Analysis on Milne-type inequalities through generalized convexity with applications

Abstract

The development of precise fractional integral inequalities is crucial for advancing mathematical methods, with convexity theory offering enhanced insights into their scope and applications. In this study, we establish new identity involving the Caputo-Fabrizio fractional integral operator. Based on this identity, we derive several Milne-type integral inequalities for (s,m)-convex functions. Our results generalize numerous classical inequalities and extend the analysis to include inequalities for bounded and Lipschitz functions. Additionally, we explore applications of these inequalities to special means and $q$-digamma functions. We also present graphical representations to demonstrate the behavior and significance of the derived inequalities.