Fractional Versions of Hermite-Hadamard, Fejér, and Schur Type Inequalities for Strongly Nonconvex Functions

Abstract

In modern world, most of the optimization problems are nonconvex which are neither convex nor concave. The objective of this research is to study a class of nonconvex functions, namely, strongly nonconvex functions. We establish inequalities of Hermite-Hadamard and Fejér type for strongly nonconvex functions in generalized sense. Moreover, we establish some fractional integral inequalities for strongly nonconvex functions in generalized sense in the setting of Riemann-Liouville integral operators.

1. Introduction

The integral and differential operators have remarkable impact on applied sciences, and the interest of researchers is increasing day by day in this research area [1, 2]. Consider a convex function defined on the interval with being constants and . The inequality

is Hermite-Hadamard’s (see [3, 4]).

The notion of convexity is very old, and it appears in Archimedes treatment of orbit length. Nowadays, convex geometry is a mathematical subject in its own right. There are several modern works on convexity that are for the studies of real analysis, linear algebra, geometry, and functional analysis. The theory of convexity helps us to solve many applied problems. In recent years, the theory of convex analysis gains huge attention of researchers due to its interesting applications in optimizations, geometry, and engineering [5, 6].

The present paper deals with a new class of convex functions and establishes inequalities of Hermite-Hadamard and Fejér. Moreover, we develop some fractional integral inequalities. See [7, 8] for more general inequalities via convexity of functions.

The classical definition of convex functions was given in [3]. Another concept which is used widely in convex analysis is -convex sets and -convexity (see [4]). By taking in the above definition, we get classical notion of convexity. After that, the strongly convex with modulus was introduced in [9]. And in [10], the notion of the strongly -convex function had been introduced. The notion of generalized convex functions had been introduced in [11, 12].

Motivated by the above researches, [13] introduced the following class of functions.

The function is strongly nonconvex in generalized sense if

holds for

Definition 1 (see [13, 14]). Consider then the RHS and the LHS Riemann-Liouville fractional integral (RL) of order with are defined by respectively, where is the Gamma function defined as It is to be noted that

The Riemann integral is reduced to classical integral for [15–18].

The definition of strong -convexity was studied in [13]. The aim of this paper is to establish the inequalities of Schur, Fejér, and Hermite-Hadamard type for the strongly nonconvex functions via RL fractional integrals.

2. Inequality of Hermite-Hadamard Type

In order to prove the inequality of Hermite-Hadamard type, the following lemma is very important.

Lemma 2 (see [19]). Let be any nonzero real number and be any positive constant. Further consider an integrable function , where which is -symmetric w.r.t. ; then, we have the following: (i)If with , (ii)If with ,

Theorem 3. Let the strongly generalized -convex function with magnitude and (·) be bounded above in Then, if is any positive real number, we have

Proof. We begin the proof by inserting and Take and , then (8) yields Multiplying (9) by and then integrating w.r.t. over the interval which is the left side of Theorem 3
Now, to obtain the left-hand side of Theorem 3, we have for , and for Combining (12) and (13), we have Multiplying (14) by and then integrating w.r.t. over the interval , we have Together (11) and (16) give the required result.

Remark 4. (i)Fixing in Theorem 3 gives Hermite-Hadamard inequality in the sense of the strongly generalized convexity(ii)Fixing and in Theorem 3, we obtain [20] (Theorem 2.1)(iii)Fixing and in Theorem 3 yields [21] (Theorem 2.1)(iv)Applying both (ii) and (iii) on Theorem 3, we obtain classical fractional version of H-H inequality

Definition 5 (see [22]). Let be any nonzero real number; then, the function is -symmetric w.r.t. if for all

Theorem 6 (inequality of Fejér type). Suppose that is a function as in Theorem 3 and an integrable, nonnegative function is symmetric w.r.t. , then

Proof. Setting in (2), Substitute and in (18), According to the given conditions of , we have Multiplying (19) by and then integrating w.r.t. over the interval , Let , then (23) becomes Now, take then by Def of , Multiply on both sides of (26) by and then integrate w.r.t. over the interval , Take in (28), then we have Similarly, we have from definition of by fixing
Combining (29) and (30), we obtain Combining (32) and (25) completes the theorem (17).

3. Fractional Integral Inequalities for Strongly Generalized -Convex Function

Lemma 7. Consider a differentiable function on , with , where a, b and . If is integrable, then holds with .

Proof. Let , and , then for generalized strongly -convex function, we have where By integration by parts, we have By combining (34), (37), and (38), we have (33). This completes the proof.

Remark 8. Setting and in Lemma 7 gives us [21] (Lemma 2.1).

Theorem 9. Let the function be as in Theorem 3.1. If is a strongly generalized -convex function on for positive and , then where

Proof. Theorem (3) gives Setting , , we have Since is a strongly generalized -convex function on , we have After combining (48) and (43), we have

Remark 10. If one takes and , then we get [21] (Theorem 2.2).

Theorem 11. Let the function be as in Theorem 3.1. If is as in Theorem 9, then where

Proof. Let : Setting , , we have Using the inequality of power mean the definition of , This completes the proof.

Remark 12. (i)Setting and in Theorem 11 gives H-H type inequality for convex functions(ii)Setting and in Theorem 11 gives H-H inequality for convex functions

4. Conclusion

In this paper, we established inequalities of Hermite-Hadamard and Fejér type for strongly generalized -convex functions. We also established some fractional integral inequalities for this class of function in the setting of RL fractional integrals. We also related our results with the existing results and proved that by fixing involved parameters, we get many previous results.

Data Availability

The data required for this research is included within this paper.

Conflicts of Interest

The authors do not have any competing interests.

Authors’ Contributions

All authors contributed equally in this paper.