Some Properties and Inequalities for the -Nonconvex Functions

Abstract

The purpose of this paper is to introduce the notion of strongly -nonconvex functions and to present some basic properties of this class of functions. We present Schur inequality, Jensen inequality, Hermite–Hadamard inequality, and weighted version of the Hermite–Hadamard inequality.

1. Introduction

Convexity, the study of convex functions, has scope in various fields of science and pure mathematics, as well as applied mathematics. From the last few decades, many extensions and generalizations of convexity have been expressed to support different research ideas in mathematics, see, for instance, [1–12].

Suppose be a convex function and , where , then the inequalityis named as the Hermite–Hadamard equality. The above inequality is a necessary condition for a function to be convex.

The generalization of inequality (1) was given by Fejér as is a convex function and is nonnegative, integrable, and symmetric about , and then

Obeidat [13] generalized the class of -convex functions, which is called strongly -convex functions.

In this article, we generalized the concept of -convex functions and defined strongly -convex functions. Moreover, we present some basic properties and results of strongly -nonconvex functions, and Jensen inequality, Schur inequality, Hermite–Hadamard inequality, and weighted version of Hermite–Hadamard-type inequalities are obtained for this class of functions.

2. Preliminaries and Some Properties

In this section, we recall some definitions from the literature which are helpful for the study of strongly -nonconvex functions.

Definition 1. (convex function). A function is said to be a convex function iffor all and .

Definition 2. (-convex function, see [2]). A function is called a h-convex function iffor all and .

Definition 3. (strongly convex function, see [14]). A function is said to be a strongly convex function with modulus iffor all and .

Definition 4. (strongly -convex function, see [15]). A function is said to be a strongly -convex function with modulus iffor all and .

Definition 5. (-convex function). A function is said to be a -convex function in the second sense iffor all and , where .

Definition 6. (-convex function, see [1]). A function is said to be a -convex function iffor all and .

Now, we give our definition of a strongly -convex function.

Definition 7. (strongly -convex function, see [13]). A function is said to be a strongly -convex function with modulus iffor all and .

Definition 8. (nonconvex set). A set is a -convex set if for all , (or ), and , where .

Definition 9. (nonconvex function). A function is called a nonconvex function iffor all and , where is a -convex set.

Definition 10. (-nonconvex function). A function is called a -nonconvex function iffor all and .

Definition 11. (strongly -nonconvex function). A function is called a strongly -nonconvex function with modulus iffor all and .
Clearly, a strongly -nonconvex function covers -convex function, -nonconvex function, h-convex function, and s-convex function.

Remark 1. Allowing , strong -convexity is obtained, and with and , strong h-convexity is obtained. Similarly, replacing , , , and , the classical convexity is retraced.

In the following, we present some properties of strongly -nonconvex functions.

Proposition 1. Let be nonnegative functions defined on interval such that where and . If is a strongly -nonconvex function, then is a strongly -nonconvex function.

Proof. If is a strongly -nonconvex function, then for every and , we haveThis completes the proof.

Proposition 2. If and are strongly -nonconvex functions and , then(i) is a strongly -nonconvex function(ii) is a strongly -nonconvex function

Proof. (i)Since and are strongly -nonconvex functions, then we have where , and this completes the proof.(ii)Since and is a strongly -nonconvex function, then we have where .

Theorem 1. Suppose for each , then is a strongly -nonconvex function with modulus , and is defined asthen is a -nonconvex function.

Proof. We start the proof withAs and , then we obtain

Remark 2. Theorem 1 is not true for arbitrary , see Example 1 in [13].

Theorem 2. Suppose for each , then is a -nonconvex function, and is defined asthen is a strongly -nonconvex function.

Proof. We start the proof withAs and , then the above inequality yieldsThis completes the proof.

Remark 3. Theorem 2 is not true for arbitrary , see Example 2 in [13].

3. Main Results

In this section, we intend to make the reformulations of the Jensen-type inequality, Hermite–Hadamard type inequalities, Fejér type inequality, and Schur type inequality for strongly -nonconvex functions.

Theorem 3. (Jensen inequality). Suppose be a strongly -nonconvex function, then the inequalityholds for all and with and .

Proof. Take and , where , such that .
Set and take a functionsupporting at , that is, and , .
Then, we havefor every .
Multiplying on both sides of the above inequality by and summing up, we obtainSince , we havewhich completes the proof.

Now, the following results are obtained.

Corollary 1. Suppose be a strongly -convex function, then the inequalityholds for all , , and with and .

Corollary 2. Suppose be a strongly -convex function, then the inequalityholds for all , , and with and .

Fixing in inequality (22), we obtain Corollary 1. Similarly, for , the inequality (22) yields Corollary 2. If we impose the above conditions with on inequality (22), we obtain the Jensen inequality for a strongly convex function.

Theorem 4. (Schur inequality). Assume that be a strongly -nonconvex function with modulus , then for all such that and , the inequalityholds for .

Proof. The proof starts with the assumption that be a strongly -nonconvex function with modulus and , then , , and .
Substituting , , and in inequality (12) yieldsUsing the property of and multiplying on both sides of the above inequality by , we obtainwhich completes the proof.

Substituting in (29), we obtain the following corollary.

Corollary 3. Assume that be a strongly -convex function with modulus , then for all such that and , the inequalityholds for .

Theorem 5. (Hermite–Hadamard inequality). Let be a strongly -nonconvex function with modulus and , then the inequalityholds.

Proof. We begin the proof by using the definition of the strongly -nonconvex function:for all and .
Integrating the above inequality with respect to “t” over yieldsNow, for the formulation of the left-hand side of inequality (33), we havewhere and .
Sincethen by integrating inequality (36) with respect to “” over , we obtainHence,which completes the proof.

Remark 4. If we set in (33), we obtain the Hermite–Hadamard inequality for a strongly convex function, see [13]. And, allowing , , and , it yields a classical Hermite–Hadamard inequality.

Theorem 6. (Fejér inequality). Let be a strongly -nonconvex function, , and be a nonnegative symmetric function about , then

Proof. Suppose is a strongly -nonconvex function, then for , we havewhere . Since is a nonnegative symmetric function about , then we haveIntegrating the above inequality with respect to “” over yieldsand since is symmetric about , we obtainHence, we obtainFor the left-hand side of inequality (40), we observe that for all , we haveAfter integrating the above inequality with respect to “” over and multiplying both sides by , which is a nonnegative and symmetric function about , we obtainThus, we obtainsince , which yieldsThis completes the proof.

Remark 5. If we allow in Theorem 6, then it reduces to the Fejér-type inequality for a strongly -convex function, and for and , we obtain the inequalities generalized in [1].

4. Conclusion

In this paper, we have introduced a strongly -nonconvex function, which is the generalization of many existing definitions. We also proved several inequalities, for example, Schur inequality, Jensen inequality, and Hermite–Hadamard inequality, for a newly defined strongly -nonconvex function. This definition can also be used to develop inequalities presented in [16–18] and references therein.

5. Future Directions

In future, we are interested to work on generalizations of stochastic -convex processes and stochastic processes. We will develop Schur inequality, Jensen inequality, and Hermite–Hadamard inequality for these generalizations.

Data Availability

All data required for this paper are included within this paper.

Conflicts of Interest

The authors do not have any conflicts of interest.

Authors’ Contributions

M. S. Saleem proposed the problem. H. Rehman proved the results. M. Imran wrote the first version of the paper. C. Wang analysed the results, revised the paper, and proposed future directions.

Acknowledgments

This research was supported by the funds of the University of Okara, Okara, Pakistan. This paper was supported by the Higher Education Commission, Pakistan.