Abstract

We obtain some Hermite-Hadamard type inequalities for products of two -convex functions via Riemann-Liouville integrals. The analogous results for -convex functions are also established.

1. Introduction

If is 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.

Since then, some refinements of the Hermite-Hadamard inequality for convex functions have been extensively obtained by a number of authors (e.g., [1–7]).

In [8], Toader defined the concept of -convexity as follows.

Definition 1 (see [8]). The function is said to be -convex, where , if for every and one has

In [3], Dragomir and Toader proved the following inequality of Hermite-Hadamard type for -convex functions.

Theorem 2 (see [3]). Let be a -convex function with ; if and , then one has the following inequality:

The notion of -convexity has been further generalized in [9] as it is stated in the following definition.

Definition 3 (see [9]). The function is said to be -convex, where , if for every and one has

In [10], Pachpatte established two new Hermite-Hadamard type inequalities for products of convex functions as follows.

Theorem 4 (see [10]). Let and be real-valued, nonnegative, and convex functions on . Then where and .

Some Hermite-Hadamard type inequalities for products of two -convex and -convex functions are established in [11].

Theorem 5 (see [11]). Let be functions such that , where . If is -convex and is -convex on for some fixed , then where

Theorem 6 (see [11]). Let be functions such that , where . If is convex and is -convex on for some fixed , then where

Some new integral inequalities involving two nonnegative and integrable functions that are related to the Hermite-Hadamard type are also proposed by many authors. In [12], Pachpatte established some Hermite-Hadamard type inequalities involving two log-convex functions. An analogous result for -convex functions is obtained by Kirmaci et al. in [13]. In [14], Sarikaya et al. presented some integral inequalities for two -convex functions.

It is remarkable that Sarikaya et al. [15] proved the following interesting inequalities of Hermite-Hadamard type involving Riemann-Liouville fractional integrals.

Theorem 7 (see [15]). Let be a positive function with and . If is a convex function on , then the following inequalities for fractional integrals hold: with .

We remark that the symbols and denote the left-sided and right-sided Riemann-Liouville fractional integrals of the order with which are defined by respectively. Here, is the Gamma function defined by .

In this paper, we obtain some new Hermite-Hadamard type inequalities for products of two -convex functions via Riemann-Liouville integrals. The analogous results for -convex functions are also given.

2. Inequalities for Products of Two Functions for Riemann-Liouville Fractional Integrals

Theorem 8. Let , , be functions such that . If is -convex and is -convex on with , then one has

Proof. Since is -convex and is -convex on , then for we get From (14), we get Multiplying both sides of the above inequality by and then integrating the resulting inequality with respect to over , we obtain Analogously, we obtain Multiplying both sides of above inequality by and then integrating the resulting inequality with respect to over , we obtain which completes the proof.

Corollary 9. With assumptions in Theorem 8, if , one gets which is just the result in Theorem 5.

Corollary 10. With assumptions in Theorem 8, one gets

Corollary 11. With assumptions in Theorem 8, if one chooses as and for all , one has

Theorem 12. Let ,  , be functions such that . If is -convex and is -convex on with , respectively, then one has

Proof. Since is -convex and is -convex on , then for we get From (23), we get Multiplying both sides of above inequality by and then integrating the resulting inequality with respect to over , we obtain Similarly, we have We get the desired result.

Corollary 13. With assumptions in Theorem 12, if , then which is just the result in Theorem 6.

Corollary 14. With assumptions in Theorem 12, one gets

3. Conclusion

In this paper, we obtain some new Hermite-Hadamard type inequalities for products of two -convex functions via Riemann-Liouville integrals. The analogous results for -convex functions are also established. An interesting topic is whether we can use the methods in this paper to establish the Hermite-Hadamard inequalities for products of two convex functions on the coordinates via Riemann-Liouville integrals.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work is supported by Youth Project of Chongqing Three Gorges University of China (no. 13QN11).