Refinements of Jensen's inequality and applications

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

doi:10.3934/math.2022297

AIMS Mathematics

2022, Volume 7, Issue 4: 5328-5346. doi: 10.3934/math.2022297

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Research article

Refinements of Jensen's inequality and applications

1.
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
2.
Department of Mathematics, University of Peshawar, Peshawar 25000, Pakistan

Received: 09 November 2021 Revised: 06 December 2021 Accepted: 23 December 2021 Published: 06 January 2022
MSC : 26A51, 26D15, 68P30

The principal aim of this research work is to establish refinements of the integral Jensen's inequality. For the intended refinements, we mainly use the notion of convexity and the concept of majorization. We derive some inequalities for power and quasi–arithmetic means while utilizing the main results. Moreover, we acquire several refinements of Hölder inequality and also an improvement of Hermite–Hadamard inequality as consequences of obtained results. Furthermore, we secure several applications of the acquired results in information theory, which consist bounds for Shannon entropy, different divergences, Bhattacharyya coefficient, triangular discrimination and various distances.
- convex function,
- Jensen's inequality,
- majorization inequality,
- information theory
Citation: Tareq Saeed, Muhammad Adil Khan, Hidayat Ullah. Refinements of Jensen's inequality and applications[J]. AIMS Mathematics, 2022, 7(4): 5328-5346. doi: 10.3934/math.2022297

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Abstract

The principal aim of this research work is to establish refinements of the integral Jensen's inequality. For the intended refinements, we mainly use the notion of convexity and the concept of majorization. We derive some inequalities for power and quasi–arithmetic means while utilizing the main results. Moreover, we acquire several refinements of Hölder inequality and also an improvement of Hermite–Hadamard inequality as consequences of obtained results. Furthermore, we secure several applications of the acquired results in information theory, which consist bounds for Shannon entropy, different divergences, Bhattacharyya coefficient, triangular discrimination and various distances.

