Novelty Hermite-Hadamard-type inequalities for various convexity functions and their applications

The classical Hermite-Hadamard inequality provides a fundamental estimate for the integral mean of convex functions and has inspired numerous extensions in mathematical analysis. In this paper, we present new generalizations of the Hermite-Hadamard inequality within the framework of fractional calculus by employing the Caputo-Fabrizio fractional integral operator, known for its non-singular kernel and smooth memory effects. Specifically, we establish refined inequalities for convex functions that are three times differentiable and extend these results to the broader class of quasi-convex functions. Furthermore, we demonstrate several applications of the newly derived inequalities to special means, specific functions, the modified Bessel function, quadrature formulas and midpoint formulas. These findings contribute to the expanding body of literature on fractional inequalities and their applications in both theoretical and applied contexts.