Fejér and Hermite-Hadamard Type Inequalities for Harmonically Convex Functions

Feixiang Chen and Shanhe Wu 1 School of Mathematics and Statistics, Chongqing Three Gorges University, Wanzhou, Chongqing 404000, China 2Department of Mathematics and Computer Science, Longyan University, Longyan, Fujian 364012, China Correspondence should be addressed to Shanhe Wu; shanhewu@gmail.com Received 21 June 2014; Accepted 23 July 2014; Published 6 August 2014 Academic Editor: Yu-Ming Chu Copyright © 2014 F. Chen and S. Wu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Introduction
Let : ⊆ R → R be a convex function and , ∈ with < ; then Inequality (1) is known in the literature as the Hermite-Hadamard inequality. Fejér [1] established the following weighted generalization of inequality (1). Theorem 1. If : [ , ] → R is a convex function, then the following inequality holds: where : [ , ] → R is positive, integrable, and symmetric with respect to = ( + )/2.
In [6], Dragomir proposed an interesting Hermite-Hadamard type inequality which refines the left hand side of inequality of (1) as follows.
Theorem 2 (see [6]). Let be a convex function defined on [ , ]. Then is convex, increasing on [0, 1], and for all ∈ [0, 1], one has where An analogous result for convex functions which refines the right hand side of inequality (1) was obtained by Yang and Hong in [7] as follows.
Theorem 3 (see [7]). Let be a convex function defined on [ , ]. Then is convex, increasing on [0, 1], and for all ∈ [0, 1], one has Yang and Tseng in [8] established the following Fejér type inequalities, which is the generalization of inequalities (3) and (5) as well as the refinement of the Fejér inequality (2). Theorem 4 (see [8] where In [9,10],İşcan and Wu gave the definition of harmonic convexity as follows. for all , ∈ and ∈ [0, 1]. If the inequality in (10) is reversed, then is said to be harmonically concave.
The following Hermite-Hadamard inequality for harmonically convex functions holds true.
In this paper, we establish a Fejér type inequality for harmonically convex functions; our main result includes, as special cases, the inequalities given by Theorems 6 and 7. Moreover, we investigate some properties of the mappings in connection to Hermite-Hadamard and Fejér type inequalities for harmonically convex functions.

Fejér Type Inequality for Harmonically Convex Functions
The following Fejér inequality for harmonically convex functions holds true.

Some Mappings in connection with Hermite-Hadamard and Fejér Inequalities for Harmonically Convex Functions
Lemma 12. Let : ⊆ R\{0} → R be a harmonically convex function and , ∈ with < , and let and hence ℎ is convex on [0, − ].
Theorem 16. Let : ⊆ R \ {0} → R be a harmonically convex function and , ∈ with < . If ∈ ( , ) and is defined by where : [ , ] → R is nonnegative and integrable and satisfies the condition of (15), then is convex and increasing on [0, 1], and This completes the proof of Theorem 16.