In this paper we give extensions of Sherman's inequality considering the class of convex functions of higher order. As particular cases, we get an extended weighted majorization inequality as well as Jensen's inequality which have direct connection to information theory. We use the obtained results to derive new estimates for Shannon's and Rényi's entropy, information energy, and some well-known measures between probability distributions. Using the Zipf-Mandelbrot law, we introduce new functionals to derive some related results.
Keywords: Abel–Gontscharoff interpolating polynomial; Entropy; Green function; Information theory; Jensen inequality; Majorization inequality; Sherman theorem; Zipf–Mandelbrot law; n-convex function; ϕ-divergence.