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A new entropy upper bound. (English) Zbl 1255.26011

Summary: Entropy, conditional entropy and mutual information for discrete-valued random variables play important roles in information theory (see [R. B. Ash, Information theory. Intersci. Tracts in Pure and Appl. Math. 19. N.Y.-Lond.-Sydney: Intersci. Publishers, a Division of John Wiley and Sons (1965; Zbl 0141.34904)] and [T. M. Cover and J. A. Thomas, Elements of information theory. 2nd ed. Wiley-Interscience. Hoboken, NJ: John Wiley & Sons. (2006; Zbl 1140.94001)]). Our purpose within this work is to present a strong upper bound for the classical Shannon entropy, refining recent results from the literature. For this purpose we have considered the work of S. Simic [Appl. Math. Lett. 22, No. 8, 1262–1265 (2009; Zbl 1173.26308)], where new entropy bounds based on a new refinement of Jensen’s inequality are presented. Our work improves the basic result of Simic through a stronger refinement of Jensen’s inequality which is then applied to information theory.

MSC:

26D15 Inequalities for sums, series and integrals
94A17 Measures of information, entropy
26D10 Inequalities involving derivatives and differential and integral operators
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