Some novel estimates of Jensen and Hermite-Hadamard inequalities for h-Godunova-Levin stochastic processes

Waqar Afzal; Thongchai Botmart; Waqar Afzal; Thongchai Botmart

doi:10.3934/math.2023366

AIMS Mathematics

2023, Volume 8, Issue 3: 7277-7291. doi: 10.3934/math.2023366

Previous Article Next Article

Research article

Some novel estimates of Jensen and Hermite-Hadamard inequalities for h-Godunova-Levin stochastic processes

Waqar Afzal ^1,2,
Thongchai Botmart ^{3
,
,}

1.
Department of Mathemtics, Government College University Lahore (GCUL), Lahore 54000, Pakistan
2.
Department of Mathematics, University of Gujrat, Gujrat 50700, Pakistan
3.
Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand

Received: 14 August 2022 Revised: 02 December 2022 Accepted: 09 December 2022 Published: 12 January 2023
MSC : 26A48, 26A51, 33B10, 39A12, 39B62

It is undeniable that convex and non-convex functions play an important role in optimization. As a result of its behavior, convexity also plays a significant role in discussing inequalities. It is clear that convexity and stochastic processes are intertwined. The stochastic process is a mathematical model that describes how systems or phenomena fluctuate randomly. Probability theory generally says that the convex function applied to the expected value of a random variable is bounded above by the expected value of the random variable's convex function. Furthermore, the deep connection between convex inequalities and stochastic processes offers a whole new perspective on the study of inequality. Although Godunova-Levin functions are well known in convex theory, their properties enable us to determine inequality terms with greater accuracy than those obtained from convex functions. In this paper, we established a more refined form of Hermite-Hadamard and Jensen type inequalities for generalized interval-valued h-Godunova-Levin stochastic processes. In addition, we provide some examples to demonstrate the validity of our main findings.
- Jensen inequality,
- Hermite-Hadamard inequality,
- Godunova-Levin stochastic process,
- interval valued functions
Citation: Waqar Afzal, Thongchai Botmart. Some novel estimates of Jensen and Hermite-Hadamard inequalities for h-Godunova-Levin stochastic processes[J]. AIMS Mathematics, 2023, 8(3): 7277-7291. doi: 10.3934/math.2023366

Related Papers:

Abstract

It is undeniable that convex and non-convex functions play an important role in optimization. As a result of its behavior, convexity also plays a significant role in discussing inequalities. It is clear that convexity and stochastic processes are intertwined. The stochastic process is a mathematical model that describes how systems or phenomena fluctuate randomly. Probability theory generally says that the convex function applied to the expected value of a random variable is bounded above by the expected value of the random variable's convex function. Furthermore, the deep connection between convex inequalities and stochastic processes offers a whole new perspective on the study of inequality. Although Godunova-Levin functions are well known in convex theory, their properties enable us to determine inequality terms with greater accuracy than those obtained from convex functions. In this paper, we established a more refined form of Hermite-Hadamard and Jensen type inequalities for generalized interval-valued h-Godunova-Levin stochastic processes. In addition, we provide some examples to demonstrate the validity of our main findings.

