%0 Journal Article
%T On Hermite-Hadamard Type Inequalities for Functions Whose Second Derivatives Absolute Values Are -Convex
%A Feixiang Chen
%A Xuefei Liu
%J ISRN Applied Mathematics
%D 2014
%R 10.1155/2014/829158
%X 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¨C9]). 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¨C14] 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
%U http://www.hindawi.com/journals/isrn.applied.mathematics/2014/829158/