Green’s Function Approach to Hermite–Hadamard–Mercer Type Fractional Inequalities and Applications

The Hermite–Hadamard–Mercer (HHM) inequality, existing in two well-established forms, plays a fundamental role in mathematical analysis. This inequality is characterized by three distinct components—namely, the left, middle, and right terms. This study is concerned to obtain novel generalized and refined HHM fractional inequalities by employing for the first time the framework of Green functions. The interesting relationships among the three defining terms are investigated, and sharp bounds are derived for the differences between the left–middle and middle–right expressions. A key feature of this work is a curvature-driven refinement of the HHM middle term, which systematically tightens the classical bounds and visibly narrows the gap between them. This refinement not only enhances the theoretical sharpness of the inequality but also provides a novel approach to how curvature and interval length influence the bounds. In addition, a thermal profile comparison between the classical and refined middle terms is included which show the improved sharpness of the obtained refinements. Also, error estimates are deduced for the difference of various means of real numbers by selecting appropriate functions and parameters within the obtained results.