References

[1]	M. Adil Khan, S. H. Wu, H. Ullah, Y. M. Chu, Discrete majorization type inequalities for convex functions on rectangles, J. Inequal. Appl., 2019 (2019), 1–18. https://doi.org/10.1186/s13660-019-1964-3 doi: 10.1186/s13660-019-1964-3
[2]	M. Adil Khan, Z. M. Al-sahwi, Y. M. Chu, New estimations for Shannon and Zipf-Mandelbrot entropies, Entropy, 20 (2018), 1–10. https://doi.org/10.3390/e20080608 doi: 10.3390/e20080608
[3]	M. Adil Khan, D. Pečarić, J. Pečarić, On Zipf-Mandelbrot entropy, J. Comput. Appl. Math., 346 (2019), 192–204. https://doi.org/10.1016/j.cam.2018.07.002 doi: 10.1016/j.cam.2018.07.002
[4]	M. Adil Khan, D. Pečarić, J. Pečarić, Bounds for Shannon and Zipf-mandelbrot entropies, Math. Methods Appl. Sci., 40 (2017), 7316–7322. https://doi.org/10.1002/mma.4531 doi: 10.1002/mma.4531
[5]	I. Ansari, K. A. Khan, A. Nosheen, D. Pečarić, J. Pečarić, Some inequalities for Csiszar divergence via theory of time scales, Adv. Differ. Equ., 2020 (2020), 1–21. https://doi.org/10.1186/s13662-020-03159-x doi: 10.1186/s13662-020-03159-x
[6]	Y. Deng, H. Ullah, M. Adil Khan, S. Iqbal, S. Wu, Refinements of Jensen's inequality via majorization results with applications in the information theory, J. Math., 2012 (2021), 1–12. https://doi.org/10.1155/2021/1951799 doi: 10.1155/2021/1951799
[7]	N. S. Barnett, P. Cerone, S. S. Dragomir, Majorisation inequalities for Stieltjes integrals, Appl. Math. Lett., 22 (2009), 416–421. https://doi.org/10.1016/j.aml.2008.06.009 doi: 10.1016/j.aml.2008.06.009
[8]	J. Borwein, A, Lewis, Convex Analysis and Nonlinear Optimization, Theory and Examples, Springer: NewYork, 2000. https://doi.org/10.1007/978-1-4757-9859-3
[9]	M. J. Cloud, B. C. Drachman, L. P. Lebedev, Inequalities with Applications to Engineering, Springer: Cham Heidelberg New York Dordrecht London, 2014.
[10]	S. S. Dragomir, Some majorization type discrete inequalities for convex functions, Math. Inequal. Appl., 7 (2004), 207–216. https://doi.org/10.7153/mia-07-23 doi: 10.7153/mia-07-23
[11]	S. Furuichi, H. R. Moradi, A. Zardadi, Some new Karamata type inequalities and their applications to some entropies, Rep. Math. Phys., 84 (2019), 201–214. https://doi.org/10.1016/S0034-4877(19)30083-7 doi: 10.1016/S0034-4877(19)30083-7
[12]	L. Horváth, D. Pečarić, J. Pečarić, Estimations of f– and Rényi divergences by using a cyclic refinement of the Jensen's inequality, Bull. Malays. Math. Sci. Soc., 42 (2019), 933–946. https://doi.org/10.1007/s40840-017-0526-4 doi: 10.1007/s40840-017-0526-4
[13]	N. Latif, D. Pečarić, J. Pečarić, Majorization, useful Csiszar divergence and useful Zipf-Mandelbrot law, Open Math., 16 (2018), 1357–1373. https://doi.org/10.1515/math-2018-0113 doi: 10.1515/math-2018-0113
[14]	L. Maligranda, J. Pečarić, L. E. Persson, Weighted Favard and Berwald inequalities, J. Math. Anal. Appl., 190 (1995), 248–262. https://doi.org/10.1006/jmaa.1995.1075 doi: 10.1006/jmaa.1995.1075
[15]	A. W. Marshall, I. Olkin, B. Arnold, Inequalities: Theory of majorization and its applications, 2nd ed., Springer Series in Statistics, Springer, New York, 2011. https://doi.org/10.1007/978-0-387-68276-1
[16]	C. P. Niculescu, L. E. Persson, Convex Functions and Their Applications. A Contemporary Approach, 2nd ed., CMS Books in Mathematics vol. 23, Springer-Verlag, New York, 2018. https://doi.org/10.1007/978-3-319-78337-6
[17]	J. Pečarić, J. Perić, New improvement of the converse Jensen inequality, Math. Inequal. Appl., 21 (2018), 217–234. https://doi.org/10.7153/mia-2018-21-17 doi: 10.7153/mia-2018-21-17
[18]	J. Pečarić, F. Proschan, Y. L. Tong, Convex Functions, Partial Orderings and Statistical Applications, Academic Press, 1992.
[19]	M. Sababheh, H. R. Moradi, S. Furuichi, Integrals refining convex inequalities, Bull. Malays. Math. Sci. Soc., 2019 (2019), 1–17.
[20]	N. Siddique, M. Imran, K. A. Khan, J. Pečarić, Majorization inequalities via Green functions and Fink's identity with applications to Shannon entropy, J. Inequal. Appl., 2020 (2020), 1–14. https://doi.org/10.1186/s13660-020-02455-0 doi: 10.1186/s13660-020-02455-0
[21]	H. Ullah, M. Adil Khan, J. Pečarić, New bounds for soft margin estimator via concavity of Gaussian weighting function, Adv. Differ. Equ., 2020 (2020), 1–10. https://doi.org/10.1186/s13662-020-03103-z doi: 10.1186/s13662-020-03103-z
[22]	S. Z. Ullah, M. Adil Khan, Y. M. Chu, Majorization theorem for strongly convex function, J. Inequal. Appl., 2019 (2019), 1–13. https://doi.org/10.1186/s13660-019-1964-3 doi: 10.1186/s13660-019-1964-3
[23]	S. Wu, M. Adil Khan, H. U. Haleemzai, Refinements of majorization inequality involving convex functions via Taylor's theorem with mean value form of the remainder, Mathematics, 7 (2019), 1–7. https://doi.org/10.3390/math7080663 doi: 10.3390/math7080663
[24]	S. Wu, M. Adil Khan, A. Basir, R. Saadati, Some majorization integral inequalities for functions defined on rectangles, J. Inequal. Appl., 2018 (2018), 1–13. https://doi.org/10.1186/s13660-018-1739-2 doi: 10.1186/s13660-018-1739-2

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