On <i>q</i>-Hermite–Hadamard Type Inequalities via <i>s</i>-Convexity and (<i>α</i>,<i>m</i>)-Convexity

Abstract

The purpose of the paper is to present new q-parametrized Hermite–Hadamard-like type integral inequalities for functions whose third quantum derivatives in absolute values are s-convex and (α,m)-convex, respectively. Two new q-integral identities are presented for three time q-differentiable functions. These lemmas are used like basic elements in our proofs, along with several important tools like q-power mean inequality, and q-Holder’s inequality. In a special case, a non-trivial example is considered for a specific parameter and this case illustrates the investigated results. We make links between these findings and several previous discoveries from the literature.

Keywords

• convex functions
• Hermite–Hadamard type inequalities
• quantum calculus