Jensen, Ostrowski and Hermite-Hadamard type inequalities for $ h $-convex stochastic processes by means of center-radius order relation

Mujahid Abbas; Waqar Afzal; Thongchai Botmart; Ahmed M. Galal; Mujahid Abbas; Waqar Afzal; Thongchai Botmart; Ahmed M. Galal

doi:10.3934/math.2023817

AIMS Mathematics

2023, Volume 8, Issue 7: 16013-16030. doi: 10.3934/math.2023817

Previous Article Next Article

Research article

Jensen, Ostrowski and Hermite-Hadamard type inequalities for $ h $-convex stochastic processes by means of center-radius order relation

1.
Department of Mathemtics, Government College University Lahore (GCUL), Lahore 54000, Pakistan
2.
Department of Medical Research, China Medical University, Taichung, Taiwan
3.
Department of Mathematics and Applied Mathematics, University of Pretoria, Lynnwood road, Pretoria 0002, South Africa
4.
Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand
5.
Department of Mechanical Engineering, College of Engineering in Wadi Alddawasir, Prince Sattam bin Abdulaziz University, Saudi Arabia
6.
Production Engineering and Mechanical Design Department, Faculty of Engineering, Mansoura University, P.O. 35516, Mansoura, Egypt

Received: 27 October 2022 Revised: 20 January 2023 Accepted: 06 February 2023 Published: 04 May 2023
MSC : 26A48, 26A51, 33B10, 39A12, 39B62

In optimization, convex and non-convex functions play an important role. Further, there is no doubt that convexity and stochastic processes are closely related. In this study, we introduce the notion of the $ h- $convex stochastic process for center-radius order in the setting of interval-valued functions ($ \mathcal{IVFS} $) which is novel in literature. By using these notions we establish Jensen, Ostrowski, and Hermite-Hadamard ($ \mathcal{H.H} $) types inequalities for generalized interval-valued $ \mathcal{CR}-h $-convex stochastic processes. Furthermore, the study provides useful examples to support its findings.
- Hermite-Hadamard,
- Ostrowski and Jensen inequalities,
- $ h $-convexity,
- stochastic $ h $-convex
Citation: Mujahid Abbas, Waqar Afzal, Thongchai Botmart, Ahmed M. Galal. Jensen, Ostrowski and Hermite-Hadamard type inequalities for $ h $-convex stochastic processes by means of center-radius order relation[J]. AIMS Mathematics, 2023, 8(7): 16013-16030. doi: 10.3934/math.2023817

Related Papers:

