New Extension of Inequalities through Extended Version of Fractional Operators for s-Convexity with Applications

Fractional integral inequalities play a significant role in both pure and applied mathematics, contributing to the advancement and extension of various mathematical techniques. An accurate formulation of such inequalities is essential to establish the existence and uniqueness of fractional methods. Additionally, convexity theory serves as a fundamental component in the study of fractional integral inequalities due to its defining characteristics and properties. Moreover, there is a strong interconnection between convexity and symmetric theories, allowing results from one to be effectively applied to the other. This correlation has become particularly evident in recent decades, further enhancing their importance in mathematical research. This article investigates the Hermite-Hadamard inequalities and their refinements by implementation of generalized fractional operators through the $s$-convex functions, which are considered in both single and double differentiable forms. The study aims to extend and refine existing inequalities with fractional operator having extended Bessel-Maitland functions as a kernel, providing a more generalized framework. By incorporating these special functions, the results encompass and improve numerous classical inequalities found in the literature, offering deeper insights and broader applicability in mathematical analysis.