Improvements of the integral Jensen inequality through the treatment of the concept of convexity of thrice differential functions

Asadullah Sohail; Muhammad Adil Khan; Xiaoye Ding; Mohamed Sharaf; Mohammed A. El-Meligy; Asadullah Sohail; Muhammad Adil Khan; Xiaoye Ding; Mohamed Sharaf; Mohammed A. El-Meligy

doi:10.3934/math.20241620

AIMS Mathematics

2024, Volume 9, Issue 12: 33973-33994. doi: 10.3934/math.20241620

Previous Article Next Article

Research article Special Issues

Improvements of the integral Jensen inequality through the treatment of the concept of convexity of thrice differential functions

1.
Department of Mathematics, University of Peshawar, Peshawar 25000, Pakistan
2.
General Education Department, Anhui Xinhua University, Hefei 230088, China
3.
Industrial Engineering Department, College of Engineering, King Saud University, P.O. Box 800, Riyadh 11421, Saudi Arabia
4.
Applied Science Research Center, Applied Science Private University, Amman, Jordan
5.
Jadara University Research Center, Jadara University, P.O. Box 733, Irbid, Jordan

Received: 14 September 2024 Revised: 11 November 2024 Accepted: 20 November 2024 Published: 02 December 2024
MSC : 26A42, 26A51, 26D15

Jensen's inequality, with its broad applications across various fields, presents an important subject for investigation and research. In this article, we introduce novel enhancements to Jensen's inequality by utilizing the convexity properties of a thrice differentiable function in the absolute sense. We conducted numerical experiments to compare our primary results with previously established findings. Additionally, we provided several applications of our major results to classical inequalities, means, and divergences.
- convex function,
- concave function,
- Jensen's inequality,
- Hölder inequality
Citation: Asadullah Sohail, Muhammad Adil Khan, Xiaoye Ding, Mohamed Sharaf, Mohammed A. El-Meligy. Improvements of the integral Jensen inequality through the treatment of the concept of convexity of thrice differential functions[J]. AIMS Mathematics, 2024, 9(12): 33973-33994. doi: 10.3934/math.20241620

Related Papers:

Abstract

Jensen's inequality, with its broad applications across various fields, presents an important subject for investigation and research. In this article, we introduce novel enhancements to Jensen's inequality by utilizing the convexity properties of a thrice differentiable function in the absolute sense. We conducted numerical experiments to compare our primary results with previously established findings. Additionally, we provided several applications of our major results to classical inequalities, means, and divergences.

