Abstract

In this paper, with a new approach, some new Hermite-Hadamard type inequalities for preinvex functions are obtained by using only the right Riemann-Liouville fractional integrals. Our results generalize previous studies. Results proved in this paper may stimulate further research in this field.

1. Introduction

Let be a finite interval of real numbers. A function is said to be convex if the inequality

holds for and (see ¹).

Convexity plays a fundamental role in many branches of pure and applied sciences. In recent years, the investigation on extended convex functions has become a deep research area. The applications of extended convex functions in establishing various inequalities have received renewed attention by many researchers. A significant generalization of convex functions is that of invex functions introduced by Hanson ². Ben-Israel and Mond ³ introduced the notions of invex sets and preinvex functions. Pini ⁴ introduced the notion of prequasiinvex functions which is a generalization of invex functions. Weir and Mond ⁵ and Noor ⁶ have studied the basic properties of the preinvex functions. For recent applications and generalizations of the preinvex functions, see ^{7, 8, 9, 10, 11, 12, 13, 14} and references therein.

Inequalities have a key role in pure and applied mathematics. A number of studies have shown that the theory of convexity has a closely relationship with the theory of inequalities. One of the most famous inequality in the literature for convex functions is known as Hermite-Hadamard integral inequality. This double inequality is stated as:

Let be a convex function on Then the following inequality holds:

Hermite-Hadamard inequality for convex functions has attracted many researchers and as gradually a remarkable of generalizations and extensions in various directions have appeared in the literature, one can see ^{15, 16, 17, 18, 19, 20, 21, 22, 23, 24} and references therein.

2. Preliminaries

Let us recall some definitions and known results concerning invexity and preinvexity.

Definition 1. ²⁵ A set is said to be invex if there exist a function such that

The invex set is also called a -connected set.

Definition 2. ⁵ Let be a function on the invex set Then, is said to be preinvex with respect to if

It is to be noted that every convex function is preinvex with respect to the map but the converse is not true in general.

In ²⁶ Noor obtained the following Hermite-Hadamard inequalities for the preinvex functions.

Theorem 1. Let be a preinvex function on the interval of real numbers and with Then the following inequality holds:

Following definitions of the left and right side Riemann-Liouville fractional integrals are well known in the literature.

Definition 3. ²⁷ Let The left and right Riemann-Liouville fractional integrals and of order are defined by

and

respectively, where is the Gamma function defined by

In ²², Sarıkaya et al. gave the fractional analogue of the inequality (1) as follows:

Theorem 2. Let be a positive function with and If is a convex function on then the following inequalities for fractional integrals hold:

with

In ²⁸, Iscan proved the following Hermite-Hadamard inequalities for preinvex functions via Riemann-Liouville fractional integrals:

Theorem 3. Let be an open invex subset with respect to and with If is a preinvex function, then the following inequalities for fractional integrals hold:

with

3. Main Results

In this section, we will obtain some generalizations of the right side of the Hermite- Hadamard type inequalities for functions whose first derivatives absolute values are preinvex via right Riemann-Liouville fractional integrals.

Lemma 1. Let be an open invex subset with respect to and with Suppose is a differentiable mapping on such that Then the following equality for the right Riemann-Liouville fractional integrals holds:

Proof. If we apply the partial integration to the right hand side of the above equality, we have

This completes the proof.

Remark 1. In Lemma 1,

1. If we take then we get the inequality given in [ ¹⁷, Lemma 3.1].

2. If we take and then we get the inequality given in [ ²⁰, Lemma A].

Theorem 4. Let be an open invex subset with respect to and with Suppose is a differentiable mapping on such that If is preinvex on then the following Riemann-Liouville fractional integral inequality holds:

Where

with

Proof. Using Lemma 1 and the preinvexity of we have

So, the proof is completed.

Remark 2. In Theorem 4,

1. If we take then we get the inequality given in [ ¹⁷, Theorem 4.1].

2. If we take and then we get the inequality given in [ ²⁹, Theorem 2.2].

Theorem 5. Let be an open invex subset with respect to and with Suppose is a differentiable mapping on such that If is preinvex on for then the following Riemann-Liouville fractional integral inequality holds:

where and are given as Theorem 4 and

Proof. Using Lemma 1, power mean inequality and the preinvexity of we have

This completes the proof.

Remark 3. In Theorem 5,

1. If we take then we get the inequality given in [ ¹⁷, Theorem 4.2].

2. If we take and then we get the inequality given in [ ²⁰, Theorem 1].

Theorem 6. Let be an open invex subset with respect to and with Suppose is a differentiable mapping on such that If is preinvex on for then the following Riemann-Liouville fractional integral inequality holds:

Where

with and

Proof. Using Lemma 1, Hölder’s integral inequality and the preinvexity of we have

This completes the proof.

Remark 4. In Theorem 6,

1. If we take then we get the inequality given in [ ¹⁷, Theorem 4.3].

2. If we take and then we get the inequality given in [ ²⁹, Theorem 2.3].

4. Conclusion

We have derived new fractional Hermite-Hadamard type integral inequalities via preinvex functions involving only the right Riemann-Liouville fractional integral. We have obtained new generalizations of the right side of the Hermite- Hadamard type inequalities for functions whose first derivatives absolute values are preinvex via right Riemann-Liouville fractional integrals. It has shown that previously known results can be obtained as special cases from our results. It is expected that idea of this article may attract interested readers.

Acknowledgements

The author is thankful to the Editor and anonymous referees for their constructive comments and valuable suggestions. This research article is supported by Kırklareli University Scientific Research Projects Coordination Unit. Project Number: KLUBAP-191.

References