Some well known inequalities for $ (h_1, h_2) $-convex stochastic process via interval set inclusion relation

Waqar Afzal; Mujahid Abbas; Sayed M. Eldin; Zareen A. Khan; Waqar Afzal; Mujahid Abbas; Sayed M. Eldin; Zareen A. Khan

doi:10.3934/math.20231015

AIMS Mathematics

2023, Volume 8, Issue 9: 19913-19932. doi: 10.3934/math.20231015

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Some well known inequalities for $ (h_1, h_2) $-convex stochastic process via interval set inclusion 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.
Center of Research, Faculty of Engineering and Technology, Future University in Egypt, New Cairo 11835, Egypt
5.
Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia

Received: 15 February 2023 Revised: 31 May 2023 Accepted: 06 June 2023 Published: 14 June 2023
MSC : 26A48, 26A51, 33B10, 39A12, 39B62

This note introduces the concept of $ (h_1, h_2) $-convex stochastic processes using interval-valued functions. First we develop Hermite-Hadmard $ (\mathbb{H.H}) $ type inequalities, then we check the results for the product of two convex stochastic process mappings, and finally we develop Ostrowski and Jensen type inequalities for $ (h_1, h_2) $-convex stochastic process. Also, we have shown that this is a more generalized and larger class of convex stochastic processes with some remark. Furthermore, we validate our main findings by providing some non-trivial examples.
- Hermite-Hadamard inequality,
- Ostrowski inequality,
- Jensen inequality,
- stochastic process,
- interval valued functions
Citation: Waqar Afzal, Mujahid Abbas, Sayed M. Eldin, Zareen A. Khan. Some well known inequalities for $ (h_1, h_2) $-convex stochastic process via interval set inclusion relation[J]. AIMS Mathematics, 2023, 8(9): 19913-19932. doi: 10.3934/math.20231015

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Abstract

This note introduces the concept of $ (h_1, h_2) $-convex stochastic processes using interval-valued functions. First we develop Hermite-Hadmard $ (\mathbb{H.H}) $ type inequalities, then we check the results for the product of two convex stochastic process mappings, and finally we develop Ostrowski and Jensen type inequalities for $ (h_1, h_2) $-convex stochastic process. Also, we have shown that this is a more generalized and larger class of convex stochastic processes with some remark. Furthermore, we validate our main findings by providing some non-trivial examples.

