Reverses of the Jensen-Type Inequalities for Signed Measures

Abstract

In this paper we derive refinements of the Jensen type inequalities in the case of real Stieltjes measure $d\lambda$, not necessarily positive, which are generalizations of Jensen's inequality and its reverses for positive measures. Furthermore, we investigate the exponential and logarithmic convexity of the difference between the left-hand and the right-hand side of these inequalities and give several examples of the families of functions for which the obtained results can be applied. The outcome is a new class of Cauchy-type means.