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Entropy 2018, 20(8), 608;

New Estimations for Shannon and Zipf–Mandelbrot Entropies

College of Science, Hunan City University, Yiyang 413000, China
Department of Mathematics, University of Peshawar, Peshawar 25000, Pakistan
Department of Mathematics, University of Sa’adah, Sa’adah 1872, Yemen
Department of Mathematics, Huzhou University, Huzhou 313000, China
Author to whom correspondence should be addressed.
Received: 9 July 2018 / Revised: 8 August 2018 / Accepted: 14 August 2018 / Published: 16 August 2018
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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. View Full-Text
Keywords: Jensen inequality; convex function; Shannon entropy; Zipf–Mandelbrot entropy Jensen inequality; convex function; Shannon entropy; Zipf–Mandelbrot entropy
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).

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Adil Khan, M.; Al-sahwi, Z.M.; Chu, Y.-M. New Estimations for Shannon and Zipf–Mandelbrot Entropies. Entropy 2018, 20, 608.

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