Some novel Kulisch-Miranker type inclusions for a generalized class of Godunova-Levin stochastic processes

Waqar Afzal; Najla M. Aloraini; Mujahid Abbas; Jong-Suk Ro; Abdullah A. Zaagan; Waqar Afzal; Najla M. Aloraini; Mujahid Abbas; Jong-Suk Ro; Abdullah A. Zaagan

doi:10.3934/math.2024249

AIMS Mathematics

2024, Volume 9, Issue 2: 5122-5146. doi: 10.3934/math.2024249

Previous Article Next Article

Research article Special Issues

Some novel Kulisch-Miranker type inclusions for a generalized class of Godunova-Levin stochastic processes

1.
Department of Mathematics, Government College University, Katchery Road, Lahore 54000, Pakistan
2.
Department of Mathematics, College of Science, Qassim University, Buraydah 52571, Saudi Arabia
3.
Department of Medical Research, China Medical University, Taichung, Taiwan
4.
School of Electrical and Electronics Engineering, Chung-Ang University, Dongjak-gu, Seoul 06974, Republic of Korea
5.
Department of Intelligent Energy and Industry, Chung-Ang University, Dongjak-gu, Seoul 06974, Republic of Korea
6.
Department of Mathematics, College of Science, Jazan University, P.O. Box. 114, Jazan 45142, Saudi Arabia

Received: 08 December 2023 Revised: 09 January 2024 Accepted: 12 January 2024 Published: 25 January 2024
MSC : 26A48, 26A51, 33B10, 39A12, 39B62

Mathematical inequalities supporting interval-valued stochastic processes are rarely addressed. Recently, Afzal et al. introduced the notion of $ \mathtt{h} $-Godunova-Levin stochastic processes and developed Hermite-Hadamard and Jensen type inequalities in the setting of interval-valued functions. This note introduces a more generalized class of Godunova-Levin stochastic process that unifies several previously published results through the use of Kulisch-Miranker type order relations that are rarely discussed in relation to stochastic processes. Further, it is the first time that fractional version of Hermite-Hadamard inequality has been developed by using interval-valued stochastic processes in conjunction with a classical operator. Moreover, we give new modified forms for Ostrowski type results and present a new way to treat Jensen type inclusions under interval stochastic processes by using a discrete sequential form. We end with an open problem regarding Milne type results and discuss the importance of different types of order relations related to inequality terms in interval-valued settings.
- Hermite-Hadamard,
- Jensen,
- Ostrowski,
- stochastic process,
- Godunova-Levin,
- fractional operator,
- mathematical operators
Citation: Waqar Afzal, Najla M. Aloraini, Mujahid Abbas, Jong-Suk Ro, Abdullah A. Zaagan. Some novel Kulisch-Miranker type inclusions for a generalized class of Godunova-Levin stochastic processes[J]. AIMS Mathematics, 2024, 9(2): 5122-5146. doi: 10.3934/math.2024249

Related Papers:

Abstract

Mathematical inequalities supporting interval-valued stochastic processes are rarely addressed. Recently, Afzal et al. introduced the notion of $ \mathtt{h} $-Godunova-Levin stochastic processes and developed Hermite-Hadamard and Jensen type inequalities in the setting of interval-valued functions. This note introduces a more generalized class of Godunova-Levin stochastic process that unifies several previously published results through the use of Kulisch-Miranker type order relations that are rarely discussed in relation to stochastic processes. Further, it is the first time that fractional version of Hermite-Hadamard inequality has been developed by using interval-valued stochastic processes in conjunction with a classical operator. Moreover, we give new modified forms for Ostrowski type results and present a new way to treat Jensen type inclusions under interval stochastic processes by using a discrete sequential form. We end with an open problem regarding Milne type results and discuss the importance of different types of order relations related to inequality terms in interval-valued settings.