Abstract

In optimization, convex and non-convex functions play an important role. Further, there is no doubt that convexity and stochastic processes are closely related. In this study, we introduce the notion of the $ h- $convex stochastic process for center-radius order in the setting of interval-valued functions ($ \mathcal{IVFS} $) which is novel in literature. By using these notions we establish Jensen, Ostrowski, and Hermite-Hadamard ($ \mathcal{H.H} $) types inequalities for generalized interval-valued $ \mathcal{CR}-h $-convex stochastic processes. Furthermore, the study provides useful examples to support its 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. E. 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]	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
[7]	H. Budak, T. Tunç, M. Sarikaya, Fractional Hermite-Hadamard-type inequalities for interval-valued functions, P. Am. Math. Soc., 148 (2020), 705–718. https://doi.org/10.1090/proc/14741 doi: 10.1090/proc/14741
[8]	S. Rashid, H. Kalsoom, Z. Hammouch, R. Ashraf, New multi-parametrized estimates having pth-order differentiability in fractional calculus for predominating $h$-convex functions in Hilbert space, Symmetry, 12 (2020), 222. https://doi.org/10.3390/sym12020222 doi: 10.3390/sym12020222
[9]	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. https://doi.org/10.1155/2022/3830324 doi: 10.1155/2022/3830324
[10]	B. Feng, M. Ghafoor, Y. M. Chu, M. I. Qureshi, X. Feng, Hermite-Hadamard and Jensen's type inequalities for modified $(p, h)$-convex functions, AIMS Math., 6 (2029), 6959–6971. https://doi.org/10.3934/math.2020446 doi: 10.3934/math.2020446
[11]	C. Park, Y. M. Chu, M. S. Saleem, Hermite-Hadamard-type inequalities for $\eta_h$-convex functions via $\Psi$-Riemann-Liouville fractional integrals, Adv. Cont. Disc. Model., 1 (2022), 1–8. https://doi.org/10.1186/s13662-022-03745-1 doi: 10.1186/s13662-022-03745-1
[12]	P. Y. Yan, Q. Li, Y. M. Chu, S. Mukhtar, S. Waheed, On some fractional integral inequalities for generalized strongly modified $h$-convex function, AIMS Math., 5 (2020), 6620–6638. https://doi.org/10.3934/math.2020426 doi: 10.3934/math.2020426
[13]	M. A. Ali, H. Budak, G. Murtaza, Y. M. Chu, Post-quantum Hermite-Hadamard type inequalities for interval-valued convex functions, J. Inequal. Appl., 1 (2021), 1–18. https://doi.org/10.1186/s13660-021-02619-6 doi: 10.1186/s13660-021-02619-6
[14]	H. Kara, M. A. Ali, H. Budak, Hermite-Hadamard-Mercer type inclusions for interval-valued functions via Riemann-Liouville fractional integrals, Turk. J. Math., 6 (2022), 2193–2207. https://doi.org/10.55730/1300-0098.3263 doi: 10.55730/1300-0098.3263
[15]	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 Math., 7 (2022), 19372–19387. https://doi.org/10.3934/math.20221064 doi: 10.3934/math.20221064
[16]	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
[17]	I. A. Baloch, Y. M. Chu, Petrovic-type inequalities for harmonic-convex functions, J. Funct. Space., 2020 (2020), 3075390. https://doi.org/10.1155/2020/3075390 doi: 10.1155/2020/3075390
[18]	E. R. Nwaeze, M. A. Khan, Y. M. Chu, Fractional inclusions of the Hermite-Hadamard type for m-polynomial convex interval-valued functions, Adv. Differ. Equ., 1 (2020), 1–17. https://doi.org/10.1186/s13662-020-02977-3 doi: 10.1186/s13662-020-02977-3
[19]	H. Kara, H. Budak, M. A. Ali, Weighted Hermite-Hadamard type inclusions for products of co-ordinated convex interval-valued functions, Adv. Differ. Equ., 1 (2021), 1–16. https://doi.org/10.1186/s13662-021-03261-8 doi: 10.1186/s13662-021-03261-8
