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

Deng, Yongping; Ullah, Hidayat; Khan, Muhammad Adil; Iqbal, Sajid; Wu, Shanhe

doi:10.1155/2021/1951799

Cited by 16 publications

(14 citation statements)

References 30 publications

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“…In the diverse elds of science, convex functions are of the greatest importance due to their dominant manners and wealthy structure [1][2][3]. In recent years, the abundant applicability of convex functions has been observed in engineering [4], di erential equations [5], epidemiology [6], information theory [7], statistics [8], optimization [9], and many others. Moreover, convex functions have some unique properties and, due to such properties, it became a focus point for researchers [10,11].…”

Section: Introductionmentioning

confidence: 99%

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Estimations of the Slater Gap via Convexity and Its Applications in Information Theory

Khan

Ullah

Saeed

et al. 2022

Mathematical Problems in Engineering

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Convexity has played a prodigious role in various areas of science through its properties and behavior. Convexity has booked record developments in the field of mathematical inequalities in the recent few years. The Slater inequality is one of the inequalities which has been acquired with the help of convexity. In this note, we obtain some estimations for the Slater gap while dealing with the notion of convexity in an extensive manner. We acquire the deliberated estimations by utilizing the definition of convex function, Jensen’s inequality for concave functions, and triangular, power mean, and Hölder inequalities. We discuss several consequences of the main results in terms of inequalities for the power means. Moreover, by utilizing the main results, we give estimations for the Csiszár and Kullback–Leibler divergences, Shannon entropy, and the Bhattacharyya coefficient. Furthermore, we present some estimations for the Zipf–Mandelbrot entropy as additional applications of the acquired results. The perception and approaches adopted in this note may pretend more research in this direction.

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Section: Introductionmentioning

confidence: 99%

“…e Jensen inequality has a variety of interesting properties and also has a very desirable structure [24]. Furthermore, there are a huge number of applications of this inequality in the different fields of science [4,7,25,26]. Due to the huge importance and applicability of this inequality, a lot of work has been carried out on it [8,27].…”

Section: Introductionmentioning

confidence: 99%

Estimations of the Slater Gap via Convexity and Its Applications in Information Theory

Khan

Ullah

Saeed

et al. 2022

Mathematical Problems in Engineering

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“…The notion of convexity has played a very fundamental role in the last century with a dynamic impact on the several areas of science including Engineering [1], Statistics [2], Economics [3], Optimization [4] and Information Theory [5], etc. The most diligent and dynamic area for the notion of convexity is the field of mathematics [6].…”

Section: Introductionmentioning

confidence: 99%

“…Some of the well-known inequalities for the class of convex functions are majorization [10], Hermite-Hadamrd [6] and Favard [11] inequalities, etc. Among these inequalities, one of the most dynamic and favorable inequalities is the Jensen inequality [5]. Jensen's inequality and convex functions have a very deep pertinency in the view that they generalize the definition of the convex function.…”

Section: Introductionmentioning

confidence: 99%

“…They also deliberated inequalities for different means and granted applications of their main results in information theory. In 2021, Deng et al [5] proved some refinements of Jensen's inequality with the help of majorization results while using the notion of convexity. Furthermore, they provided some refinements for classical inequalities and also presented applications of main results.…”

Section: Introductionmentioning

confidence: 99%

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Determination of Bounds for the Jensen Gap and Its Applications

2021

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The Jensen inequality has been reported as one of the most consequential inequalities that has a lot of applications in diverse fields of science. For this reason, the Jensen inequality has become one of the most discussed developmental inequalities in the current literature on mathematical inequalities. The main intention of this article is to find some novel bounds for the Jensen difference while using some classes of twice differentiable convex functions. We obtain the proposed bounds by utilizing the power mean and Höilder inequalities, the notion of convexity and the prominent Jensen inequality for concave function. We deduce several inequalities for power and quasi-arithmetic means as a consequence of main results. Furthermore, we also establish different improvements for Hölder inequality with the help of obtained results. Moreover, we present some applications of the main results in information theory.

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Some estimations of the Jensen difference and applications

Khan

Ullah

Saeed

2022

Math Methods in App Sciences

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The Jensen inequality is one of the most favorable inequalities during the last few decades due to its expressive characteristics and properties. This inequality has gained a very dominant position in the several fields of science as a result of its extensive applications. In the present note, we use an interesting approach for the determination of estimates for the Jensen difference. We acquire several estimates for the Jensen difference while utilizing the convex function definition, Jensen's inequality for concave functions, power mean, and Hölder inequalities. For the real visualizations of the acquired estimates, we provide some particular examples. We get different improvements for the Hölder inequality while taking specified functions in the received estimates. Moreover, we deduce some more results from the main work in the form of improvements for the Hermite-Hadamard inequality. Furthermore, various relations for the famous quasi-arithmetic and power means are acquired with the help of established estimates. At the end, we present various applications of the main work in information theory. The intended applications provide some new bounds for the Csiszár and Rényi divergences, Shannon entropy, and Bhattacharyya co-efficient.

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

Cited by 16 publications

References 30 publications

Estimations of the Slater Gap via Convexity and Its Applications in Information Theory

Estimations of the Slater Gap via Convexity and Its Applications in Information Theory

Determination of Bounds for the Jensen Gap and Its Applications

Some estimations of the Jensen difference and applications

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