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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 560586, 14 pages
doi:10.1155/2012/560586
Research Article

On Integral Inequalities of Hermite-Hadamard Type for s-Geometrically Convex Functions

Tian-Yu Zhang,1 Ai-Ping Ji,1 and Feng Qi2

1College of Mathematics, Inner Mongolia University for Nationalities, Inner Mongolia Autonomous Region, Tongliao City 028043, China
2School of Mathematics and Informatics, Henan Polytechnic University, Jiaozuo City, Henan Province, 454010, China

Received 12 May 2012; Accepted 19 May 2012

Academic Editor: Yonghong Yao

Copyright © 2012 Tian-Yu 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

The authors introduce the concept of the s-geometrically convex functions. By the well-known Hölder inequality, they establish some integral inequalities of Hermite-Hadamard type related to the s-geometrically convex functions and apply these inequalities to special means.

1. Introduction

We firstly list several definitions and some known results.

Definition 1.1. A function is said to be convex if for and .

Definition 1.2 (see [1]). A function is said to be -convex if for some , where , and .
If   , the -convex function becomes a convex function on .

Theorem 1.3 ([2], Theorem 2.2). Let be a differentiable mapping on , , . (i) If is a convex function on , then (ii) If is a convex function on , for , then

Theorem 1.4 ([3], Theorems 2.3 and 2.4). Let be differentiable on , , . If   is convex on , for , then

Theorem 1.5 ([4], Theorem 1–4). Let be differentiable on , , . If is   -convex on , for   , then

Theorem 1.6 ([5], Theorem 4). Let be differentiable on , , , and . If is  -convex on for some , and , such that then

Theorem 1.7 ([6], Theorems 2.2–2.4). Let be differentiable on , , , and . (i)If   is -convex on for some , then (ii)If is a -convex function on for some , then where .(iii)If is  s-convex on   for some , then Now we introduce the definition of the -geometrically convex function.

Definition 1.8. A function is said to be a geometrically convex function if for and .

Definition 1.9. A function is said to be a -geometrically convex function if for some , where and .
If   , the -geometrically convex function becomes a geometrically convex function on .
In this paper, we will establish some integral inequalities of Hermite-Hadamard type related to the -geometrically convex functions and then apply these inequalities to special means.

2. A Lemma

In order to prove our results, we need the following lemma.

Lemma 2.1. Let be differentiable on , and , with . If , then

Proof. Integrating by part and changing variables of integration yields This completes the proof of Lemma 2.1.

3. Main Results

Theorem 3.1. Let   be differentiable on ,   , with , and . If is   -geometrically convex and monotonically decreasing on for and , then where

Proof. (1) Since is -geometrically convex and monotonically decreasing on , from Lemma 2.1 and Hölder inequality, we have If , then (i)If , by (3.7), we obtain that (ii)If , by (3.7), we obtain that (iii)If , by (3.7), we obtain that From (3.6) to (3.10),  (3.1) holds.
(2) Since is -geometrically convex and monotonically decreasing on , from Lemma 2.1 and Hölder inequality, we have (i)If , by (3.7), we have (ii)If , by (3.7), we have (iii)If , by (3.7), we have From (3.11) to (3.14), (3.2) holds. This completes the required proof.

Applying Theorem 3.1 to , respectively, results in the following corollary.

Corollary 3.2. Let be differentiable on , with , and . If is   -geometrically convex and monotonically decreasing on for   , then(i) when , one has (ii) when , one has where , , , are same with (3.3)–(3.5).

Theorem 3.3. Let   be differentiable on ,   ,  with , and . If     is   -geometrically convex and monotonically decreasing on ,  for  and   , then where and is the same as in (3.4).

Proof. (1) Since   is  s-geometrically convex and monotonically decreasing on , from Lemma 2.1 and Hölder inequality, we have (i) If , we have (ii) If , we have (iii) If , we have From (3.20) to (3.23), (3.17) holds.
(2) Since   is   -geometrically convex and monotonically decreasing on , from Lemma 2.1 and Hölder inequality, we have From (3.24) and (3.21) to (3.23), (3.18) holds. This completes the proof.

If taking in Theorem 3.3, we can derive the following corollary.

Corollary 3.4. Let be differentiable on ,    ,  with ,  and . If is geometrically convex and monotonically decreasing on   for , then where , , and are the same as in Theorem 3.3.

4. Application to Special Means

Let be the arithmetic, logarithmic, generalized logarithmic means for respectively.

Let , , and then the function is monotonically decreasing on . For , we have Hence, is   -geometrically convex on for .

Theorem 4.1. Let   , , and . Then In particular, if , one has

Proof. Let . Then and By Theorem 3.1, Theorem 4.1 is thus proved.

Theorem 4.2. Let , , and . Then one has

Proof. Let . Then and Using Theorem 3.3, Theorem 4.2 is thus proved.

Acknowledgments

The research was supported by Science Research Funding of Inner Mongolia University for Nationalities (Project no. NMD1103).

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