Some Inequalities of Hermite-Hadamard Type for Functions Whose 3rd Derivatives Are P-Convex

Abstract

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

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Xi, B. , Wang, S. and Qi, F. (2012) Some Inequalities of Hermite-Hadamard Type for Functions Whose 3rd Derivatives Are P-Convex. Applied Mathematics, 3, 1898-1902. doi: 10.4236/am.2012.312260.

1. Introduction

The following definition is well known in the literature.

Definition 1.1. A function is said to be convex if

(1.1)

holds for all.

In [1], the concept of the so-called -convex functions was introduced as follows.

Definition 1.2. ([1]) We say that a map

belongs to the class if it is nonnegative and satisfies

(1.2)

for all.

In [2], S. S. Dragomir proved the following theorems.

Theorem 1.1. ([2]) Let be a differentiable mapping on and. If

is convex on, then

(1.3)

Theorem 1.2. ([2]) Let be a differentiable mapping on and. If is convex on for, then

(1.4)

Theorem 1.3. ([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

(2.1)

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

(3.1)

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

(2.2)

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

(2.3)

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

(2.5)

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.

Conflicts of Interest

The authors declare no conflicts of interest.