On Fractional Milne-Mercer Bounds for $s$-Convex Functions with Applications

Abstract

Fractional integral inequalities play a fundamental role in modern analysisand have significant applications in the qualitative study of fractionaldifferential and integral equations. Motivated by the need for sharperestimates in fractional settings, this paper establishes several newMilne-Mercer type inequalities involving the Riemann-Liouville fractionalintegral operators. By employing the theory of $s$-convex functions, wederive a unified framework that extends and generalizes a variety ofclassical Mercer-type inequalities available in the literature. Furthermore,refined estimates are obtained for bounded functions and functionssatisfying Lipschitz-type conditions. The proposed results not only recovernumerous known inequalities as particular cases but also provide strongerbounds within the fractional calculus setting. To illustrate theapplicability and effectiveness of the theoretical findings, graphicalexamples are presented. In addition, applications to quadrature formulas, $q$-digamma functions, and modified Bessel functions are discussed,demonstrating the broad utility of the developed inequalities. The resultscontribute to the ongoing development of fractional integral inequalitiesand offer new tools for the analysis of special functions and relatedproblems in applied mathematics.