New Bounds on Hermite–Hadamard–Mercer-Type Inequalities: Applications and Computational Analysis

In mathematical analysis, the theory of inequalities plays a fundamental role due to its wide-ranging applications in various fields of the physical sciences. In this paper, we develop new Hermite–Hadamard–Mercer-type inequalities involving a broad class of fractional integral operators, including both classical and Caputo–Fabrizio fractional integrals. To this end, we first establish a general fractional and classical Hermite–Hadamard–Mercer identity, which serves as the foundation for our main results. Furthermore, by utilizing this identity and the properties of s-convex functions, we derive several new inequalities via Hölder’s inequality, the power-mean inequality, and Young’s inequality. Some known results are recovered as special cases, and various particular instances are discussed in detail. Moreover, graphical simulations are provided to support the theoretical findings. Applications to the midpoint formula, the modified Bessel function, and the q-digamma function are also presented.