References

[1]	R. E. Moore, Interval analysis, Englewood Cliffs: Prentice-Hall, 1966.
[2]	J. M. Snyder, Interval analysis for computer graphics, Proceedings of the 19th annual conference on computer graphics and interactive techniques, 1992,121–130.
[3]	N. A. Gasilov, Ş. Emrah Amrahov, Solving a nonhomogeneous linear system of interval differential equations, Soft Comput., 22 (2018), 3817–3828.
[4]	D. Singh, B. A. Dar, Sufficiency and duality in non-smooth interval valued programming problems, J. Ind. Manag. Optim., 15 (2019), 647–665. https://doi.org/10.3934/jimo.2018063 doi: 10.3934/jimo.2018063
[5]	E. de Weerdt, Q. P. Chu, J. A. Mulder, Neural network output optimization using interval analysis, IEEE T. Neural Networ., 20 (2009), 638–653. http://doi.org/10.1109/TNN.2008.2011267 doi: 10.1109/TNN.2008.2011267
[6]	S. S. Dragomir, J. Pecaric, L. E. Persson, Some inequalities of Hadamard type, Soochow J. Math., 21 (1995), 335–341.
[7]	A. Almutairi, A. Kılıçman, New refinements of the Hadamard inequality on coordinated convex function, J. Inequal. Appl., 2019 (2019), 192. https://doi.org/10.1186/s13660-019-2143-2 doi: 10.1186/s13660-019-2143-2
[8]	X. N. Leng, T. Feng, X. Z. Meng. Stochastic inequalities and applications to dynamics analysis of a novel sivs epidemic model with jumps, J. Inequal. Appl., 2017 (2017), 138. https://doi.org/10.1186/s13660-017-1418-8 doi: 10.1186/s13660-017-1418-8
[9]	S. S. Dragomir, Inequalities of Hermite-Hadamard type for functions of selfadjoint operators and matrices, J. Math. Inequal., 11 (2017), 241–259. http://doi.org/10.7153/jmi-11-23 doi: 10.7153/jmi-11-23
[10]	K. Nikodem, On convex stochastic processes, Aeq. Math., 20 (1980), 184–197. http://doi.org/10.1007/BF02190513 doi: 10.1007/BF02190513
[11]	A. Skowronski, On some properties ofj-convex stochastic processes, Aeq. Math., 44 (1992), 249–258. http://doi.org/10.1007/BF01830983 doi: 10.1007/BF01830983
[12]	D. Kotrys, Hermite–Hadamard inequality for convex stochastic processes, Aequat. Math., 83 (2012), 143–152. http://doi.org/10.1007/s00010-011-0090-1 doi: 10.1007/s00010-011-0090-1
[13]	D. Kotrys, Remarks on strongly convex stochastic processes, Aequat. Math., 86 (2013), 91–98. http://doi.org/10.1007/s00010-012-0163-9 doi: 10.1007/s00010-012-0163-9
[14]	H. Agahi, Refinements of mean-square stochastic integral inequalities on convex stochastic processes, Aequat. Math., 90 (2016), 765–772. http://doi.org/10.1007/s00010-015-0378-7 doi: 10.1007/s00010-015-0378-7
[15]	H. Agahi, A. Babakhani, On fractional stochastic inequalities related to Hermite–Hadamard and Jensen types for convex stochastic processes, Aequat. Math., 90 (2016), 1035–1043. http://doi.org/10.1007/s00010-016-0425-z doi: 10.1007/s00010-016-0425-z
[16]	W. Afzal, K. Shabbir, T. Botmart, S. Treanţă, Some new estimates of well known inequalities for $(h_1, h_2)$-Godunova-Levin functions by means of center-radius order relation, AIMS Mathematics, 8 (2022), 3101–3119. https://doi.org/10.3934/math.2023160 doi: 10.3934/math.2023160
[17]	H. Agahi, M. Yadollahzadeh, On stochastic pseudo-integrals with applications, Stat. Probabil. Lett., 124 (2017), 41–48. https://doi.org/10.1016/j.spl.2017.01.001 doi: 10.1016/j.spl.2017.01.001
[18]	F. Ma, W. Nazeer, M. Ghafoor, Hermite-Hadamard, Jensen, and fractional integral inequalities for generalized-convex stochastic processes, J. Math., 2021 (2021), 5524780. http://doi.org/10.1155/2021/5524780 doi: 10.1155/2021/5524780
[19]	W. Afzal, K. Shabbir, S. Treanţă, K. Nonlaopon, Jensen and Hermite-Hadamard type inclusions for harmonical h-Godunova-Levin functions, AIMS Mathematics, 8 (2022), 3303–3321. https://doi.org/10.3934/math.2023170 doi: 10.3934/math.2023170
