Jensen-type inequalities for convex and $m$-convex functions via fractional calculus

TL;DR

This paper develops new Jensen-type inequalities for m-convex functions using fractional calculus and applies them to generalized integral operators, extending known results for convex functions.

Contribution

It introduces novel Jensen-type inequalities for m-convex functions via fractional calculus, broadening the scope of classical inequalities.

Findings

New inequalities for m-convex functions derived

Applications to generalized Riemann-Liouville integral operators

Special case for m=1 yields new convex function inequalities

Abstract

Inequalities play an important role in pure and applied mathematics. In particular, Jensen's inequality, one of the most famous inequalities, plays a main role in the study of the existence and uniqueness of initial and boundary value problems for differential equations. In this work we prove some new Jensen-type inequalities for $m$-convex functions, and we apply them to generalized Riemann-Liouville-type integral operators. It is remarkable that, if we consider $m=1$, we obtain new inequalities for convex functions.