Fejér and Hermite-Hadamard Type Inequalities for Harmonically Convex Functions

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).