New Fractional Refinements of Harmonic Hermite-Hadamard Mercer type Inequalities Via Support Line

Abstract

In this research, we first provide new and refined fractional integral Mercer inequalities for harmonic convex functions by deploying the idea of line of support. Thus, these refinements allow us to develop new extensions for integral inequalities pertaining harmonic convex functions. We also provide some new fractional auxiliary equalities in Mercer sense. By employing Mercer's harmonic convexity on them, we exhibits new fractional  Mercer variants of  trapezoid and  midpoint type inequalities.  We prove new Hermite-Hadamard (H-H) type inequalities with special functions involving fractional integral operators. For the development of these new integral inequalities, we use Power-mean, H\"{o}lder's and improved H\"{o}lder integral inequalities. We unveiled complicated integrals into simple forms by involving hypergeometric functions. Visual illustrations demonstrate the accuracy and supremacy of the offered technique. As an application, new bounds regarding hypergeometric functions as well as special means of $\mathbb{R}$ (real numbers) and quadrature rule are exemplified to show the applicability and validity of the offered technique.