Generalization of Jensen's and Jensen-Steffensen's inequalities by generalized majorization theorem

Abstract

In this paper, we use generalized majorization theorem and give the generalizations of Jensen’s and Jensen-Steffensen’s inequalities. We present the generalization of converse of Jensen’s inequality. We give bounds for the identities related to the generalization of Jensen’s inequality by using ˇ Cebyˇsev functionals. We also give Gr¨uss and Ostrowski types inequalities for these functionals. We present mean value theorems and n-exponential convexity which leads to exponential convexity and log - convexity for these functionals. We give some families of functions which enable us to construct a large families of functions that are exponentially convex and also give classes of means.