[20]	T. Abdeljawad, S. Rashid, H. Khan, Y. M. Chu, New Hermite-Hadamard-type inequalities for-convex fuzzy-interval-valued functions, Adv. Differ. Equ., 1 (2021), 1–20. https://doi.org/10.1186/s13662-020-02782-y doi: 10.1186/s13662-020-02782-y
[21]	M. B. Khan, M. A. Noor, K. I. Noor, Y. M. Chu, On new fractional integral inequalities for p-convexity within interval-valued functions, Adv. Differ. Equ., 1 (2020), 1–17. https://doi.org/10.1186/s13662-021-03245-8 doi: 10.1186/s13662-021-03245-8
[22]	G. Sana, M. B. Khan, M. A. Noor, Y. M. Chu, Harmonically convex fuzzy-interval-valued functions and fuzzy-interval Riemann-Liouville fractional integral inequalities, Int. J. Comput. Intell. Syst., 14 (2021), 1809–1822. https://doi.org/10.2991/ijcis.d.210620.001 doi: 10.2991/ijcis.d.210620.001
[23]	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., 1 (2021), 1401–1418. https://dx.doi.org/10.2991/ijcis.d.210409.001 doi: 10.2991/ijcis.d.210409.001
[24]	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
[25]	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
[26]	V. Stojiljkovic, Hermite Hadamard type inequalities involving $(kp)$ fractional operator with ($\alpha$, h- m)- p convexity, Eur. J. Pure. Appl. Math., 16 (2023), 503–522. https://doi.org/10.29020/nybg.ejpam.v16i1.4689 doi: 10.29020/nybg.ejpam.v16i1.4689
[27]	V. Stojiljkovic, A new conformable fractional derivative and applications, Seleccion. Mat., 9 (2022), 370–380. http://dx.doi.org/10.17268/sel.mat.2022.02.12 doi: 10.17268/sel.mat.2022.02.12
[28]	G. Mani, R. Ramaswamy, A. J. Gnanaprakasam, V. Stojiljkovic, Z. M. Fadail, S. Radenovic, Application of fixed point results in the setting of F-contraction and simulation function in the setting of bipolar metric space, AIMS Math., 8 (2023), 3269–3285. http://dx.doi.org/2010.3934/math.2023168
[29]	V. Stojiljković, R. Ramaswamy, O. A. A. Abdelnaby, S. Radenovic, Riemann-Liouville fractional inclusions for convex functions using interval valued setting, Mathematics, 10 (2022), 3491. https://doi.org/10.3390/math10193491 doi: 10.3390/math10193491
[30]	W. Afzal, K. Shabbir, S. Treanţă, K. Nonlaopon, Jensen and Hermite-Hadamard type inclusions for harmonical h-Godunova-Levin functions, AIMS Math., 8 (2022), 3303–3321. https://doi.org/10.3934/math.2023170 doi: 10.3934/math.2023170
[31]	K. Nikodem, On convex stochastic processes, Aequationes Math., 2 (1998), 427–446. https://dx.doi.org/10.1007/BF02190513 doi: 10.1007/BF02190513
[32]	M. Shaked, J. G. Shanthikumar, Stochastic convexity and its applications, Adv. Appl. Probab., 1 (1980), 184–197. https://dx.doi.org/10.1006ADA170112
[33]	A. Skowronski, On some properties of $j$-convex stochastic processes, Aequationes Math., 2 (1992), 249–258. https://dx.doi.org/10.1007/BF01830983 doi: 10.1007/BF01830983
[34]	D. Kotrys, Hermite-Hadamard inequality for convex stochastic processes, Aequationes Math., 83 (2012), 143–151. https://dx.doi.org/10.1007/s00010-011-0090-1 doi: 10.1007/s00010-011-0090-1
[35]	S. Varoşanec, On $h-$convexity, J. Math. Anal. Appl., 326 (2007), 303–311. https://dx.doi.org/10.1016/j.jmaa.2006.02.086
[36]	D. Barraez, L. Gonzalez, N. Merentes, On $h$-convex stochastic processes, Math. Aeterna, 5 (2015), 571–581.
[37]	J. El-Achky, S. Taoufiki, On $(p-h)$-convex stochastic processes, J. Interdiscip. Math., 2 (2022), 1–12. https://doi.org/10.1080/09720502.2021.1938994 doi: 10.1080/09720502.2021.1938994
[38]	W. Afzal, T. Botmart, Some novel estimates of Jensen and Hermite-Hadamard inequalities for $h$-Godunova-Levin stochastic processes, AIMS Math., 8 (2023), 7277–7291. https://doi.org/10.3934/math.2023366 doi: 10.3934/math.2023366