[20]	H. Zhou, MS. Saleem, M. Ghafoor, J. J. Li, Generalization of-convex stochastic processes and some classical inequalities, Math. Probl. Eng., 2020 (2020), 1583807.
[21]	T. Saeed, W. Afzal, K. Shabbir, S. Treanţă, M. D. Sen, Some novel estimates of Hermite-Hadamard and Jensen type inequalities for $(h_1, h_2)$-convex functions pertaining to total order relation, Mathematics, 10 (2022), 4777. https://doi.org/10.3390/math10244777 doi: 10.3390/math10244777
[22]	W. Afzal, A. A. Lupaş, K. Shabbir, Hermite-Hadamard and Jensen-type inequalities for harmonical $(h_1, h_2)$-Godunova Levin interval-valued functions, Mathematics, 10 (2022), 2970. https://doi.org/10.3390/math10162970 doi: 10.3390/math10162970
[23]	W. Afzal, W. Nazeer, T. Botmart, S. Treanţă, Some properties and inequalities for generalized class of harmonical Godunova-Levin function via center radius order relation, AIMS Mathematics, 8 (2023), 1696–1712. https://doi.org/10.3934/math.2023087 doi: 10.3934/math.2023087
[24]	S. Varoşanec, On h-convexity, J Math. Anal. Appl., 326 (2007), 303–311.
[25]	M. Bombardelli, S. Varoşanec, Properties of h-convex functions related to the Hermite–Hadamard–Fejér inequalities, Comput. Math. Appl., 58 (2009), 1869–1877. https://doi.org/10.1016/j.camwa.2009.07.073 doi: 10.1016/j.camwa.2009.07.073
[26]	W. Afzal, M. Abbas, J. E. Macias-Diaz, S. Treanţă, Some h-Godunova–Levin function inequalities using center radius (Cr) order, Fractal Fract., 6 (2022), 518. https://doi.org/10.3390/fractalfract6090518 doi: 10.3390/fractalfract6090518
[27]	W. Afzal, K. Shabbir, T. Botmart, Generalized version of Jensen and Hermite-Hadamard inequalities for interval-valued $(h_1, h_2)$-Godunova-Levin functions, AIMS Mathematics, 7 (2022), 19372–19387. https://doi.org/10.3934/math.20221064 doi: 10.3934/math.20221064
[28]	X. J. Zhang, K. Shabbir, W. Afzal, H. Xiao, D. Lin, Hermite Hadamard and Jensen-type inequalities via Riemann integral operator for a generalized class of Godunova Levin functions, J. Math, 2022 (2022), 3830324. http://dx.doi.org/10.1155/2022/3830324 doi: 10.1155/2022/3830324
[29]	T. Saeed, W. Afzal, M. Abbas, S. Treanţă, M. D. Sen, Some new generalizations of integral inequalities for harmonical cr-$(h_1, h_2)$-Godunova Levin functions and applications, Mathematics, 10 (2022), 4540. https://doi.org/10.3390/math10234540 doi: 10.3390/math10234540
[30]	S. S. Dragomir, Inequalities of Hermite-Hadamard type for h-convex functions on linear spaces, Proyecciones J. Math., 34 (2015), 323–341.
[31]	D. Barráez, L. González, N. Merentes, On h-convex stochastic processes, Math. Aeterna, 5 (2015), 571–581.
[32]	D. F. Zhao, T. Q. An, G. J. Ye, W. Liu, New jensen and Hermite-Hadamard type inequalities for $h$-convex interval-valued functions, J. Inequal. Appl., 2018 (2018), 302. https://doi.org/10.1186/s13660-018-1896-3 doi: 10.1186/s13660-018-1896-3
[33]	O. Almutairi, A. Kiliıcman, Some integral inequalities for $h$-Godunova-Levin preinvexity, Symmetry, 11 (2019), 1500. https://doi.org/10.3390/sym11121500 doi: 10.3390/sym11121500
[34]	A. Rakhlin, O. Shamir, K. Sridharan, Making gradient descent optimal for strongly convex stochastic optimization, 2011. Available from: https://arXiv.org/abs/1109.5647
[35]	H. Q. Jin, Z. Q. Xu, X. Y. Zhou, A convex stochastic optimization problem arising from portfolio selection, Int. J. Math.,, 18 (2008), 171–183. https://doi.org/10.1111/j.1467-9965.2007.00327.x

Reader Comments

Your name:*

Email:*
© 2023 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)