Strongly Reciprocally (p,h)-Convex Functions And Some Inequalities

Abstract

This thesis is a detailed study of convex functions, focusing on h-convex, p convex, and stronglyireciprocally (p,h)-convex functions of higheriorder cases.iThe mainiobjective ofithis research isito enhance both theiunderstanding and analysis of these generalizediconvex functions, with a focus on their connections to established mathematicaliinequalities, such as theiHermite-Hadamardiinequality, Fejériinequality, and fractionaliintegraliinequalities. The foundational concept of convexity, its basic properties, and applications are reexamined. The utility of convex functions in numerous areas ranging from theoretical research to practical applications of pure and appliedisciences including, economic models, mathematical analysis, and opti mizationiproblems is highlighted. The role of inequality theory is also pointed out for convex functions. The classical convexity is extended to stronglyireciprocally (p, h)-convex functions of higheriorder instances, enabling a deeper insight into how convex functions can be applied to mathematicaliinequalities, suchias theiHermite Hadamardiinequality, the Fejériinequality, andifractional integraliinequalities are explored. A novel mapping Mw F(x) for h-convex functions is explored along a series of useful results. These results are formulated through lemmas and propositions, which are then utilized to derive some generalized Fejér-typeiinequalities and an improved variant of Hermite-Hadamard inequalities. A second mapping, Hw F(x) for h-convex function is introduced, with the further contribution of some results that leads to the derivation of a significant theorem. These findings contribute new tools for mathematical analysis and significantly broaden the scope of known results to generalize convex functions and inequalities.