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 (φ \,-\,h) ( ϕ - h ) integrals and (φ \,-\,h) ( ϕ - h ) derivatives, respectively. Then we investigate some implicit integral inequalities for (φ \,-\,h) ( ϕ - 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 ħ -convex functions defined on the non-negative part of the real line.