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.
Keywords: Entropy; Green function; Jensen’s inequality; Zipf–Mandelbrot law; f-divergence.