Refinements of Jensen's inequality and applications

Saeed, Tareq; Khan, Muhammad Adil; Ullah, Hidayat

doi:10.3934/math.2022297

Cited by 11 publications

(5 citation statements)

References 19 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Jensen's inequality has a very strong relationship with ordinary convexity in the sense that it generalizes the definition of convex function. Another important fact about this inequality is that it is the origin of many other classical inequalities [23].…”

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

Self Cite

View full text Add to dashboard Cite

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.

show abstract

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

Self Cite

View full text Add to dashboard Cite

show abstract

“…Moreover, they compared their main results with some other earlier established improvements of Jensen's inequality and showed that the obtained improvements provided better estimations for the Jensen difference. Saeed et al [60] provided refinements of the celebrated integral Jensen inequality via the theory of majorization. With the help of these, refinements of the Hölder and Hermite-Hadamard inequalities were provided.…”

Section: Remarkmentioning

confidence: 99%

Estimations of the Jensen Gap and Their Applications Based on 6-Convexity

et al. 2023

Self Cite

View full text Add to dashboard Cite

The main purpose of this manuscript is to present some new estimations of the Jensen gap in a discrete sense along with their applications. The proposed estimations for the Jensen gap are provided with the help of the notion of 6-convex functions. Some numerical experiments are performed to determine the significance and correctness of the intended estimates. Several outcomes of the main results are discussed for the Hölder inequality and the power and quasi-arithmetic means. Furthermore, some applications are presented in information theory, which provide some bounds for the divergences, Bhattacharyya coefficient, Shannon entropy, and Zipf–Mandelbrot entropy.

show abstract

“…Furthermore, they also granted applications of the improved Jensen inequality in information theory. Saeed et al 36 established some refinements of the Jensen inequality by the support of majorization results. They discussed some consequences of their refinements for means, Hölder and Hermite‐Hadamard inequalities and also presented application for divergences and Shannon entropy.…”

Section: Introductionmentioning

confidence: 97%

Some estimations of the Jensen difference and applications

Khan

Ullah

Saeed

2022

Math Methods in App Sciences

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Refinements of Jensen's inequality and applications

Cited by 11 publications

References 19 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

Estimations of the Jensen Gap and Their Applications Based on 6-Convexity

Some estimations of the Jensen difference and applications

Contact Info

Product

Resources

About