Journal of Applied Mathematics

Volume 2014 (2014), Article ID 386806, 6 pages

http://dx.doi.org/10.1155/2014/386806

Research Article

Fejér and Hermite-Hadamard Type Inequalities for Harmonically Convex Functions

Feixiang Chen¹ and Shanhe Wu²

¹School of Mathematics and Statistics, Chongqing Three Gorges University, Wanzhou, Chongqing 404000, China

²Department of Mathematics and Computer Science, Longyan University, Longyan, Fujian 364012, China

Received 21 June 2014; Accepted 23 July 2014; Published 6 August 2014

Academic Editor: Yu-Ming Chu

Copyright © 2014 Feixiang Chen and Shanhe 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.

Abstract.

We establish a Fejér type inequality for harmonically convex functions. Our results are the generalizations of some known results. Moreover, some properties of the mappings in connection with Hermite-Hadamard and Fejér type inequalities for harmonically convex functions are also considered.

1. Introduction

Let 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 is a convex function, then the following inequality holds:

where is positive, integrable, and symmetric with respect to .

Some generalizations, refinements, variations, and improvements of inequalities (1) and (2) were investigated by Wu [2], Chen and Liu [3], Sarikaya and Ogunmez [4], and Xiao et al. [5], respectively.

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 , and for all , 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 , and for all , one has

where

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]). If is convex on , is positive, integrable, and symmetric about . Then and are convex, increasing on , and for all , one has

where

In [9, 10], İşcan and Wu gave the definition of harmonic convexity as follows.

Definition 5. Let be a real interval. A function is said to be harmonically convex if

for all and . 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.

Theorem 6 (see [9]). Let be a harmonically convex function and with . If , then one has

In [10], İşcan and Wu established the following Hermite-Hadamard inequalities for harmonically convex functions via the Riemann-Liouville fractional integral.

Theorem 7 (see [10]). Let be a function such that , where with . If is a harmonically convex function on , then the following inequalities for fractional integrals hold:

where and .

The Riemann-Liouville fractional integrals and of order with are defined by

where is the Gamma function defined by .

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.

2. Fejér Type Inequality for Harmonically Convex Functions

The following Fejér inequality for harmonically convex functions holds true.

Theorem 8. Let be a harmonically convex function and with . If , then one has

where is nonnegative and integrable and satisfies

Proof. Since is a harmonically convex function on , we have, for all ,

Choosing and , we have

Since is nonnegative and satisfies the condition of (15), we obtain

Integrating both sides of the above inequalities with respect to over , we obtain

The proof of Theorem 8 is completed.

Remark 9. Putting in Theorem 8, we obtain inequality (11).

Remark 10. Choosing

in Theorem 8, it is easy to observe that .

Since

where , which implies that inequality (14) can be transformed to inequality (12) under an appropriate selection of .

Remark 11. In Theorem 8, taking , where , is nonnegative, integrable, and symmetric with respect to . Then inequality (14) becomes

3. Some Mappings in connection with Hermite-Hadamard and Fejér Inequalities for Harmonically Convex Functions

Lemma 12. Let be a harmonically convex function and with , and let

. Then is convex, increasing on , and for all ,

Proof. Firstly, for , we have

and hence is convex on .

Next, if , it follows from the harmonic convexity of that

It is easy to observe that

Thus inequality (24) holds.

Finally, for , since is convex, it follows from (24) that

and hence, , which means that is increasing on . This completes the proof of Lemma 12.

Theorem 13. Let be a harmonically convex function and with . If and is defined by

then is convex and increasing on , and

Proof. It follows from Lemma 12 that

is convex and increasing on . Hence is convex and increasing on . Further, inequality (30) can be deduced from (24). Theorem 13 is proved.

Theorem 14. Let be a harmonically convex function and with . If and is defined by

then is convex and increasing on , and

Proof. We note that if is convex and is linear, then the composition is convex. It follows from Lemma 12 that

and are increasing on and , respectively. Hence,

is convex and increasing on . We infer that is convex and increasing on . Furthermore, inequality (33) follows directly from (24). The proof of Theorem 14 is completed.

Theorem 15. Let be a harmonically convex function and with . If and is defined by

where is nonnegative and integrable and satisfies the condition of (15), then is convex and increasing on , and

Proof. From Lemma 12 we obtain that

is convex and increasing on . Since is nonnegative and satisfies , it follows that is convex and increasing on , while inequality (37) can be deduced from (24). Theorem 15 is proved.

Theorem 16. Let be a harmonically convex function and with . If and is defined by

where is nonnegative and integrable and satisfies the condition of (15), then is convex and increasing on , and

Proof. By using the same method as in the proof of Theorem 14, we obtain from Lemma 12 that

is convex and increasing on . Since is nonnegative and satisfies , we deduce that is convex and increasing on . Inequality (40) follows from (24) and the identity

This completes the proof of Theorem 16.

Remark 17. If we put

in inequalities (37) and (40), respectively, we obtain the refined versions of inequality (12).

Conflict of Interests

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

Acknowledgments

The present investigation was supported, in part, by the Youth Project of Chongqing Three Gorges University of China (no. 13QN11) and, in part, by the Foundation of Scientific Research Project of Fujian Province Education Department of China (no. JK2012049).

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