The well-known Hermite-Hadamard inequality for convex functions is extensively studied for different kinds of integrals and derivatives. This paper investigates some of its variants for $q-h$-integrals using properties of convex functions. Inequalities for $q$-integrals that have been published in recent years can be extracted from the main results of this paper.

This paper investigates some well-known inequalities for $q$- $h$-integrals. These include Hölder, Ostrowski, Grüss and Opial type inequalities. Refinement of the Hadamard inequality for $q$- $h$-integrals is also established by applying the definition of strongly convex functions. From main theorems, $q$-Hölder, $q$-Ostrowski and $q$-Grüss inequalities can be obtained in particular cases.

For generalizations of concepts of different fields fractional derivative operators as well as fractional integral operators are useful notions. Our aim in this paper is to discuss boundedness of the integral operators which contain Mittag-Leffler function in their kernels. The results are obtained for strongly $(\alpha ,h-m)$-convex functions which hold for different kinds of convex functions at the same time. They also give improvements/refinements of many already published results.