[39]	M. Vivas-Cortez, M. S. Saleem, S. Sajid, Fractional version of Hermite-Hadamard-Mercer inequalities for convex stochastic processes via $\Psi_k$-Riemann-Liouville fractional integrals and its applications, Appl. Math., 16 (2022), 695–709. http://dx.doi.org/10.18576/amis/22nuevoformat20(1)2 doi: 10.18576/amis/22nuevoformat20(1)2
[40]	W. Afzal, E. Y. Prosviryakov, S. M. El-Deeb, Y. Almalki, Some new estimates of HermiteHadamard, Ostrowski and Jensen-type inclusions for $h$-convex stochastic process via interval-valued functions, Symmetry., 15 (2023), 831. https://doi.org/10.3390/sym15040831 doi: 10.3390/sym15040831
[41]	J. El-Achky, D. Gretete, M. Barmaki, Inequalities of Hermite-Hadamard type for stochastic process whose fourth derivatives absolute are quasi-convex, $P$-convex, $s$-convex and $h$-convex, J. Interdiscip. Math., 3 (2021), 1–17. https://doi.org/10.1080/09720502.2021.1887607 doi: 10.1080/09720502.2021.1887607
[42]	N. Sharma, R. Mishra, A. Hamdi, Hermite-Hadamard type integral inequalities for multidimensional general $h$-harmonic preinvex stochastic processes, Commun. Stat.-Theor. M., 4 (2020), 1–41. https://doi.org/10.1080/03610926.2020.1865403 doi: 10.1080/03610926.2020.1865403
[43]	H. Zhou, M. S. Saleem, M. Ghafoor, J. Li, Generalization of-convex stochastic processes and some classical inequalities, Math. Probl. Eng., 2020 (2020), 1583807. https://doi.org/10.1155/2020/1583807 doi: 10.1155/2020/1583807
[44]	W. Afzal, S. M. Eldin, W. Nazeer, A. M. Galal, Some integral inequalities for harmonical $cr$-$h$-Godunova-Levin stochastic processes, AIMS Math., 8 (2023), 13473–13491. https://doi.org/10.3934/math.2023683 doi: 10.3934/math.2023683
[45]	H. Budak, M. Z. Sarikaya, On generalized stochastic fractional integrals and related inequalities, Theor. Appl., 5 (2018), 471–481. https://doi.org/10.15559/18-VMSTA117 doi: 10.15559/18-VMSTA117
[46]	M. Tunc, Ostrowski-type inequalities via h-convex functions with applications to special means, J. Inequal. Appl., 1 (2013), 1–10. https://doi.org/10.1186/1029-242X-2013-326 doi: 10.1186/1029-242X-2013-326
[47]	L. Gonzales, J. Materano, M. V. Lopez, Ostrowski-type inequalities via h-convex stochastic processes, JP J. Math. Sci., 6 (2013), 15–29.
[48]	A. K. Bhunia, S. S. Samanta, A study of interval metric and its application in multi-objective optimization with interval objectives, Comput. Ind. Eng., 74 (2014), 169–178. https://doi.org/10.3390/math10122089 doi: 10.3390/math10122089
[49]	W. Liu, F. Shi, G. Ye, D. Zhao, The properties of harmonically $cr$-$h$-convex function and its applications, Mathematics, 10 (2022), 2089. https://doi.org/10.1016/j.cie.2014.05.014 doi: 10.1016/j.cie.2014.05.014
[50]	W. Afzal, M. Abbas, J. E. Macias-Diaz, S. Treanţă, Some H-Godunova-Levin unction inequalities using center radius (Cr) order, Fractal Fract., 6 (2022), 518. https://doi.org/10.3390/fractalfract6090518 doi: 10.3390/fractalfract6090518
[51]	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 Math., 8 (2023), 1696–1712. https://doi.org/10.3934/math.2023087 doi: 10.3934/math.2023087
[52]	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 Math., 8 (2022), 3101–3119. https://doi.org/10.3934/math.2023160 doi: 10.3934/math.2023160
[53]	P. Cerone, S. S. Dragomir, Ostrowski type inequalities for functions whose derivatives satisfy certain convexity assumptions, Demonstr. Math., 37 (2004), 299–308. https://doi.org/10.3390/fractalfract6090518 doi: 10.3390/fractalfract6090518

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)