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Integral Inequalities of Hermite–Hadamard Type for m-AH Convex Functions

Tian-Yu Zhang*, Feng Qi

College of Mathematics, Inner Mongolia University for Nationalities, Tongliao City, Inner Mongolia Autonomous Region, China

Abstract: In the paper, the authors introduce the concept “m-AH convex functions” and establish some inequalities of Hermite-Hadamard type for m-AH convex functions.

Keywords: Hermite-Hadamard’s inequality, m-AH convex function, Hölder’s inequality

Received May 19, 2014; Revised June 01, 2014; Accepted June 03, 2014

Citation: Tian-Yu Zhang, Feng Qi, Integral Inequalities of Hermite–Hadamard Type for m-AH Convex Functions. Turkish Journal of Analysis and Number Theory, 2 (3). 60-64. January 2014. doi:10.12691/tjant-2-3-1

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

[picture] | (1.1)

holds for all and .

Definition 1.2. A function is said to be geometrically convex if

[picture] | (1.2)

holds for all and .

Definition 1.3 ([3]). A function is said to be -convex if

[picture] | (1.3)

holds for all,, and some .

Definition 1.4 ([11]). Let be a positive function on and . If

[picture] | (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

[picture] | (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

[picture] | (1.6)

(ii) If is convex on for , then

[picture] | (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

[picture] | (1.8)

[picture] | (1.9)

Theorem 1.3 ([[4], Theorem 2]). Let be-convex and . If for , then

[picture] | (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

[picture] | (1.11)

In this paper, we will introduce the concept “-AH convex functions” and establish some inequalities of Hermite-Hadamard 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

[picture] | (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

[picture] | (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 for-AH convex functions, we find the following lemmas.

Lemma 2.1. Let be differentiable on , with and . Then

[picture] | (2.3)

Proof. Integrating by part gives

[picture] | (2.4)

Similarly, we have

[picture] | (2.5)

[picture] | (2.6)

[picture] | (2.7)

From (2.4)-(2.7), the identity (2.3) follows. The proof is complete.

Lemma 2.2. For and , we have

[picture] | (2.8)

and

[picture] | (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

[picture] | (3.1)

where

[picture] | (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

[picture] | (3.3)

(2) if , then

[picture] | (3.4)

(3) if , then

[picture] | (3.5)

Theorem 3.2. Let be differentiable, for . If is an -AH convex function on for , , and , then

[picture] | (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

[picture] | (3.7)

(2) if , then

[picture] | (3.8)

(3) if , then

[picture] | (3.9)

Theorem 3.3. Let be differentiable, for . If is an -AH convex function on for , , and , then

[picture] | (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

[picture] | (3.11)

(2) if , then

[picture] | (3.12)

(3) if , then

[picture] | (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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