Abstract and Applied Analysis

Volume 2014 (2014), Article ID 851271, 5 pages

http://dx.doi.org/10.1155/2014/851271

Research Article

Hermite-Hadamard Type Inequalities for Superquadratic Functions via Fractional Integrals

Guangzhou Li¹ and Feixiang Chen²

¹Department of Sports and Health, Chongqing Three Gorges University, Wanzhou, Chongqing 404000, China

²School of Mathematics and Statistics, Chongqing Three Gorges University, Wanzhou, Chongqing 404000, China

Received 3 July 2014; Accepted 28 July 2014; Published 14 August 2014

Academic Editor: Praveen Agarwal

Copyright © 2014 Guangzhou Li and Feixiang Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We use basic properties of superquadratic functions to obtain some new Hermite-Hadamard type inequalities via Riemann-Liouville fractional integrals. For superquadratic functions which are also convex, we get refinements of existing results.

1. Introduction

Let be a convex function and with ; then

is known as the Hermite-Hadamard inequality.

This remarkable result is well known in the literature as the Hermite-Hadamard inequality. Recently, the generalizations, refinements, and improvements of the classical Hermite-Hadamard inequality have been the subject of intensive research.

Definition 1 (see [1]). A function is superquadratic provided that for all there exists a constant such that

for all .

Theorem 2 (see [1]). The inequality

holds for all probability measures and all nonnegative, -integrable functions , if and only if is superquadratic.

The discrete version of the above theorem is also used in the sequel.

Lemma 3 (see [2]). Suppose that is superquadratic. Let , , and let , where and . Then

Nonnegative superquadratic functions are much better behaved as we see next.

Lemma 4 (see [1]). Let be a superquadratic function with as in Definition 1. Then one gets the following:

(1);(2)if , then whenever is differentiable at ;(3)if , then is convex and .

The Hermite-Hadamard inequalities for superquadratic functions are established by Banic et al. in [3].

Theorem 5 (see [3]). Let be a superquadratic function and ; then

It is remarkable that Sarikaya et al. [4] proved the following interesting inequalities of Hermite-Hadamard type involving Riemann-Liouville fractional integrals.

Theorem 6 (see [4]). Let be a positive function with and . If is a convex function on , then the following inequalities for fractional integrals hold:

with .

We remark that the symbols and denote the left-sided and right-sided Riemann-Liouville fractional integrals of the order with which are defined by

respectively. Here, is the gamma function defined by .

Fractional integral operators are widely used to solve differential equations and integral equations. So a lot of work has been obtained on the theory and applications of fractional integral operators.

For more results concerning the fractional integral operators, we refer the reader to [5–10] and references cited therein.

In this paper, we establish some new Hermite-Hadamard type inequalities for superquadratic functions via Riemann-Liouville fractional integrals which refine the inequalities of (6) for superquadratic functions which are also convex.

2. Main Results

From Lemma 3 for , we get for , , for the superquadratic function that

hold, and therefore

Let ; we get that

Therefore, by replacing in (9)

we get that

Theorem 7. Let be a superquadratic integrable function on with . Then

with .

Proof. By inequality (12), we get that

By the change of variable , we get

Therefore

We have completed the proof.

Corollary 8. Putting in Theorem 7 gives

Let ; we get that

Therefore, by replacing in (9)

we get that

Theorem 9. Let be a superquadratic integrable function on with . Then

with .

Proof. By inequality (20), we get that

The proof is completed.

Corollary 10. Choosing in Theorem 9, one has

Corollary 11. Let be defined as in Theorem 7; one gets

Corollary 12. Taking in Corollary 11, one obtains

3. Conclusion

In this note, we obtain some new Hermite-Hadamard type inequalities for superquadratic functions via Riemann-Liouville fractional integrals. For superquadratic functions which are also convex, we get refinements of known results. The concept of superquadratic functions in several variables is introduced in [11]. An interesting topic is whether we can use the methods in this paper to establish the Hermite-Hadamard inequalities for superquadratic functions in several variables via Riemann-Liouville integrals.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work is supported by the Youth Project of Chongqing Three Gorges University of China (no. 13QN11).

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