On Hermite-Hadamard Type Inequalities for Riemann-Liouville Fractional Integrals via Two Kinds of Convexity

Feixiang Chen School of Mathematics and Statistics, Chongqing Three Gorges University, Wanzhou, Chongqing 404000, China Correspondence should be addressed to Feixiang Chen; cfx2002@126.com Received 16 February 2014; Revised 20 May 2014; Accepted 3 June 2014; Published 15 June 2014 Academic Editor: Annamaria Barbagallo Copyright © 2014 Feixiang Chen.This is an open access article distributed under theCreativeCommonsAttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We obtain someHermite-Hadamard type inequalities for products of twom-convex functions via Riemann-Liouville integrals.The analogous results for (α,m)-convex functions are also established.


Introduction
If  :  →  is a convex function on the interval , then for any ,  ∈  with  ̸ =  we have the following double inequality: This remarkable result is well known in the literature as the Hermite-Hadamard inequality.Since then, some refinements of the Hermite-Hadamard inequality for convex functions have been extensively obtained by a number of authors (e.g., [1][2][3][4][5][6][7]).
In [8], Toader defined the concept of -convexity as follows.
Some Hermite-Hadamard type inequalities for products of two -convex and (, )-convex functions are established in [11].
Theorem 5 (see [11]). where Theorem 6 (see [11]). where )  () Some new integral inequalities involving two nonnegative and integrable functions that are related to the Hermite-Hadamard type are also proposed by many authors.In [12], Pachpatte established some Hermite-Hadamard type inequalities involving two log-convex functions.An analogous result for -convex functions is obtained by Kirmaci et al. in [13].In [14], Sarikaya et al. presented some integral inequalities for two ℎ-convex functions.
It is remarkable that Sarikaya et al. [15] proved the following interesting inequalities of Hermite-Hadamard type involving Riemann-Liouville fractional integrals.
We remark that the symbols    + and    −  denote the leftsided and right-sided Riemann-Liouville fractional integrals of the order  ≥ 0 with  ≥ 0 which are defined by respectively.Here, Γ() is the Gamma function defined by In this paper, we obtain some new Hermite-Hadamard type inequalities for products of two -convex functions via Riemann-Liouville integrals.The analogous results for (, )-convex functions are also given.

Inequalities for Products of Two Functions
for Riemann-Liouville Fractional Integrals Proof.Since  is  1 -convex and  is  2 -convex on [, ], then for  ∈ [0, 1] we get From ( 14), we get Multiplying both sides of the above inequality by  −1 and then integrating the resulting inequality with respect to  over [0, 1], we obtain Analogously, we obtain Multiplying both sides of above inequality by  −1 and then integrating the resulting inequality with respect to  over [0, 1], we obtain (21) , respectively, then one has Proof.Since  is ( 1 ,  1 )-convex and  is ( 2 ,  2 )-convex on [, ], then for  ∈ [0, 1] we get From ( 23), we get Multiplying both sides of above inequality by  −1 and then integrating the resulting inequality with respect to  over [0, 1], we obtain Similarly, we have We get the desired result.

Conclusion
In this paper, we obtain some new Hermite-Hadamard type inequalities for products of two -convex functions via Riemann-Liouville integrals.The analogous results for (, )-convex functions are also established.An interesting topic is whether we can use the methods in this paper to establish the Hermite-Hadamard inequalities for products of two convex functions on the coordinates via Riemann-Liouville integrals.