Generalized Error Bounds for Mercer-Type Inequalities in Fractional Integrals with Applications

Fractional integral inequalities have emerged as powerful and versatile tools in advancing both pure and applied mathematics in recent years. Numerous researchers have recently introduced various generalized inequalities involving fractional integral operators. In this paper, we develop new Hermite-Hadamard-Mercer type inequalities that extend the classical Hermite-Hadamard framework to the setting of the Riemann-Liouville fractional integral. Additionally, we establish Hermite-Hadamard-Mercer type inequalities for functions whose absolute second derivatives are convex. Moreover, we present a new midpoint-type inequality based on the well-known power-mean inequality. Several applications are also provided, including new results related to special means, the midpoint formula, quadrature formulas, the $q$-digamma function, and modified Bessel functions.