Some Improvements of Jensen’s Inequality via 4-Convexity and Applications

Ullah, Hidayat; Khan, Muhammad Adil; Saeed, Tareq; Sayed, Zaid Mohammed Mohammed Mahdi

doi:10.1155/2022/2157375

Cited by 9 publications

(11 citation statements)

References 17 publications

(21 reference statements)

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“…Remark 3. The same type of estimate stated on the right side of (19) is also obtained in inequality (9) in [41] for the absolute Jensen difference.…”

Section: Resultssupporting

confidence: 59%

“…Moreover, inequalities are frequently utilized in mathematical analysis and proof techniques [4,18]. Furthermore, they allow us to establish bounds, estimate limits, and prove the existence or non-existence of certain mathematical objects or properties [19]. In mathematical analysis, calculus, and advanced proof-based mathematics, inequalities are essential elements [20][21][22].…”

Section: Introductionmentioning

confidence: 99%

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Derivation of Bounds for Majorization Differences by a Novel Method and Its Applications in Information Theory

Basir,

Khan,

Ullah

et al. 2023

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In the recent era of research developments, mathematical inequalities and their applications perform a very consequential role in different aspects, and they provide an engaging area for research activities. In this paper, we propose a new approach for the improvement of the classical majorization inequality and its weighted versions in a discrete sense. The proposed improvements give several estimates for the majorization differences. Some earlier improvements of the Jensen and Slater inequalities are deduced as direct consequences of the obtained results. We also discuss the conditions under which the main results give better estimates for the majorization differences. Applications of the acquired results are also presented in information theory.

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“…Remark 3. The same type of estimate stated on the right side of (19) is also obtained in inequality (9) in [41] for the absolute Jensen difference.…”

Section: Resultssupporting

confidence: 59%

Section: Introductionmentioning

confidence: 99%

Derivation of Bounds for Majorization Differences by a Novel Method and Its Applications in Information Theory

Basir,

Khan,

Ullah

et al. 2023

Axioms

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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%

“…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]. In 1980, Slater [28] presented a companion inequality to the aforementioned inequality while using a convex function, which is well known as Slater's inequality in the literature.…”

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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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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“…In addition to this, they also granted some applications of main results for different divergences. Ullah et al 35 provided some new improvements for the Jensen inequality by using 4‐convexity. They gave improvements for the Hölder inequality, Hermite‐Hadamard inequality, and power and quasi‐arithmetic means.…”

Section: Introductionmentioning

confidence: 99%

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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Some Improvements of Jensen’s Inequality via 4-Convexity and Applications

Cited by 9 publications

References 17 publications

Derivation of Bounds for Majorization Differences by a Novel Method and Its Applications in Information Theory

Derivation of Bounds for Majorization Differences by a Novel Method and Its Applications in Information Theory

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

Some estimations of the Jensen difference and applications

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