AMApplied Mathematics2152-7385Scientific Research Publishing10.4236/am.2012.312260AM-25448ArticlesPhysics&Mathematics Some Inequalities of Hermite-Hadamard Type for Functions Whose 3rd Derivatives Are <i>P</i>-Convex oyanXi1ShuhongWang1FengQi2*College of Mathematics, Inner Mongolia University for Nationalities, Tongliao, ChinaSchool of Mathematics and Informatics, Henan Polytechnic University, Jiaozuo, China* E-mail:qifeng618@gmail.com(FQ);12122012031218981902September 30, 2012October 30, 2012 November 7, 2012© Copyright 2014 by authors and Scientific Research Publishing Inc. 2014This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/

In the paper, the authors establish some new Hermite-Hadamard type inequalities for functions whose 3rd derivatives are P-convex.

Integral Inequality; Hermite-Hadamard’s Integral Inequality; <i>P</i>-Convex Function; Derivative
1. Introduction

The following definition is well known in the literature.

Definition 1.1. A function is said to be convex if

holds for all .

In , the concept of the so-called -convex functions was introduced as follows.

Definition 1.2. () We say that a map belongs to the class if it is nonnegative and satisfies

for all .

In , S. S. Dragomir proved the following theorems.

Theorem 1.1. () Let be a differentiable mapping on and . If is convex on , then

Theorem 1.2. () Let be a differentiable mapping on and . If is convex on for , then

Theorem 1.3. (, Theorems 2) Let be an absolutely continuous function on such that for . If is quasi-convex on , then For more information and recent developments on this topic, please refer to [4-14] and closely related references therein.

The concepts of various convex functions have indeed found important places in contemporary mathematics as can be seen in a large number of research articles and books devoted to the field these days.

In this paper, we will establish some new HermiteHadamard type inequalities for functions whose rd derivatives are P-convex.

2. A Lemma

In this section, we establish an integral identity.

Lemma 2.1. Let be a three times differentiable mapping on and . If , then

Proof. Integrating by part and changing variable of definite integral yield and The proof of Lemma 2.1 is complete.

3. Hermite-Hadamard’s Type Inequalities for P-Convex Functions

Theorem 3.1. Let be differentiable on , , and If is -convex on for , then

Proof. Since is a -convex function on , by Lemma 2.1 and Hölder’s inequality, we obtain The proof of Theorem 3.1 is complete.

Corollary 3.1.1. Under the conditions of Theorem 3.1, if , we have Theorem 3.2. Let be differentiable on , , and . If is -convex on for , then

Proof. From Lemma 2.1, Hölder’s inequality, and the -convexity of on , we drive Theorem 3.2 is proved.

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

Proof. From Lemma 2.1, Hölder’s inequality, and the -convexity of on , we have Theorem 3.3 is thus proved.

Theorem 3.4. Let be differentiable on , , and If for is -convex on and , then

Proof. Using Lemma 2.1, Hölder’s inequality, and the -convexity of on yields The proof of Theorem 3.4 is complete.

Corollary 3.3.1. Under the conditions of Theorem 3.4(1) if , then (2) if , then (3) if , then Finally we would like to note that these Hermite-Hadamard type inequalities obtained in this paper can be applied to the fields of integral inequalities, approximation theory, special means theory, optimization theory, information theory, and numerical analysis, as done before by a number of mathematicians.

4. Acknowledgements

The first two authors were partially supported by the Science Research Funding of Inner Mongolia University for Nationalities under Grant No. NMD1103.