The main aim of this paper is to give an improvement of the recent result on the sharpness of the Jensen inequality. The results given here are obtained using different Green functions and considering the case of the real Stieltjes measure, not necessarily positive. Finally, some applications involving various types of f-divergences and Zipf–Mandelbrot law are presented.

Motivated by some results about reverses of the Jensen inequality for positive measure, in this paper we give generalizations of those results for real Stieltjes measure $d\lambda$ which is not necessarily positive using several Green functions. Utilizing these results we define some new mean value theorems of Lagrange and Cauchy types, and derive some new Cauchy-type means.

We derive the Levinson type generalization of the Jensen and the converse Jensen inequality for real Stieltjes measure, not necessarily positive. As a consequence, also the Levinson type generalization of the Hermite-Hadamard inequality is obtained. Similarly, we derive the Levinson type generalization of Giaccardi’s inequality. The obtained results are then applied for establishing new mean-value theorems. The results from this paper represent a generalization of several recent results.

Motivated by some recent investigations about the sharpness of the Jensen inequality, this paper deals with the sharpness of the converse of the Jensen inequality. These results are then used for deriving new inequalities for different types of generalized f-divergences. As divergences measure the differences between probability distributions, these new inequalities are then applied on the Zipf–Mandelbrot law as a special kind of a probability distribution.

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