Keywords: HermiteHadamard’s inequality, mAH convex function, Hölder’s inequality
Turkish Journal of Analysis and Number Theory, 2014 2 (3),
pp 6064.
DOI: 10.12691/tjant231
Received May 19, 2014; Revised June 01, 2014; Accepted June 03, 2014
Copyright © 2013 Science and Education Publishing. All Rights Reserved.
1. Introduction
Throughout this paper, we use the following notations:
, , .
We first recall several definitions.
Definition 1.1. A functionis said to be convex if
 (1.1) 
holds for all and .
Definition 1.2. A function is said to be geometrically convex if
 (1.2) 
holds for all and .
Definition 1.3 (^{[3]}). A function is said to be convex if
 (1.3) 
holds for all,, and some .
Definition 1.4 (^{[11]}). Let be a positive function on and . If
 (1.4) 
holds for all , , then we say that the functionis geometrically convex on.
Definition 1.5 (^{[12]}). A function is said to be convex where , if we have
 (1.5) 
for all , .
We now recall some inequalities of Hermite Hadamard type.
Theorem1.1 ([^{[1]}. Theorem 2.2]). Let be a differentiable mapping on and with .
(i) If is convex on , then
 (1.6) 
(ii) If is convex on for , then
 (1.7) 
Theorem 1.2 ([^{[2]}, Theorems 2.3 and 2.4]). Let be differentiable onand with . If is a convex function on for , then
 (1.8) 
 (1.9) 
Theorem 1.3 ([^{[4]}, Theorem 2]). Let beconvex and . If for , then
 (1.10) 
Theorem 1.4. ([^{[6]}, Theorem 2.2]). Let be an open real interval and let be a differentiable function on such that for . If is convex on for some given numbers and , then
 (1.11) 
In this paper, we will introduce the concept “AH convex functions” and establish some inequalities of HermiteHadamard type for AH convex functions.
2. Definition and lemmas
The concept of AH convex function may be introduced as follows.
Definition 2.1. A function is said to be AH convex if for all and the inequality
 (2.1) 
holds. If the inequality (2.1) is reversed then is said to be AH concave function.
Definition 2.2. A function is said to be AH convex for some given number , if the inequality
 (2.2) 
holds for all and . if the inequality (2.2) reverses, then is said to be AH concave.
When , the AH convex function is AH convex function on .
In order to establish some inequalities of Hermite– Hadamard type forAH convex functions, we find the following lemmas.
Lemma 2.1. Let be differentiable on , with and . Then
 (2.3) 
Proof. Integrating by part gives
 (2.4) 
Similarly, we have
 (2.5) 
 (2.6) 
 (2.7) 
From (2.4)(2.7), the identity (2.3) follows. The proof is complete.
Lemma 2.2. For and , we have
 (2.8) 
and
 (2.9) 
3. Main Results
In this section, we will present several Hermite Hadamard type inequalities for the AH convex functions.
Theorem 3.1. Let be differentiable, for . If is an AH convex function on for , then
 (3.1) 
where
 (3.2) 
Proof. By Lemmas 2.1 and 2.2 and the AH convexity of on , we have
So the inequality (3.1) holds, which complete the proof.
Corollary . Under the conditions of Theorem 3.1,
(1) if , then
 (3.3) 
(2) if , then
 (3.4) 
(3) if , then
 (3.5) 
Theorem 3.2. Let be differentiable, for . If is an AH convex function on for , , and , then
 (3.6) 
where , are defined in (3.2).
Proof. By Lemma 2.1 and 2.2, the AH convexity of on , and Hölder’s inequality, we have
So the inequality (3.6) holds, which complete the proof.
Corollary . Under the conditions of Theorem 3.2,
(1) if , then
 (3.7) 
(2) if , then
 (3.8) 
(3) if , then
 (3.9) 
Theorem 3.3. Let be differentiable, for . If is an AH convex function on for , , and , then
 (3.10) 
Proof. By Lemma 2.1 and 2.2, the AH convexity of on , and Hölder’s inequality, we have
So the inequality (3.10) holds, which completes the proof.
Corollary . Under the conditions of Theorem 3.3,
if , then
 (3.11) 
(2) if , then
 (3.12) 
(3) if , then
 (3.13) 
Acknowledgement
This work was partially supported by the NNSF under Grant No. 11361038 of and by the Foundation of the Research Program of Science and Technology at Universities of Inner Mongolia Autonomous Region under Grant No. NJZY14191, .
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