Next Article in Journal
Information Geometry of Randomized Quantum State Tomography
Previous Article in Journal
Quantum Quantifiers for an Atom System Interacting with a Quantum Field Based on Pseudoharmonic Oscillator States
Article Menu
Issue 8 (August) cover image

Export Article

Open AccessArticle
Entropy 2018, 20(8), 608; https://doi.org/10.3390/e20080608

New Estimations for Shannon and Zipf–Mandelbrot Entropies

1
College of Science, Hunan City University, Yiyang 413000, China
2
Department of Mathematics, University of Peshawar, Peshawar 25000, Pakistan
3
Department of Mathematics, University of Sa’adah, Sa’adah 1872, Yemen
4
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
Full-Text   |   PDF [242 KB, uploaded 18 August 2018]

Abstract

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).
SciFeed

Share & Cite This Article

MDPI and ACS Style

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

Show more citation formats Show less citations formats

Note that from the first issue of 2016, MDPI journals use article numbers instead of page numbers. See further details here.

Related Articles

Article Metrics

Article Access Statistics

1

Comments

[Return to top]
Entropy EISSN 1099-4300 Published by MDPI AG, Basel, Switzerland RSS E-Mail Table of Contents Alert
Back to Top