Generalizations of cyclic refinements of Jensen's inequality by Lidstone's polynomial with applications in information theory

AbstractTo procure inequalities for divergences between probability distributions, Jensen’s inequality is the key to success. Shannon, Relative and Zipf-Mandelbrot entropies have many applications in many applied sciences, such as, in information theory, biology and economics, etc. We consider discrete and continuous cyclic refinements of Jensen’s inequality and extend them from convex function to higher order convex function by means of different new Green functions by employing Hermite interpolating polynomial whose error term is approximated by Peano’s kernal. As an application of our obtained results, we give new bounds for Shannon, Relative and Zipf-Mandelbrot entropies.

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A new refinement of Jensen’s inequality with applications in information theory

Open Mathematics ◽

10.1515/math-2020-0123 ◽

2020 ◽

Vol 18 (1) ◽

pp. 1748-1759

Author(s):

Lei Xiao ◽

Guoxiang Lu

Keyword(s):

Information Theory ◽

Jensen’S Inequality ◽

Shannon’S Entropy ◽

Jensen's Inequality ◽

Shannon's Entropy

Abstract In this paper, we present a new refinement of Jensen’s inequality with applications in information theory. The refinement of Jensen’s inequality is obtained based on the general functional in the work of Popescu et al. As the applications in information theory, we provide new tighter bounds for Shannon’s entropy and some f-divergences.

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Reversing Jensen’s Inequality for Information-Theoretic Analyses

Information ◽

10.3390/info13010039 ◽

2022 ◽

Vol 13 (1) ◽

pp. 39

Author(s):

Neri Merhav

Keyword(s):

Information Theory ◽

Jensen’S Inequality ◽

Jensen Inequality ◽

Concave Functions ◽

Jensen's Inequality ◽

Information Theoretic ◽

Main Ideas

In this work, we propose both an improvement and extensions of a reverse Jensen inequality due to Wunder et al. (2021). The new proposed inequalities are fairly tight and reasonably easy to use in a wide variety of situations, as demonstrated in several application examples that are relevant to information theory. Moreover, the main ideas behind the derivations turn out to be applicable to generate bounds to expectations of multivariate convex/concave functions, as well as functions that are not necessarily convex or concave.

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Uniform Treatment of Jensen’s Inequality by Montgomery Identity

Journal of Mathematics ◽

10.1155/2021/5564647 ◽

2021 ◽

Vol 2021 ◽

pp. 1-17

Author(s):

Tahir Rasheed ◽

Saad Ihsan Butt ◽

Đilda Pečarić ◽

Josip Pečarić ◽

Ahmet Ocak Akdemir

Keyword(s):

Information Theory ◽

Integral Inequality ◽

Convex Functions ◽

Jensen’S Inequality ◽

Fractional Integrals ◽

Jensen's Inequality ◽

Hadamard Inequality ◽

Montgomery Identity ◽

Discrete Inequality ◽

Uniform Treatment

We generalize Jensen’s integral inequality for real Stieltjes measure by using Montgomery identity under the effect of n − convex functions; also, we give different versions of Jensen’s discrete inequality along with its converses for real weights. As an application, we give generalized variants of Hermite–Hadamard inequality. Montgomery identity has a great importance as many inequalities can be obtained from Montgomery identity in q − calculus and fractional integrals. Also, we give applications in information theory for our obtained results, especially for Zipf and Hybrid Zipf–Mandelbrot entropies.

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Refinements of Jensen's inequality and applications

AIMS Mathematics ◽

10.3934/math.2022297 ◽

2022 ◽

Vol 7 (4) ◽

pp. 5328-5346

Author(s):

Tareq Saeed ◽

◽

Muhammad Adil Khan ◽

Hidayat Ullah ◽

Keyword(s):

Information Theory ◽

Shannon Entropy ◽

Research Work ◽

Jensen’S Inequality ◽

Hölder Inequality ◽

Jensen's Inequality ◽

Arithmetic Means ◽

Hadamard Inequality ◽

Bhattacharyya Coefficient

<abstract><p>The principal aim of this research work is to establish refinements of the integral Jensen's inequality. For the intended refinements, we mainly use the notion of convexity and the concept of majorization. We derive some inequalities for power and quasi–arithmetic means while utilizing the main results. Moreover, we acquire several refinements of Hölder inequality and also an improvement of Hermite–Hadamard inequality as consequences of obtained results. Furthermore, we secure several applications of the acquired results in information theory, which consist bounds for Shannon entropy, different divergences, Bhattacharyya coefficient, triangular discrimination and various distances.</p></abstract>

