Generalization of quantum calculus and corresponding Hermite–Hadamard inequalities

Abstract The aim of this paper is first to introduce generalizations of quantum integrals and derivatives which are called (\, -\, h) (ϕ - h) integrals and (\, -\, h) (ϕ - h) derivatives, respectively. Then we investigate some implicit integral inequalities for (\, -\, 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.