#### Abstract

We extend some estimates of the right-hand side of Hermite-Hadamard-type inequalities for functions whose second derivatives absolute values are -convex. Applications to some special means are considered.

#### 1. Introduction

Let be a convex function defined on the interval of real numbers and with . The following double inequality is known in the literature as the Hermite-Hadamard inequality. Both inequalities hold in the reversed direction if is concave. We note that Hermite-Hadamard inequality may be regarded as a refinement of the concept of convexity and it follows easily from Jensen's inequality. Note that some of the classical inequalities for means can be derived from (1.1) for appropriate particular selections of the function . Both inequalities hold in the reversed direction if is concave (see [1]).

It is well known that the Hermite-Hadamard inequality plays an important role in nonlinear analysis. Over the last decade, this classical inequality has been improved and generalized in a number of ways; there have been a large number of research papers written on this subject, (see, [2β13]) and the references therein. In [13] Dragomir and Agarwal established the following results connected with the right-hand side of (1.1) as well as applied them for some elementary inequalities for real numbers and numerical integration.

Theorem 1.1. *Assume that with and is a differentiable function on . If is convex on , then the following inequality holds:
*

Theorem 1.2. *Assume that with and is a differentiable function on . Assume with . If is convex on , then the following inequality holds:
*

In [1] Pearce and PeΔariΔ proved the following theorem.

Theorem 1.3. *Let be a differentiable function on with . If is convex on , for , then the following inequality holds:
*

Recall that the function is said to be quasiconvex if for every we have

The generalizations of the Theorems 1.1 and 1.2 are introduced by Ion in [14] for quasiconvex functions and are given in [6] to differentiable -convex functions. Then, Alomari et al. in [2] improved the results in [14] and Theorem 1.3, for twice differentiable quasiconvex functions.

On the other hand, Dragomir et al. in [11] defined the following class of functions.

*Definition 1.4. *Let be an interval. The function is said to be -convex (or belong to the class ) if it is nonnegative and, for all and , satisfies the inequality

Note that contain all nonnegative convex and quasiconvex functions. Since then numerous articles have appeared in the literature reflecting further applications in this category, see [3, 6, 12, 15, 16] and references therein. Ozdemir and Yildiz in [15] proved the following results.

Theorem 1.5. *Let be a twice differentiable function on and with . If is -convex, , then the following inequality holds:
*

Corollary 1.6. *If in Theorem 1.5 one chooses , one obtains
*

Theorem 1.7. *Let be a twice differentiable function on and with . If is -convex, and , then the following inequality holds:
*

Corollary 1.8. *If in Theorem 1.7 one chooses , one obtains
*

The main purpose of this paper is to establish the refinements of results in [15]. Applications for special means are considered.

#### 2. Main Results

In order to prove our main theorems, we need the following Lemma in [5] throughout this paper.

Lemma 2.1. *Suppose that is a twice differentiable function on , the interior of . Assume that , with and , is integrable on . Then, the following equality holds:
*

In the following theorem, we will propose some new upper bound for the right-hand side of (1.1) for -convex functions, which is better than the inequality had done in [15].

Theorem 2.2. *Let be a twice differentiable function on such that is a -convex function on . Suppose that with and . Then, the following inequality holds:
*

*Proof. *Since is a -convex function, by using Lemma 2.1 we get

An immediate consequence of Theorem 2.2 is as follows.

Corollary 2.3. *Let be as in Theorem 2.2, if in addition*

*
(i) , then one has*

*
(ii) , then one has*

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

Theorem 2.4. *Let be a differentiable function on . Assume that , such that is a -convex function on . Suppose that with and . Then, the following inequality holds:
**
where .*

*Proof. *By assumption, Lemma 2.1 and HΓΆlder's inequality, we have
where . We note that the Beta and Gamma functions are defined, respectively, as follows
and are used to evaluate the integral . Indeed, by setting , we get
and using property
of Beta function, we obtain
where and the proof is completed.

The following corollary is an immediate consequence of Theorem 2.4.

Corollary 2.5. *Let be as in Theorem 2.4, if in addition*

*
(i) , then one has*

*
(ii) , then one has*

Another similar result may be extended in the following theorem.

Theorem 2.6. *Let be a differentiable function on . Assume that such that is a -convex function on . Suppose that with and . Then, the following inequality holds:
*

* Proof. *Suppose that . From Lemma 2.1 and using well-known power mean inequality, we get
which completes the proof.

Corollary 2.7. *Let be as in Theorem 2.6, if in addition*(i)*, then (2.4) holds,*(ii)*, (2.5) holds. *

#### 3. Applications to Special Means

Now, we consider the applications of our theorems to the special means. We consider the means for arbitrary real numbers , (). We take the following

*
(1) Arithmetic mean:*

*
(2) Logarithmic mean:*

*
(3) Generalized log-mean:*

Now, using the results of Section 2, we give some applications for special means of real numbers.

Proposition 3.1. *Let , , and , . Then, one has
*

*Proof. *The assertion follows from Theorem 2.2 applied to the -convex function , .

Proposition 3.2. *Let , , and . Then, for all one has
**
where . *

*Proof. *The assertion follows from Theorem 2.4 applied to the -convex function , .

Proposition 3.3. *Let , , and , . Then, for all one has
*

*Proof. *The assertion follows from Theorem 2.6 applied to the -convex function , .