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SUPERADDITIVITY OF SOME FUNCTIONALS ASSOCIATED WITH JENSEN’S INEQUALITY FOR CONVEX FUNCTIONS ON LINEAR SPACES WITH APPLICATIONS

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972709001105 ◽

2010 ◽

Vol 82 (1) ◽

pp. 44-61 ◽

Cited By ~ 3

Author(s):

S. S. DRAGOMIR

Keyword(s):

Information Theory ◽

Convex Functions ◽

Convex Sets ◽

Jensen’S Inequality ◽

Jensen's Inequality ◽

Norm Inequalities ◽

Linear Spaces ◽

Normed Linear Spaces

AbstractSome new results related to Jensen’s celebrated inequality for convex functions defined on convex sets in linear spaces are given. Applications for norm inequalities in normed linear spaces and f-divergences in information theory are provided as well.

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Inequalities for the normalized Jensen functional with applications

Mathematica Slovaca ◽

10.1515/ms-2015-0121 ◽

2016 ◽

Vol 66 (1) ◽

Cited By ~ 1

Author(s):

Hossein Dehghan

Keyword(s):

Information Theory ◽

Jensen’S Inequality ◽

Jensen's Inequality ◽

Jensen Functional ◽

New Inequalities

AbstractIn this paper we present some new inequalities for the normalized Jensen functional which will much improve the famous Jensen’s inequality. We also give some applications in Analysis and Information Theory. Our results refine and generalize the corresponding ones announced in [DRAGOMIR, S. S.:

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Refinements of Jensen’s Inequality via Majorization Results with Applications in the Information Theory

Journal of Mathematics ◽

10.1155/2021/1951799 ◽

2021 ◽

Vol 2021 ◽

pp. 1-12

Author(s):

Yongping Deng ◽

Hidayat Ullah ◽

Muhammad Adil Khan ◽

Sajid Iqbal ◽

Shanhe Wu

Keyword(s):

Information Theory ◽

Shannon Entropy ◽

Probability Distributions ◽

Jensen’S Inequality ◽

Jensen Inequality ◽

Jensen's Inequality ◽

Power Means ◽

Quasiarithmetic Means

In this study, we present some new refinements of the Jensen inequality with the help of majorization results. We use the concept of convexity along with the theory of majorization and obtain refinements of the Jensen inequality. Moreover, as consequences of the refined Jensen inequality, we derive some bounds for power means and quasiarithmetic means. Furthermore, as applications of the refined Jensen inequality, we give some bounds for divergences, Shannon entropy, and various distances associated with probability distributions.

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New Estimates for Csiszár Divergence and Zipf–Mandelbrot Entropy via Jensen–Mercer’s Inequality

Complexity ◽

10.1155/2020/8928691 ◽

2020 ◽

Vol 2020 ◽

pp. 1-8

Author(s):

Muhammad Adil Khan ◽

Zakir Husain ◽

Yu-Ming Chu

Keyword(s):

Information Theory ◽

Jensen’S Inequality ◽

Jensen's Inequality

Jensen’s inequality is one of the fundamental inequalities which has several applications in almost every field of science. In 2003, Mercer gave a variant of Jensen’s inequality which is known as Jensen–Mercer’s inequality. The purpose of this article is to propose new bounds for Csiszár and related divergences by means of Jensen–Mercer’s inequality. Also, we investigate several new bounds for Zipf–Mandelbrot entropy. The idea of this article may further stimulate research on information theory with the help of Jensen–Mercer’s inequality.

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SOME REVERSES OF THE JENSEN INEQUALITY WITH APPLICATIONS

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972712001098 ◽

2013 ◽

Vol 87 (2) ◽

pp. 177-194 ◽

Cited By ~ 5

Author(s):

S. S. DRAGOMIR

Keyword(s):

Information Theory ◽

Convex Functions ◽

Hölder’S Inequality ◽

General Setting ◽

Jensen’S Inequality ◽

Jensen Inequality ◽

Lebesgue Integral ◽

Jensen's Inequality ◽

Hölder's Inequality ◽

Divergence Measures

AbstractTwo new reverses of the celebrated Jensen’s inequality for convex functions in the general setting of the Lebesgue integral, with applications to means, Hölder’s inequality and$f$-divergence measures in information theory, are given.

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