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International Journal of Mathematics and Mathematical Sciences

Volume 2014, Article ID 136035, 20 pages

http://dx.doi.org/10.1155/2014/136035
Research Article

Applying GG-Convex Function to Hermite-Hadamard Inequalities Involving Hadamard Fractional Integrals

1Department of Mathematics, Guizhou University, Guiyang, Guizhou 550025, China

2School of Mathematics and Computer Science, Guizhou Normal College, Guiyang, Guizhou 550018, China

Received 1 February 2014; Revised 7 June 2014; Accepted 10 June 2014; Published 14 July 2014

Academic Editor: Hari M. Srivastava

Copyright © 2014 Zhi Zhang et al. 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

By virtue of fractional integral identities, incomplete beta function, useful series, and inequalities, we apply the concept of GG-convex function to derive new type Hermite-Hadamard inequalities involving Hadamard fractional integrals. Finally, some applications to special means of real numbers are demonstrated.

1. Introduction

Fractional calculus played an important role in various fields such as electricity, biology, economics, and signal and image processing [18]. The fractional Hermite-Hadamard inequality gives a lower and an upper estimation for both right-hand and left-hand integrals average of any convex function defined on a compact interval, involving the midpoint and the endpoints of the domain.

As we know, Set [9] firstly studied fractional Ostrowski inequalities involving Riemann-Liouville fractional integrals. Then, Sarikaya et al. [10] studied Hermite-Hadamard type inequalities involving Riemann-Liouville fractional integrals. Further, our group go on studying fractional version Hermite-Hadamard inequality involving Riemann-Liouville and Hadamard fractional integrals for all kinds of functions [1119].

Recently, Wang et al. [16, 17] established the following two powerful fractional integral identities involving Hadamard fractional integrals.

Lemma 1 (see [12, Lemma 3.1]). Let be a differentiable mapping on . If , then the following equality for fractional integrals holds: where the symbols and are defined by where is the Gamma function.

Lemma 2 (see [11, Lemma 2.1]). Let be a differentiable mapping on with . If , then the following equality for fractional integrals holds: where

Remark 3. It is remarkable that Professor Srivastava et al. [20] give some further refinements and extensions of the Hermite-Hadamard inequalities in variables. In the forthcoming works, we will try to extend to study fractional version Hermite-Hadamard inequalities in variables based on such fundamental results.

Next, we recall the following basic concepts and results in our previous papers.

Definition 4 (see [21, 22]). Let . A function is said to be GG-convex on if, for every and , one has

Remark 5. By the arithmetic-geometric mean inequality, we have Linking (5) and (6), we obtain which appears in the standard definition of GA-convex function [21]. So GG-convex function is GA-convex function.

Lemma 6 (see [19, Lemma 2.5]). For , one has

Lemma 7 (see [13, Lemma 2.1]). For and , one has where .

Lemma 8 (see [13, Lemma 2.2]). For and , one has

Lemma 9 (see [13, Lemma 2.3]). For and , one has

In the present paper, we will use the above concepts and lemmas to derive some new fractional Hermite-Hadamard inequalities involving Hadamard fractional integrals.

2. Main Results Based on Lemma 1

Now we are ready to state the following main results in this section.

Theorem 10. Let be a differentiable mapping. If is measurable and is GG-convex on for some fixed and , , then the following integrals hold:

Proof. Noting Definition 4 and Lemmas 1 and 6, we have The proof is done.

Theorem 11. Let be a differentiable mapping. If is measurable and , is GG-convex on for some fired and , , then the following integrals hold: where .

Proof. By using Definition 4 and Lemmas 1 and 6, we have The proof is done.

Theorem 12. Let be a differentiable mapping. If is measurable and is GG-convex on for some fired , and , , then the following integrals hold:

Proof. By using Definition 4 and Lemmas 1, 7, 8, and 9, we have The proof is done.

Theorem 13. Let be a differentiable mapping. If is measurable and , is GG-convex on , for some fired , , and , , then the following integrals hold: where .

Proof. By using Definition 4 and Lemmas 1, 7, 8, and 9, we have The proof is done.

3. Main Results Based on Lemma 2

Theorem 14. Let be a differentiable mapping. If is measurable and is GG-convex on , for some fired and , , then the following integrals hold:

Proof. By using Definition 4 and Lemmas 2 and 6, we have