On Hermite-Hadamard Type Inequalities for Riemann-Liouville Fractional Integrals via Two Kinds of Convexity

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