Modified Hermite - Hadamard Inequalities: Analytical Extensions and Computational Applications

Abstract

The Hermite–Hadamard inequality is a cornerstone of convex analysis, providing a two-sided estimate that links the midpoint value, the integral average, and the endpoint average of a convex function. Over the past century, numerous modifications and refinements have been developed to extend its applicability to broader function classes, fractional integral operators, weighted settings, and numerical quadrature. This paper presents a unified analytical review of the most significant modified Hermite–Hadamard inequalities and evaluates their computational performance. The study covers generalizations based on ss-convex, h-convex, harmonically convex, quasi-convex, m-convex, and (α,m)-convex functions; fractional versions using Riemann–Liouville integrals; derivative-based error bounds; weighted and Fejér-type formulations; and interval subdivision techniques. A systematic computational benchmarking methodology is employed using polynomial, exponential, reciprocal, and radical test functions, with error metrics including bound width, relative position, and convergence under subdivision. The results show that no single modification is universally superior; rather, the choice depends on the function class, available derivative information, required accuracy, and computational cost. Practical selection guidelines are provided, and open problems are identified, including the need for unified error estimates, variable-order extensions, and multidimensional generalizations. This paper aims to serve as a comprehensive resource for researchers and practitioners interested in the theory and applications of Hermite–Hadamard-type inequalities.