The main purpose of this paper is to find new estimations for the Shannon and Zipf–Mandelbrot entropies. We apply some refinements of the Jensen inequality to obtain different bounds for these entropies. Initially, we use a precise convex function in the refinement of the Jensen inequality and then tamper the weight and domain of the function to obtain general bounds for the Shannon entropy (SE
). As particular cases of these general bounds, we derive some bounds for the Shannon entropy (SE
) which are, in fact, the applications of some other well-known refinements of the Jensen inequality. Finally, we derive different estimations for the Zipf–Mandelbrot entropy (ZME
) by using the new bounds of the Shannon entropy for the Zipf–Mandelbrot law (ZML
). We also discuss particular cases and the bounds related to two different parametrics of the Zipf–Mandelbrot entropy. At the end of the paper we give some applications in linguistics.
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