Keywords: RiemannLiouville fractional integral, (m, h1, h2)convex function, integral inequality of HermiteHadamard type
Turkish Journal of Analysis and Number Theory, 2014 2 (1),
pp 2328.
DOI: 10.12691/tjant216
Copyright © 2013 Science and Education Publishing. All Rights Reserved.
1. Introduction
The following definitions are well known in the literature.
Definition 1.1. A function is said to be convex if
 (1) 
holds for all and . If the inequality (1) reverses, then is said to be concave on .
The wellknown HermiteHadamard inequality reads that for every convex function ,we have
, where with . If is concave, the above inequalities reverse.
Definition 1.2. (^{[2]}) For , a function is said to be convex if
 (2) 
holds for all and . If the above inequality (2) reverses, then is said to be concave on .
Definition 1.3. (^{[6]}) Let , be an interval, and . A function is said to be convex if the inequality
 (3) 
If the above inequality (3) reverses, then is said to be concave on .
Definition 1.4. (^{[10]}) For, if
is valid for all and ,then we say that is a convex function on .
There have been many inequalities of HermiteHadamard type for the above convex functions. Some of them may be recited as follows.
Theorem 1.1. (^{[2]}) Let be a differentiable function on , where and . If is a convex function on , then
Theorem 1.2. (^{[3]}) Letis a differentiable function on ,is convex function on , where ，then
Theorem 1.3. (^{[4]}) Letis a differentiable function on , where ，if is convex function on , then
Theorem 1.4. (^{[5]}) Letisconvex function, where , if ，then
Definition 1.6. (^{[1]}) Let ，The RiemannLiouville integrals and of order with are defined by
and
Respectively, where and is the classical Euler gamma function be defined by
Theorem 1.5. (^{[12]}) Letbe a positive function with , If is a convex function on then
with .
Theorem 1.6. (^{[12]}) Let be a differentiable mapping on with , If is convex on , then
In this paper, motivated by the above results, we will establish a RiemannLiouville fractional integral identity involving a differentiable mapping and present some new inequalities of HermiteHadamard type involving RiemannLiouville fractional integrals for (m, h_{1}, h_{2})convex functions.
2. A Definition and A Lemma
In the most recent paper ^{[11]}, Maksa and Palés introduced even more general notion of convexity. More precisely, convex functions are defined as solutions f of the functional inequality
where and are given functions.
We first introduce a definition of convex functions.
Definition 2.1. Assume, and .Then is said to be (m, h_{1}, h_{2})convex if the inequality
holds for all and . If the above inequality reverses, then is said to be (m, h_{1}, h_{2})concave on .
Let be a differentiable function on and . Denote by
Specially, when , we have
Lemma 2.1. Let be a differentiable function on such that . Then
Proof. Letting. By integration by parts, we have
Similarly, we obtain
and
The proof of Lemma 2.1 is complete.
3. New Inequalities for (m, h_{1}, h_{2})Convex Functions
Theorem 3.1. Let be a differentiable function such that for . If is convex on for and , then
where for and is the classical Beta function which may be defined by
Proof. From Lemma 2.1, Hölder inequality, and the (m, h_{1}, h_{2})convexity of , we obtain
where
and
The proof of Theorem 3.1 is complete.
Corollary 3.1.1. Under the conditions of Theorem 3.1, if then
Furthermore, if then
Corollary 3.1.2. Under the conditions of Corollary 3.1.1, if then
Specially, if , then
Corollary 3.1.3. Under the conditions of Theorem 3.1,if and , then
Specially, if then
Theorem 3.2. Let be a differentiable function such that and . If is convex on for and , then
where where is as given in Theorem 3.1.
Proof. From Lemma 2.1, Hölder inequality, and the convexity of , we obtain
The proof of Theorem 3.2 is complete.
Corollary 3.2.1. Under the conditions of Theorem 3.2, (1) if and , then
(2) if and , then
(3) if and , then
(4) if and , then
Acknowledgement
This work was partially supported by the NNSF under Grant No. 11361038 of China and by the Foundation of the Research Program of Science and Technology at Universities of Inner Mongolia Autonomous Region under Grant No. NJZY13159, China.
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