Properties and Riemann-Liouville fractional Hermite-Hadamard inequalities for the generalized ( α , m ) $(\alpha,m)$ -preinvex functions

Abstract

The authors first introduce the concepts of generalized $(\alpha,m)$ -preinvex function, generalized quasi m-preinvex function and explicitly $(\alpha, m)$ -preinvex function, and then provide some interesting properties for the newly introduced functions. The more important point is that we give a necessary and sufficient condition respecting the relationship between the generalized $(\alpha, m)$ -preinvex function and the generalized quasi m-preinvex function. Second, a new Riemann-Liouville fractional integral identity involving twice differentiable function on m-invex is found. By using this identity, we establish the right-sided new Hermite-Hadamard-type inequalities via Riemann-Liouville fractional integrals for generalized $(\alpha,m)$ -preinvex mappings. These inequalities can be viewed as generalization of several previously known results.