References

[1]	S. N. Malik, M. Raza, Q. Xin, J. Sokol, R. Manzoor, S. Zainab, On convex functions associated with symmetric cardioid domain, Symmetry, 13 (2021), 2321. https://doi.org/10.3390/sym13122321 doi: 10.3390/sym13122321
[2]	H. N. Shi, W. S. Du, Schur-power convexity of a completely symmetric function dual, Symmetry 11 (2019), 897. https://doi.org/10.3390/sym11070897 doi: 10.3390/sym11070897
[3]	M. Mnif, H. Pham, Stochastic optimization under constraints, Stoch. Proc. Appl., 93 (2001), 149–180. https://doi.org/10.1016/S0304-4149(00)00089-2 doi: 10.1016/S0304-4149(00)00089-2
[4]	A. Liu, V. K. Lau, B. Kananian, Stochastic successive convex approximation for non-convex constrained stochastic optimization, IEEE Trans. Signal Proc., 67 (2019), 4189–4203. https://doi.org/10.1109/TSP.2019.2925601 doi: 10.1109/TSP.2019.2925601
[5]	W. Afzal, M. Abbas, J. E. Macias-Diaz, S. Treanţă, Some $\mathtt{H}$-Godunova-Levin Function inequalities using center radius (Cr) order relation, Fractal Fract., 6 (2022), 518. https://doi.org/10.3390/fractalfract6090518 doi: 10.3390/fractalfract6090518
[6]	V. Preda, L. I. Catana, Tsallis Log-Scale-Location models. moments, Gini index and some stochastic orders, Mathematics, 9 (2021), 1216. https://doi.org/10.3390/math9111216 doi: 10.3390/math9111216
[7]	W. Afzal, A. A. Lupaş, K. Shabbir, Hermite-Hadamard and Jensen-type inequalities for harmonical $(\mathtt{h_1, h_2})$-Godunova Levin interval-valued functions, Mathematics, 10 (2022), 2970. https://doi.org/10.3390/math10162970 doi: 10.3390/math10162970
[8]	T. Saeed, W. Afzal, K. Shabbir, S. Treanţă, M. De la 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
[9]	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. http://dx.doi.org/10.3934/math.2023087 doi: 10.3934/math.2023087
[10]	R. E. Moore, Method and Applications of Interval Analysis, Philadelphia: Society for Industrial and Applied Mathematics, 1979.
[11]	W. Afzal, M. Abbas, W. Hamali, A. M. Mahnashi, M. D. Sen, Hermite-Hadamard-Type inequalities via Caputo-Fabrizio fractional integral for $h$-Godunova-Levin and $(h_1, h_2)$-Convex functions, Fractal Fract., 7 (2023), 687. https://doi.org/10.3390/fractalfract7090687 doi: 10.3390/fractalfract7090687
[12]	K. Nikodem, On convex stochastic processes, Aequat. Math., 20 (1980), 184–197.
[13]	A. Skowroński, On some properties of J-convex stochastic processes, Aequat. Math., 44 (1992), 249–258.
[14]	L. Li, Z. Hao, On Hermite-Hadmard inequlity for $h$-convex stochastic processes, Aequat. Math., 91 (2017), 909–920. https://doi.org/10.1007/s00010-017-0488-5 doi: 10.1007/s00010-017-0488-5
[15]	H. Budak, M. Z. Sarikaya, A new Hermite-Hadamard inequality for $h$-convex stochastic processes, RGMIA Res. Rep. Collect., 19 (2016), 30.
[16]	N. Okur, ˙I. ˙Işcan, E. Y. Dizdar, Hermite-Hadmard type inequalities for harmonically convex stochastic processes, Int. Econ. Adm. Stud., 18 (2018), 281–292.
[17]	L. González, D. Kotrys, K. Nikodem, Separation by convex and strongly convex stochastic processes, Publ. Math. Debrecen, 3 (2016), 365–372.
[18]	O. Almutairi, A. Kiliçman, Generalized Fejer-Hermite-Hadamard type via generalized $(h-m)$-convexity on fractal sets and applications, Chaos Solit. Fract., 147 (2021), 110938. https://doi.org/10.1016/j.chaos.2021.110938 doi: 10.1016/j.chaos.2021.110938
[19]	H. Zhou, M. S. Saleem, M. Ghafoor, J. Li, Generalization of h-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
[20]	H. Fu, M. S. Saleem, W. Nazeer, M. Ghafoor, P. Li, On Hermite-Hadamard type inequalities for $\eta$-polynomial convex stochastic processes, AIMS Math., 6 (2021), 6322–6339. http://dx.doi.org/10.3934/math.2021371 doi: 10.3934/math.2021371
[21]	M. Tunc, Ostrowski-type inequalities via h-convex functions with applications to special means, J. Inequal. Appl., 2013 (2013), 326. http://dx.doi.org/10.1186/1029-242X-2013-326 doi: 10.1186/1029-242X-2013-326
