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