Generalization of quantum calculus and corresponding Hermite-Hadamard inequalities

The aim of this paper is first to introduce generalizations of quantum integrals and derivatives which are called (phi-h) integrals and (phi-h) derivatives, respectively. Then we investigate some implicit integral inequalities for (phi-h) integrals. Different classes of convex functions are used to prove these inequalities for symmetric functions. Under certain assumptions, Hermite-Hadamard-type inequalities for q-integrals are deduced. The results presented herein are applicable to convex, m-convex, and & hstrok;-convex functions defined on the non-negative part of the real line.