Abstract and Applied Analysis

Volume 2012 (2012), Article ID 560586, 14 pages

http://dx.doi.org/10.1155/2012/560586

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

^{1}College of Mathematics, Inner Mongolia University for Nationalities, Inner Mongolia Autonomous Region, Tongliao City 028043, China^{2}School 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. *Letbe 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).

#### References

- H. Hudzik and L. Maligranda, “Some remarks on $s$-convex functions,”
*Aequationes Mathematicae*, vol. 48, no. 1, pp. 100–111, 1994. View at Publisher · View at Google Scholar - S. S. Dragomir and R. P. Agarwal, “Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula,”
*Applied Mathematics Letters*, vol. 11, no. 5, pp. 91–95, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - U. S. Kirmaci, “Inequalities for differentiable mappings and applications to special means of real numbers and to midpoint formula,”
*Applied Mathematics and Computation*, vol. 147, no. 1, pp. 137–146, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - U. S. Kirmaci, M. Klaričić Bakula, M. E. Özdemir, and J. Pečarić, “Hadamard-type inequalities for $s$-convex functions,”
*Applied Mathematics and Computation*, vol. 193, no. 1, pp. 26–35, 2007. View at Publisher · View at Google Scholar - S. Hussain, M. I. Bhatti, and M. Iqbal, “Hadamard-type inequalities for $s$-convex functions. I,”
*Punjab University. Journal of Mathematics*, vol. 41, pp. 51–60, 2009. View at Google Scholar - M. W. Alomari, M. Darus, and U. S. Kirmaci, “Some inequalities of Hermite-Hadamard type for $s$-convex functions,”
*Acta Mathematica Scientia B*, vol. 31, no. 4, pp. 1643–1652, 2011. View at Publisher · View at Google Scholar