Some estimations of the Jensen difference and applications

• Published in: Mathematical methods in the applied sciences Vol. 46; no. 5; pp. 5863 - 5892
• Main Authors: Adil Khan, Muhammad, Ullah, Hidayat, Saeed, Tareq
• Format: Journal Article
• Language: English
• Published: Freiburg Wiley Subscription Services, Inc 30.03.2023
• Subjects: convex functions; Entropy (Information theory); Estimates; Hermite‐Hadamard inequality; Hölder inequality; Inequalities; Inequality; Information theory; Jensen's inequality; power means
• ISSN: 0170-4214; 1099-1476
• DOI: 10.1002/mma.8873

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.