[22]	L. Gonzales, J. Materano, M. V. Lopez, Ostrowski-Type inequalities via $h$-convex stochastic processes, JP J. Math. Sci., 15 (2016), 15–29.
[23]	Y. An, G. Ye, D. Zhao, W. Liu, Hermite-Hadamard type inequalities for interval $(\mathtt h_1, \mathtt h_2)$-Convex functions, Mathematics, 7 (2019), 436. http://dx.doi.org/10.3390/math7050436 doi: 10.3390/math7050436
[24]	Y. Chalco-Cano, A. Flores-Franulič, H. Román-Flores, Ostrowski-type inequalities for interval-valued functions using generalized Hukuhara derivative, Comput. Appl. Math., 31 (2012), 457–472. http://dx.doi.org/10.1590/S1807-03022012000300002 doi: 10.1590/S1807-03022012000300002
[25]	L. Chen, M. S. Saleem, M. S. Zahoor, R. Bano, Some inequalities related to Interval-Valued $\eta_{h}$-Convex functions, J. Math., 2021 (2021), 6617074. http://dx.doi.org/10.1155/2021/6617074 doi: 10.1155/2021/6617074
[26]	H. Budak, A. Kashuri, S. Butt, Fractional Ostrowski type inequalities for interval valued functions, Filomat, 36 (2022), 2531–2540. https://doi.org/10.2298/FIL2208531B doi: 10.2298/FIL2208531B
[27]	H. Bai, M. S. Saleem, W. Nazeer, M. S. Zahoor, T. Zhao, Hermite-Hadamard- and Jensen-type inequalities for interval $(\mathtt h_1, \mathtt h_2)$ nonconvex function, J. Math., 2020 (2020), 3945384. https://doi.org/10.1155/2020/3945384 doi: 10.1155/2020/3945384
[28]	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. https://doi.org/10.1007/s00010-016-0425-z doi: 10.1007/s00010-016-0425-z
[29]	J. E. H. Hernandez, On $(m, \mathtt h_1, \mathtt h_2)$-G-convex dominated stochastic processes, Kragujev. J. Math., 46 (2022), 215–227.
[30]	M. Vivas-Cortez, C. Garcıa, Ostrowski type inequalities for functions whose derivatives are $(m, \mathtt h_1, \mathtt h_2)$-convex, Appl. Math. Inf. Sci., 11 (2017), 79–86. https://doi.org/10.18576/amis/110110 doi: 10.18576/amis/110110
[31]	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. http://dx.doi.org/10.3934/math.2023366 doi: 10.3934/math.2023366
[32]	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., 8 (2023), 13793–13794. http://dx.doi.org/10.3934/math.20221064 doi: 10.3934/math.20221064
[33]	X. 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
[34]	W. Afzal, M. Abbas, S. M. Eldin, Z. A. Khan, Some well known inequalities for $ (h_1, h_2) $-convex stochastic process via interval set inclusion relation, AIMS Math., 8 (2023), 19913–19932. http://dx.doi.org/10.3934/math.20231015 doi: 10.3934/math.20231015
[35]	T. Saeed, W. Afzal, M. Abbas, S. Treanţă, M. De la Sen, Some new generalizations of integral inequalities for Harmonical Cr-$(h_1, h_2)$-Godunova-Levin functions and applications, Mathematics, 10 (2022), 4540. https://doi.org/10.3390/math10234540 doi: 10.3390/math10234540
[36]	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. http://dx.doi.org/10.3934/math.2023683 doi: 10.3934/math.2023683
[37]	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 (2023), 3101–3119. http://dx.doi.org/10.3934/math.2023160 doi: 10.3934/math.2023160
[38]	W. Afzal, E. Prosviryakov, S. M. El-Deeb, Y. Almalki, Some new estimates of Hermite-Hadamard, 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
[39]	T. Du, Y. Peng, Hermite-Hadamard type inequalities for multiplicative Riemann-Liouville fractional integrals, J. Comput. Appl. Math., 440 (2014), 115582. https://doi.org/10.1016/j.cam.2023.115582 doi: 10.1016/j.cam.2023.115582
[40]	T. Du, T. Zhou, On the fractional double integral inclusion relations having exponential kernels via interval-valued co-ordinated convex mappings, Chaos Solit. Fract., 156 (2022), 111846. https://doi.org/10.1016/j.chaos.2022.111846 doi: 10.1016/j.chaos.2022.111846
[41]	T. Zhou, Z. Yuan, T. Du, On the fractional integral inclusions having exponential kernels for interval-valued convex functions, Math. Sci., 17 (2023), 107–120. https://doi.org/10.1007/s40096-021-00445-x doi: 10.1007/s40096-021-00445-x
[42]	T. Du, C. Luo, Z. Cao, On the Bullen-type inequalities via generalized fractional integrals and their application, Fractals, 29 (2021), 2150188. https://doi.org/10.1142/S0218348X21501887 doi: 10.1142/S0218348X21501887