References

[1]	R. E. Moore, Interval analysis, Englewood Cliffs, Prentice-Hall, 1966.
[2]	N. A. Gasilov, Ş. E. Amrahov, Solving a nonhomogeneous linear system of interval differential equations, Soft Comput., 22 (2018), 3817–3828.
[3]	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
[4]	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
[5]	K. Nikodem, On convex stochastic processes, Aequationes Math., 2 (1998), 427–446. https://dx.doi.org/10.1007/BF02190513 doi: 10.1007/BF02190513
[6]	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
[7]	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
[8]	N. Okur, I. Işcan, E. Y. Dizdar, Hermite-Hadamard type inequalities for p-convex stochastic processes, Int. J. Optim. Control Theor. Appl., 9 (2019), 148–153. https://doi.org/10.11121/ijocta.01.2019.00602 doi: 10.11121/ijocta.01.2019.00602
[9]	E. Set, M. Tomar, S. Maden, Hermite-Hadamard type inequalities for s-convex stochastic processes in the second sense, Turk. J. Anal. Number Theor., 2 (2014), 202–207. https://dx.doi.org/10.12691/tjant-2-6-3 doi: 10.12691/tjant-2-6-3
[10]	H. Budak, M. Z. Sarikaya, A new Hermite-Hadamard inequality for $h$-convex stochastic processes, RGMIA Res. Rep. Collect., 19 (2016), 30. http://dx.doi.org/10.20852/ntmsci.2019.376 doi: 10.20852/ntmsci.2019.376
[11]	D. Barraez, L. Gonzalez, N. Merentes, On $h$-convex stochastic processes, Math. Aeterna, 5 (2015), 571–581.
[12]	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
[13]	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
[14]	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
[15]	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
[16]	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
[17]	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
[18]	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
[19]	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
[20]	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
[21]	M. Abbas, W. Afzal, T. Botmart, A. M. Galal, Jensen, Ostrowski and Hermite-Hadamard type inequalities for -convex stochastic processes by means of center-radius order relation, AIMS Math., 8 (2023), 16013–16030.. http://dx.doi.org/2010.3934/math.2023817
[22]	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
[23]	L. Gonzales, J. Materano, M. V. Lopez, Ostrowski-type inequalities via $h$-convex stochastic processes, JP J. Math. Sci., 6 (2013), 15–29.
[24]	D. Zhao, T. An, G. Ye, W. Liu, New Jensen and Hermite-Hadamard type inequalities for h-convex interval-valued functions, J. Inequal. Appl., 1 (2018), 1–14. https://doi.org/10.1186/s13660-018-1896-3 doi: 10.1186/s13660-018-1896-3
[25]	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
[26]	J. E. H. Hernandez, On $(m, h_1, h_2)$-G-convex dominated stochastic processes, Kragujev. J. Math., 46 (2022), 215–227. https://doi.org/10.3934/math.2023366 doi: 10.3934/math.2023366
[27]	M. J. Vivas-Cortez, On $(m, h_1, h_2)$-convex stochastic processes using fractional integral operator, Appl. Math. Inform. Sci., 12 (2018), 45–53. http://dx.doi.org/10.18576/amis/120104 doi: 10.18576/amis/120104
[28]	H. Fu, M. S. Saleem, W. Nazeer, M. Ghafoor, P. Li, On Hermite-Hadamard type inequalities for n-polynomial convex stochastic processes, AIMS Math., 6 (2021), 6322–6339. http://dx.doi.org/10.3934/math.2021371 doi: 10.3934/math.2021371
[29]	H. Agahi, A. Babakhani, On fractional stochastic inequalities related to Hermite-Hadamard and Jensen types for convex stochastic processes, Aequationes Math., 90 (2016), 1035–1043. http://dx.doi.org/10.3934/math.2021371 doi: 10.3934/math.2021371
[30]	D. Kotrys, Remarks on Jensen, Hermite-Hadamard and Fejer inequalities for strongly convex stochastic processes, Math. Aeterna, 5 (2015), 104.
[31]	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
[32]	L. Li, Z. Hao, On Hermite-Hadamard inequality for $h$-convex stochastic processes, Aequationes Math., 91 (2017), 909–920. http://dx.doi.org/10.1007/s00010-017-0488-5 doi: 10.1007/s00010-017-0488-5
[33]	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
[34]	M. U. Awan, N. Akhtar, S. Iftikhar, M. A. Noor, Y. M. Chu, New Hermite-Hadamard type inequalities for n-polynomial harmonically convex functions, J. Inequal. Appl., 1 (2020), 1–12. https://doi.org/10.1186/s13660-020-02393-x doi: 10.1186/s13660-020-02393-x
[35]	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), 84. https://doi.org/10.1186/s13660-021-02619-6 doi: 10.1186/s13660-021-02619-6
[36]	A. Iqbal, M. A. Khan, M. Suleman, Y. M. Chu, The right Riemann-Liouville fractional Hermite-Hadamard type inequalities derived from Green's function, AIP Adv., 10 (2020), 045032. https://doi.org/10.1063/1.5143908 doi: 10.1063/1.5143908
[37]	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
[38]	T. Abdeljawad, S. Rashid, H. Khan, Y. M. Chu, On new fractional integral inequalities for p-convexity within interval-valued functions, Adv. Differ. Equ., 2020 (2020), 330. https://doi.org/10.1186/s13662-020-02782-y doi: 10.1186/s13662-020-02782-y
[39]	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
[40]	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
[41]	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
[42]	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
[43]	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
[44]	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
[45]	K. Ahmad, M. A. Khan, S. Khan, A. Ali, Y. M. Chu, New estimation of Zipf-Mandelbrot and Shannon entropies via refinements of Jensen's inequality, AIP Adv., 11 (2021), 015147. https://doi.org/10.1063/5.0039672 doi: 10.1063/5.0039672
[46]	M. A. Khan, J. Pecaric, Y. M. Chu, Refinements of Jensen's and McShane's inequalities with applications, AIMS Math., 5 (2020), 4391–4945. https://doi.org/10.3934/math.2020315 doi: 10.3934/math.2020315
[47]	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
[48]	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
[49]	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
[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, 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
[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]	F. Li, J. Liu, Y. Yan, J. Rong, J. Yi, A time-variant reliability analysis method based on the stochastic process discretization under random and interval variables, Symmetry, 13 (2021), 568. https://doi.org/10.3390/sym13040568 doi: 10.3390/sym13040568
[54]	C. Wang, W. Gao, C. Song, N. Zhang, Stochastic interval analysis of natural frequency and mode shape of structures with uncertainties, J. Sound Vib., 333 (2014), 2483–2503. https://doi.org/10.1016/j.jsv.2013.12.015 doi: 10.1016/j.jsv.2013.12.015
[55]	S. Wang, G. H. Huang, B. T. Yang, An interval-valued fuzzy-stochastic programming approach and its application to municipal solid waste management, Environ. Modell. Softw., 29 (2012), 24–36. https://doi.org/10.1016/j.jsv.2013.12.015 doi: 10.1016/j.jsv.2013.12.015
[56]	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/10.1016/j.envsoft.2011.10.007 doi: 10.1016/j.envsoft.2011.10.007

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