References

[1]	S. A. Azar, Jensen's inequality in finance, Int. Adv. Econ. Res., 14 (2008), 433–440. https://doi.org/10.1007/s11294-008-9172-9 doi: 10.1007/s11294-008-9172-9
[2]	A. Basir, M. A. Khan, H. Ullah, Y. Almalki, S. Chasreechai, T. Sitthiwirattham, Derivation of bounds for majorization differences by a novel method and its applications in information theory, Axioms, 12 (2023), 1–17. https://doi.org/10.3390/axioms12090885 doi: 10.3390/axioms12090885
[3]	A. Basir, M. A. Khan, H. Ullah, Y. Almalki, C. Metpattarahiran, T. Sitthiwirattham, Improvements of integral majorization inequality with applications to divergences, Axioms, 13 (2024), 1–14. https://doi.org/10.3390/axioms13010021 doi: 10.3390/axioms13010021
[4]	S. B. Chen, S. Rashid, M. A. Noor, Z. Hammouch, Y. M. Chu, New fractional approaches for $n$-polynomial $P$-convexity with applications in special function theory, Adv. Differ. Equ., 2020 (2020), 543. https://doi.org/10.1186/s13662-020-03000-5 doi: 10.1186/s13662-020-03000-5
[5]	S. B. Chen, S. Rashid, Z. Hammouch, M. A. Noor, R. Ashraf, Y. M. Chu, Integral inequalities via Raina's fractional integrals operator with respect to a monotone function, Adv. Differ. Equ., 2020 (2020), 647. https://doi.org/10.1186/s13662-020-03108-8 doi: 10.1186/s13662-020-03108-8
[6]	Y. M. Chu, T. H. Zhao, Concavity of the error function with respect to Hölder means, Math. Inequal. Appl., 19 (2016), 589–595. https://doi.org/10.7153/mia-19-43 doi: 10.7153/mia-19-43
[7]	S. S. Dragomir, A new refinement of Jensen's inequality in linear spaces with applications, Math. Comput. Model., 52 (2010), 1497–1505. https://doi.org/10.1016/j.mcm.2010.05.035 doi: 10.1016/j.mcm.2010.05.035
[8]	M. R. Delavar, S. S. Dragomir, On $\eta$-convexity, Math. Inequal. Appl., 20 (2017), 203–216. https://doi.org/10.7153/mia-20-14 doi: 10.7153/mia-20-14
[9]	S. S. Dragomir, C. Pearce, Selected topics on Hermite-Hadamard inequalities and applications, RGMIA Monographs, Victoria University, 2000. Available from: https://ssrn.com/abstract=3158351.
[10]	H. Hudzik, L. Maligranda, Some remarks on $s$-convex functions, Aequationes Math., 48 (1994), 100–111. https://doi.org/10.1007/BF01837981 doi: 10.1007/BF01837981
[11]	S. Ivelić, J. Pečarić, Generalizations of converse Jensen's inequality and related results, J. Math. Inequal., 5 (2011), 43–60.
[12]	Y. Khurshid, M. A. Khan, Y. M. Chu, Ostrowski type inequalities involving conformable integrals via preinvex functions, AIP Adv., 10 (2020), 055204. https://doi.org/10.1063/5.0008964 doi: 10.1063/5.0008964
[13]	M. A. Khan, S. Khan, S. Erden, M. Samraiz, A new approach for the derivation of bounds for the Jensen difference, Math. Methods Appl. Sci., 45 (2022), 36–48. https://doi.org/10.1002/mma.7757 doi: 10.1002/mma.7757
[14]	M. B. Khan, M. A. Noor, L. Abdullah, Y. M. Chu, Some new classes of preinvex fuzzy-interval-valued functions and inequalities, Int. J. Comput. Intell. Syst., 14 (2021), 1403–1418. https://doi.org/10.2991/ijcis.d.210409.001 doi: 10.2991/ijcis.d.210409.001
[15]	M. B. Khan, M. A. Noor, K. I. Noor, Y. M. Chu, Higher-order strongly preinvex fuzzy mappings and fuzzy mixed variational-like inequalities, Int. J. Comput. Intell. Syst., 14 (2021), 1856–1870. https://doi.org/10.2991/ijcis.d.210616.001 doi: 10.2991/ijcis.d.210616.001
[16]	M. A. Khan, A. Sohail, H. Ullah, T. Saeed, Estimations of the Jensen gap and their applications based on 6-convexity, Mathematics, 11 (2023), 1–25. https://doi.org/10.3390/math11081957 doi: 10.3390/math11081957
[17]	M. A. 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
[18]	V. Lakshmikantham, A. S. Vatsala, Theory of differential and integral inequalities with initial time difference and applications, In: Analytic and geometric inequalities and applications, Dordrecht: Springer, 1999, 191–203. https://doi.org/10.1007/978-94-011-4577-0_12
[19]	G. X. Lu, New refinements of Jensen's inequality and entropy upper bounds, J. Math. Inequal., 12 (2018), 403–421. https://doi.org/10.7153/jmi-2018-12-30 doi: 10.7153/jmi-2018-12-30
[20]	A. W. Marshall, I. Olkin, B. C. Arnold, Inequalities: theory of majorization and its applications, 2 Eds., New York: Springer, 2011. https://doi.org/10.1007/978-0-387-68276-1
[21]	R. Mikić, J. Pečarić, J. Pečarić, Inequalities of the Jensen and Edmundson-Lah-Ribarić type for 3-convex functions with applications, J. Math. Inequal., 12 (2018), 677–692. https://doi.org/10.7153/jmi-2018-12-52 doi: 10.7153/jmi-2018-12-52
[22]	J. Nicholson, Otto Hölder and the development of group theory and Galois theory, University of Oxford, 1993.
[23]	T. Rasheed, S. I. Butt, Ð. Pečarić. J. Pečarić, A. O. Akdemir, Uniform treatment of Jensen's inequality by Montgomery identity, J. Math., 2021 (2021), 5564647. https://doi.org/10.1155/2021/5564647 doi: 10.1155/2021/5564647
[24]	S. Rashid, H. Kalsoom, Z. Hammouch, R. Ashraf, D. Baleanu, Y. M. Chu, New multi-parametrized estimates having $pth$-order differentiability in fractional calculus for predominating $\hslash$-convex functions in Hilbert space, Symmetry, 12 (2020), 1–24. https://doi.org/10.3390/sym12020222 doi: 10.3390/sym12020222
[25]	S. Sezer, Z. Eken, G. Tinaztepe, G. Adilov, $P$-convex functions and some of their properties, Numer. Funct. Anal. Optim., 42 (2021), 443–459. https://doi.org/10.1080/01630563.2021.1884876 doi: 10.1080/01630563.2021.1884876
[26]	H. Ullah, M. A. Khan, J. Pečarić, New bounds for soft margin estimator via concavity of Guassian 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
[27]	H. Ullah, M. A. Khan, T. Saeed, Determination of bounds for the Jensen gap and its applications, Mathematics, 9 (2021), 1–29. https://doi.org/10.3390/math9233132 doi: 10.3390/math9233132
[28]	S. Varošanec, On $h$-convexity, J. Math. Anal. Appl., 326 (2007), 303–311. https://doi.org/10.1016/j.jmaa.2006.02.086 doi: 10.1016/j.jmaa.2006.02.086
[29]	C. C. White, D. P. Harrington, Application of Jensen's inequality to adaptive suboptimal design, J. Optim. Theory Appl., 32 (1980), 89–99. https://doi.org/10.1007/BF00934845 doi: 10.1007/BF00934845
[30]	X. X. You, M. A. Khan, H. Ullah, T. Saeed, Improvements of Slater's inequality by means of $4$-convexity and its applications, Mathematics, 10 (2022), 1–19. https://doi.org/10.3390/math10081274 doi: 10.3390/math10081274
[31]	T. H. Zhao, L. Shi, Y. M. Chu, Convexity and concavity of the modified Bessel functions of the first kind with respect to Hölder means, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. (RACSAM), 114 (2020), 1–14. https://doi.org/10.1007/s13398-020-00825-3 doi: 10.1007/s13398-020-00825-3
[32]	T. H. Zhao, M. K. Wang, Y. M. Chu, Monotonicity and convexity involving generalized elliptic integral of the first kind, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. (RACSAM), 115 (2021), 1–13. https://doi.org/10.1007/s13398-020-00992-3 doi: 10.1007/s13398-020-00992-3
[33]	T. H. Zhao, M. K. Wang, W. Zhang, Y. M. Chu, Quadratic transformation inequalities for Gaussian hypergeometric function, J. Inequal. Appl., 2018 (2018), 1–15. https://doi.org/10.1186/s13660-018-1848-y doi: 10.1186/s13660-018-1848-y

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)