Reverses of the Jensen-Type Inequalities for Signed Measures

In this paper we derive refinements of the Jensen type inequalities in the case of real Stieltjes measure dλ, 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.