Normalized Jensen Functional, Superquadracity and Related Inequalities

Abstract

In this paper we generalize the inequality

$$ MJ_n (f,x,q) \geqslant J_n (f,x,p) \geqslant mJ_n (f,x,q) $$

where

$$ J_n (f,x,p) = \sum\limits_{i = 1}^n {p_i f(x_i ) - f\left( {\sum\limits_{i = 1}^n {p_i x_i } } \right)} , $$

obtained by S.S. Dragomir for convex functions. We show that for the class of functions that we call superquadratic, strictly positive lower bounds of J _n (f, x, p)—mJ _n (f, x, q) and strictly negative upper bounds of J _n (f, x, p)∔MJ _n (f, x, q) exist when the functions are also nonnegative. We also provide cases where we can improve the bounds m and M for convex functions and superquadratic functions. Finally, an inequality related to the Čebyšev functional and superquadracity is also given.