Abstract

Some new Hermite-Hadamard type inequalities are obtained for functions whose second derivatives absolute values are -convex.

1. Introduction

Let be a convex function on the interval ; then for any with we have the following double inequality: This remarkable result is well known in the literature as the Hermite-Hadamard inequality. Note that some of the classical inequalities for means can be derived from (1) for appropriate particular selections of the mapping . Both inequalities hold in the reversed direction if is concave. Some refinements of the Hermite-Hadamard inequality on convex functions have been extensively investigated by a number of authors (e.g., [1–9]).

Definition 1 (see [10]). A function is said to be -convex on , if the inequality holds for all with and for some fixed .

It can be easily seen that for , -convexity reduces to ordinary convexity of functions defined on .

In [11], Dragomir and Fitzpatrick proved a variant of Hermite-Hadamard inequality which holds for the -convex functions.

Theorem 2 (see [11]). Suppose that is an s-convex function in the second sense, where and let , . If , then the following inequality holds:

The constant is the best possible in the second inequality in (3).

Definition 3 (see [4]). We say that is -convex function or that belongs to the class , if is nonnegative and for all and one has

Remark 4. Applying Definition 1 for , we get Definition 3.
Along this paper we consider a real interval , and we denote that is the interior of .
In [12], Barani et al. introduced the following theorems for twice differentiable -convex functions.

Theorem 5 (see [12]). Let be a twice differentiable function on such that is a -convex function on . Suppose that with and . Then, the following inequality holds:

Theorem 6 (see [12]). Let be a twice differentiable function on . Assume that , such that is a -convex function on . Suppose that with and . Then, the following inequality holds:

Theorem 7 (see [12]). Let be a twice differentiable function on . Assume that such that is a -convex function on . Suppose that with and . Then, the following inequality holds:

For recent results and generalizations concerning Hermite-Hadamard's inequality for twice differentiable functions see [10, 12–14] and the references given therein.

In this paper, we establish some new inequalities of Hadamard's type for the class of -convex functions in the second sense.

2. The New Hermite-Hadamard Type Inequalities

Lemma 8 (see [13]). Let be a twice differentiable mapping on , where with . If , then the following equality holds:

Theorem 9. Let be a differentiable mapping on , such that , where with . If is -convex on for some fixed ; then the following inequality holds:

Proof. From Lemma 8, we have Because is -convex, we have which completes the proof.

The corresponding version for powers of the absolute value of the second derivative is incorporated in the following theorems.

Theorem 10. Let be a differentiable mapping on , such that , where with . If is -convex on for some fixed and , then the following inequality holds: where .

Proof. From Lemma 8 and using the Hölder inequality, we have We note that the Beta and Gamma functions are defined, respectively, as follows: Then and we get the desired result.

Theorem 11. Let be a differentiable mapping on , such that , where with . If is -convex on for some fixed and , then the following inequality holds:

Proof. From Lemma 8 and using the well-known power-mean inequality, we have This proves the theorem.

Remark 12. Applying Theorem 9 for , we get the result of Theorem 5.

Remark 13. Applying Theorem 10 for , we get the result of Theorem 6.

Remark 14. Applying Theorem 11 for , we get the result in Theorem 7.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work is supported by Youth Project of Chongqing Three Gorges University of China (No. 13QN11).