[43]	U. Kulish, W. Miranker, Computer Arithmetic in Theory and Practice, New York: Academic Press, 2014.
[44]	F. Jarad, S. K. Sahoo, K. S. Nisar, S. Treanţă, H. Emadifar, T. Botmart, New stochastic fractional integral and related inequalities of Jensen-Mercer and Hermite-Hadamard-Mercer type for convex stochastic processes, J. Inequal. Appl., 2023 (2023), 51. https://doi.org/10.1186/s13660-023-02944-y doi: 10.1186/s13660-023-02944-y
[45]	X. Liu, G. Ye, D. Zhao, W. Liu, Fractional Hermite-Hadamard-type inequalities for interval-valued functions, J. Inequal. Appl., 2019 (2019), 266. https://doi.org/10.1186/s13660-019-2217-1 doi: 10.1186/s13660-019-2217-1
[46]	L. González, N. Merentes. M. Valera-López, Some estimates on the Hermite-Hadamard inequality through convex and quasi-convex stochastic processes, Math. Aeterna, 5 (2015), 745–767.
[47]	E. Set, M. Tomar, S. Maden, Hermite-Hadamard type inequalities for s-convex stochastic processes in the second sense, Turkish J. Anal. Number Theory, 2 (2014), 202–207. https://doi.org/10.12691/tjant-2-6-3 doi: 10.12691/tjant-2-6-3
[48]	D. Barráez, L. González, N. Merentes, A. Moros, On h-convex stochastic processes, Math. Aeterna, 5 (2015), 571–581.
[49]	B. Feng, M. Ghafoor, Y. M. Chu, M. I. Qureshi, X. Feng, C. Yao, et al., Hermite-Hadamard and Jensen's type inequalities for modified (p, h)-convex functions, AIMS Math., 5 (2020), 6959–6971. http://dx.doi.org/10.3934/math.2020446 doi: 10.3934/math.2020446
[50]	M. B. Khan, H. M. Srivastava, P. O. Mohammed, K. Nonlaopon, Y. S. Hamed, Some new estimates on coordinates of left and right convex interval-valued functions based on pseudo order relation, Symmetry, 14 (2022), 473. https://doi.org/10.3390/sym14030473 doi: 10.3390/sym14030473
[51]	M. B. Khan, M. A. Noor, K. I. Noor, Y. M. Chu, New Hermite-Hadamard-type inequalities for-convex fuzzy-interval-valued functions, Adv. Differ. Equ., 2021 (2021), 149. https://doi.org/10.1186/s13662-021-03245-8 doi: 10.1186/s13662-021-03245-8
[52]	Y. Almalki, W. Afzal, Some new estimates of Hermite-Hadamard inequalities for harmonical Cr-h-convex functions via generalized fractional integral operator on set-valued mappings, Mathematics, 11 (2023), 4041. https://doi.org/10.3390/math11194041 doi: 10.3390/math11194041
[53]	M. B. Khan, M. A. Noor, M. Noor, M. Al‐Shomrani, L. Abdullah, Some novel inequalities for LR-h-convex interval‐valued functions by means of pseudo‐order relation, Math. Meth. App Sci., 45 (2022), 1310–1340. https://doi.org/10.1002/mma.7855 doi: 10.1002/mma.7855
[54]	W. Liu, F. Shi, G. Ye, D. Zhao, Some inequalities for Cr-Log-h- convex functions, J. Inequal. Appl., 2022 (2022), 160. https://doi.org/10.1186/s13660-022-02900-2 doi: 10.1186/s13660-022-02900-2
[55]	W. Afzal, K. Shabbir, M. Arshad, J. K. K. Asamoah, A. M. Galal, Some novel estimates of integral inequalities for a generalized class of harmonical convex mappings by means of center-radius order relation, J. Math., 2023 (2023), 8865992. https://doi.org/10.1155/2023/8865992 doi: 10.1155/2023/8865992
[56]	M. B. Khan, G. Santos-García, M. A. Noor, M. S. Soliman, Some new concepts related to fuzzy fractional calculus for up and down convex fuzzy-number valued functions and inequalities, Chaos Solit. Fract., 164 (2022), 112692. https://doi.org/10.1016/j.chaos.2022.112692 doi: 10.1016/j.chaos.2022.112692
[57]	H. Román-Flores, V. Ayala, A. Flores-Franulič, Milne type inequality and interval orders, Comput. Appl. Math., 40 (2021), 130. https://doi.org/10.1007/s40314-021-01500-y doi: 10.1007/s40314-021-01500-y
[58]	M. Abbas, W. Afzal, T. Botmart, A. M. Galal, Ostrowski and Hermite-Hadamard type inequalities for $ h $-convex stochastic processes by means of center-radius order relation, AIMS Math., 8 (2023), 16013–16030. http://dx.doi.org/10.3934/math.2023817 doi: 10.3934/math.2023817

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)