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Determination of Bounds for the Jensen Gap and Its Applications

Author

Listed:
  • Hidayat Ullah

    (Department of Mathematics, University of Peshawar, Peshawar 25000, Pakistan)

  • Muhammad Adil Khan

    (Department of Mathematics, University of Peshawar, Peshawar 25000, Pakistan)

  • Tareq Saeed

    (Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia)

Abstract

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.

Suggested Citation

  • Hidayat Ullah & Muhammad Adil Khan & Tareq Saeed, 2021. "Determination of Bounds for the Jensen Gap and Its Applications," Mathematics, MDPI, vol. 9(23), pages 1-29, December.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:23:p:3132-:d:695377
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    References listed on IDEAS

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    1. Lin, Qian, 2019. "Jensen inequality for superlinear expectations," Statistics & Probability Letters, Elsevier, vol. 151(C), pages 79-83.
    2. Yu-Ming Chu & Miao-Kun Wang & Zi-Kui Wang, 2011. "A Sharp Double Inequality between Harmonic and Identric Means," Abstract and Applied Analysis, Hindawi, vol. 2011, pages 1-7, October.
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    Cited by:

    1. Shanhe Wu & Muhammad Adil Khan & Tareq Saeed & Zaid Mohammed Mohammed Mahdi Sayed, 2022. "A Refined Jensen Inequality Connected to an Arbitrary Positive Finite Sequence," Mathematics, MDPI, vol. 10(24), pages 1-10, December.
    2. Xuexiao You & Muhammad Adil Khan & Hidayat Ullah & Tareq Saeed, 2022. "Improvements of Slater’s Inequality by Means of 4-Convexity and Its Applications," Mathematics, MDPI, vol. 10(8), pages 1-19, April.
    3. Muhammad Adil Khan & Asadullah Sohail & Hidayat Ullah & Tareq Saeed, 2023. "Estimations of the Jensen Gap and Their Applications Based on 6-Convexity," Mathematics, MDPI, vol. 11(8), pages 1-25, April.

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