Advanced Analysis of Hermite-Hadamard Mercer Type Inequalities for Twice Differentiable Functions with Application

In recent years, fractional integral inequalities have emerged as powerful and widely applicable instruments in both pure and applied mathematics. Their versatility has motivated numerous researchers to propose broader and more general forms of classical inequalities, many of which rely on different types of fractional integral operators. In this work, we investigate Hermite-Jensen Mercer-type inequalities in the framework of s-convex functions by utilizing the Caputo-Fabrizio fractional integral operator, which is distinguished by its non-singular exponential kernel. By establishing a new fractional integral identity for twice-differentiable mappings, we derive a series of generalized inequalities that extend and unify several classical results. In addition, applications to the q-digamma function, and the modified Bessel function are provided. Numerical computations and graphical illustrations are included to visualize the behaviour of the established bounds under various parameters. Many established results are recovered as special cases by taking appropriate parameter values.

کلیدواژه ها:

Hermite-Hadamard Mercer Inequalities ، Jensen's Mercer ، Fractional integrals ، Hö lder's inequality