Estimation of f-divergence and Shannon entropy by Levinson type inequalities for higher order convex functions via Taylor polynomial

Volume 21, Issue 4, pp 322--334 http://dx.doi.org/10.22436/jmcs.021.04.05

Publication Date: May 15, 2020 Submission Date: October 12, 2019 Revision Date: February 07, 2020 Accteptance Date: March 13, 2020

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Authors

Muhammad Adeel - Department of Mathematics, University of Sargodha, Sargodha-40100, Pakistan. - Department of Mathematics, University of Central Punjab, Lahore, Pakistan. Khuram Ali Khan - Department of Mathematics, University of Sargodha, Sargodha-40100, Pakistan. Ðilda Pečarić - University of Croatia, Ilica 242, Zagreb, Croatia. Josip Pečarić - RUDN University, Moscow, Russia.

Abstract

In this paper, Levinson-type inequalities are generalized by using Taylor polynomial for the class of \(k\)-convex \((k \geq 3)\) functions. Bounds for the remainders in new generalized identities involving data points of two types are given by using Čebyšev, Grúss and Ostrowski-type inequalities. In seek of applications of our results to information theory, new generalizations based on \(f\)-divergence estimates are also proven. Moreover, some inequalities for Shannon entropies are deduced as well.

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Keywords

• Information theory
• convex functions
• Levinson's inequality

MSC

•  94A15
•  94A17
•  26D15

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