On $\mathscr{M}$-convex functions

Abstract

In this article, we introduce the notion of $\mathscr{M}$-convex functions, $\log$-$\mathscr{M}$-convex functions and the notion of quasi $\mathscr{M}$-convex functions. We derive some new analogues of Hermite-Hadamard like inequalities associated with $\mathscr{M}$-convex functions by using the concepts of ordinary, fractional and quantum calculus. The main results of this paper may be useful where bounds for natural phenomena described by integrals such as mechanical work are frequently required. These results are also helpful in the field of numerical analysis where error analysis is required.

Keywords

• convex
• $\mathscr{m}$-convex
• $\log$-$\mathscr{m}$-convex
• quasi $\mathscr{m}$-convex
• hermite-hadamard inequality
• fractional
• quantum