On Jensen’s Inequality and Related Results

Abstract

A real valued function Ψ on S (S is a set that is convex) is called convex if epigraph of Ψ is a convex set. Alternatively, for any two points x,y ∈ S the line segment µx + (1 − µ)y (µ ∈ [0,1]) joining these two points on the graph of the function Ψ lies above or on the graph. The well known king of inequalities, that is, the Jensen’s inequality is the generalization of the above result. Integral, functional, probabilistic and many other indispensable forms of this fundamental result can be found in literature. In the dissertation, we introduce some advancements in Jensen’s type inequalities. The m-exponential convexity and the log-convexity have been investigated. Positive functionals are used to investigate them. The positive functionals are defined in the form of the difference of two sides of the refined and some known inequalities. We also give an idea of logarithmic and m-exponentially concave functions and apply this concept on the linear functional associated with the Jensen’s inequality for generalized Choquet integral. The consequences of obtained results provide us interesting applications in the probability. We also deduce an interesting result of information theory. We discuss Cauchy and Lagrange type mean value theorems which lead us to Stolarsky type means. We also draw some interesting results associated with the refined Jessen’s inequality and Lupa¸s-Beesack-Peˇcari´c (LBP) type inequality for m(M)-ψ-convex functions.