Hermite-Hadamard-Fejer type Integral Inequalities via Generalized Fractional Integral Operators

Abstract

Fractional calculus is a generalization of ordinary calculus. The origin of fractional calculus is as old as the calculus. The first formulation of an operator deals with the fractional integration was due to Abel (1823), Litnikov (1868) and Sonin (1869). After that a lot of such operators came into being by several mathematicians for example Prabhakar (1971), Shukla and Prajapati (2007) and Srivastava and Tomovski (2009). In (2012) Salim and Faraj introduced a very general fractional integral operator via generalized Mittag-Leffler function. In receent decades many authors have produced interesting results about the Hermite-Hadamard inequality. The Hermite-Hadamard inequality is equivalently de fined by the convex functions. Therefore we are motivated to study and produce new fractional integral inequalities of the Hermite-Hadamard and Hermite-Hadamard Fejer type. We are interested to give such inequalities for convex functions and related functions that includes m-convex functions, harmonically convex functions and p-convex functions. It is remarkable to mention that several published results are special cases of presented in this thesis. Moreover given results contained in them selves many new fractional inequalities in particular for fractional operator defined by Prabhkar, Shukla and Prajapati, Srivastava and Tomovski and also for classical Riemann-Liouville fractional integral operator.