Generalized convex functions with some integral properties

Abstract

Convex function, its generalizations and inequalities involving convex functions have many applications in various fields of science. Aim of this dissertation is to study the several generalized convex functions in one dimension as well as in two dimensions and the inequalities via classical and several generalized fractional integrals with some applications. In this thesis, we established several inequalities via conformable and new conformable fractional integrals for p-convex functions. Inequalities involving Katugampola fractional integrals are also proved for s-convex functions in second sense and m-convex functions and some application to special mean are also given. Some mean value theorems are given for p-convex functions and s-convex functions in first sense. Classical integral inequalities are obtained for some new class of convex functions called exponentially p-convex functions and exponentially s-convex functions in second sense with some applications. Riemann−Liouville fractional integrals inequalities are proved for (s,p)-convex functions and some classical integral inequalities are obtained for co-ordinated (s,p)-convex functions. Furthermore, we also obtained Hermite-Hadamard and Fej´er type inequalities for co-ordinated harmonically convex functions in Riemann−Liouville fractional integrals and Katugampola fractional integrals.