, then for any is a generalized strongly -convex function of higher order.(iii)If be generalized strongly -convex function of higher order with on and , then is also generalized strongly -convex functions of higher order.

Proof. (i)Take , then by definition of and , we obtain where and (ii)Let , then by definition, we obtain where and (iii)It directly follows from (i) and (ii).

Proposition 2. Let for be collection of generalized strongly -convex functions of higher order. Then, supremum functionis also a generalized strongly -convex function of higher order.

Proof. Fix and . For every , we havewhich implies in turn thatThis justifies the convexity of supremum function.

Proposition 3. Let be a generalized strongly -convex function of higher order, thenis also a generalized strongly -convex function.

Proof. Take any and Denote , where , thenimplying that is also a generalized strongly -convex function of higher order.

4. Main Results

In this section, we will develop the Hermite–Hadamard-, Fejér-, and Schur-type inequalities for generalized strongly -convex function of higher order.

Theorem 1. (Hermite–Hadamard-type inequality). Let be a generalized strongly -convex function of higher order with and such that is bounded above , then the function satisfies the following:

Proof. We begin the proof by proving the left-hand side of theorem.
Let be a generalized strongly -convex function of higher order for all , we set , and we haveIntegrating above equation with respect to over [0, 1], we haveTake and , we obtainWe obtainwhich is the left-hand side of theorem.
To prove right-hand side, we take for all . By definition, we haveIntegrating the above inequality with respect to over [0, 1], we getAfter solving this, we getAlso in the similar way, we obtainFrom (A) and (B), we haveThis implies thatwhich is the right-hand side of theorem. This completes the proof.

Remark 2. Imposing some conditions on Theorem 1, we obtain different versions of Hermite–Hadamard inequality [30, 33]:(1)If we take , gives the Hermite–Hadamard inequality for generalized strongly -convex functions.(2)If we take and , Theorem 1 gives Hermite–Hadamard inequality for -convex functions.(3)If we take , then we obtain the Hermite–Hadamard-type inequality of strongly convex function of higher order [34].

Theorem 2. (Schur-type inequality). Let be a generalized strongly convex function of higher order with and , be a bifunction, then for all such that and , the following inequality holds:

Proof. Let be a generalized strongly -convex function of higher order with and , thenTake , and in inequality (26), we haveMultiplying (36) by on both sides, we get

Remark 3. If we take , then we obtain the Schur-type inequality of a generalized strongly -convex function [30].
Now we will derive the weighted version of Hermite–Hadamard inequality via a generalized strongly convex function of higher order.

Theorem 3. (Fejér-type inequality). Let be a generalized strongly function of higher order with and provided is bounded from above on . Also suppose that is nonnegative, integrable, and symmetric with respect to then

Proof. Let us prove the left-hand side of theorem. Take and be a generalized strongly -convex function of higher order. We take for all in (26):TakeSince be a nonnegative, integrable, and -symmetric with respect , thenfor all . Multiplying (40) by ,Choose and rearranging the above equation yields thatFor the right-hand side of theorem, we choose Multiplying on both sides by and integrating w.r.t over [0, 1], we getSimilarly, we getBy adding (46) and (47), we getWe getThis completes the proof.

Remark 4. Imposing some conditions on Theorem 3, we get different versions of Hermite–Fejér-type inequality:(1)If we put and , then we obtain the Fejér-type inequality of a generalized strongly -convex function [30].(2)For in the Theorem 3, we obtain Theorem 1.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that there are no conflicts of interest.

Authors’ Contributions

Chahn Yong Jung improved the presentation of the paper. Muhammad Shoaib Saleem supervised this work. Yu-Ming Chu revised the paper and arranged funding. Nazia Jahangir proved the main results. Huma Akhtar wrote the first draft of the paper.

Acknowledgments

The research was supported by the National Natural Science Foundation of China (Grant nos. 11971142, 11871202, 61673169, 11701176, 11626101, and 11601485).