A comprehensive review of Grüss-type fractional integral inequality

Muhammad Tariq; Sotiris K. Ntouyas; Hijaz Ahmad; Asif Ali Shaikh; Bandar Almohsen; Evren Hincal; Muhammad Tariq; Sotiris K. Ntouyas; Hijaz Ahmad; Asif Ali Shaikh; Bandar Almohsen; Evren Hincal

doi:10.3934/math.2024112

AIMS Mathematics

2024, Volume 9, Issue 1: 2244-2281. doi: 10.3934/math.2024112

Previous Article Next Article

Review

A comprehensive review of Grüss-type fractional integral inequality

1.
Department of Basic Sciences and Related Studies, Mehran UET, Jamshoro 76062, Pakistan
2.
Department of Mathematics, University of Ioannina, Ioannina 45110, Greece
3.
Near East University, Operational Research Center in Healthcare, TRNC Mersin 10, Nicosia, 99138, Turkey
4.
Center for Applied Mathematics and Bioinformatics, Gulf University for Science and Technology, Kuwait
5.
Department of Computer Science and Mathematics, Lebanese American University, Beirut, Lebanon
6.
Department of Mathematics, Faculty of Arts and Sciences Near East University, Mersin 99138, Turkey
7.
Department of Mathematics, College of Science, King Saud University, Riyadh 11451, Saudi Arabia

Received: 23 October 2023 Revised: 28 November 2023 Accepted: 06 December 2023 Published: 21 December 2023
MSC : 26A33, 26A51, 26D07, 26D10, 26D15

A survey of results on Grüss-type inequalities associated with a variety of fractional integral and differential operators is presented. The fractional differential operators includes, Riemann-Liouville fractional integral operators, Riemann-Liouville fractional integrals of a function with respect to another function, Katugampola fractional integral operators, Hadamard's fractional integral operators, $ k $-fractional integral operators, Raina's fractional integral operators, tempered fractional integral operators, conformable fractional integrals operators, proportional fractional integrals operators, generalized Riemann-Liouville fractional integral operators, Caputo-Fabrizio fractional integrals operators, Saigo fractional integral operators, quantum integral operators, and Hilfer fractional differential operators.
Citation: Muhammad Tariq, Sotiris K. Ntouyas, Hijaz Ahmad, Asif Ali Shaikh, Bandar Almohsen, Evren Hincal. A comprehensive review of Grüss-type fractional integral inequality[J]. AIMS Mathematics, 2024, 9(1): 2244-2281. doi: 10.3934/math.2024112

Related Papers:

Abstract

A survey of results on Grüss-type inequalities associated with a variety of fractional integral and differential operators is presented. The fractional differential operators includes, Riemann-Liouville fractional integral operators, Riemann-Liouville fractional integrals of a function with respect to another function, Katugampola fractional integral operators, Hadamard's fractional integral operators, $ k $-fractional integral operators, Raina's fractional integral operators, tempered fractional integral operators, conformable fractional integrals operators, proportional fractional integrals operators, generalized Riemann-Liouville fractional integral operators, Caputo-Fabrizio fractional integrals operators, Saigo fractional integral operators, quantum integral operators, and Hilfer fractional differential operators.

References

[1]	F. Cingano, Trends in income inequality and its impact on economic growth, OECD Soc. Employ. Migr. Working Pap., 2014. https://doi.org/10.1787/5jxrjncwxv6j-en doi: 10.1787/5jxrjncwxv6j-en
[2]	M. J. Cloud, B. C. Drachman, L. P. Lebedev, Inequalities with applications to engineering, Springer International Publishing, 2014.
[3]	R. P. Bapat, Applications of inequality in information theory to matrices, Linear Algebra Appl., 78 (1986), 107–117. https://doi.org/10.1016/0024-3795(86)90018-2 doi: 10.1016/0024-3795(86)90018-2
[4]	C. J. Thompson, Inequality with applications in statistical mechanics, J. Math. Phys., 6 (1965), 1812–1813. https://doi.org/10.1063/1.1704727 doi: 10.1063/1.1704727
[5]	S. I. Butt, L. Horváth, D. Pečarić, J. Pečarić, Cyclic improvements of jensen's inequalities: Cyclic inequalities in information theory, Monogr. Inequal., 18 (2020).
[6]	T. Rasheed, S. I. Butt, D. Pečarić, J. Pečarić, Generalized cyclic Jensen and information inequalities, Chaos Soliton. Fract., 163 (2022), 112602. https://doi.org/10.1016/j.chaos.2022.112602 doi: 10.1016/j.chaos.2022.112602
[7]	S. I. Butt, D. Pečarić, J. Pečarić, Several Jensen-Gruss inequalities with applications in information theory, Ukrain. Mate. Zh., 74 (2023), 1654–1672. https://doi.org/10.37863/umzh.v74i12.6554 doi: 10.37863/umzh.v74i12.6554
[8]	N. Mehmood, S. I. Butt, D. Pečarić, J. Pečarić, Generalizations of cyclic refinements of Jensen's inequality by Lidstone's polynomial with applications in information theory, J. Math. Inequal., 14 (2019), 249–271. https://doi.org/10.7153/jmi-2020-14-17 doi: 10.7153/jmi-2020-14-17
[9]	M. Tariq, S. K. Ntouyas, A. A. Shaikh, A comprehensive review of the Hermite-Hadamard inequality pertaining to fractional integral operators, Mathematics, 11 (2023), 1953. https://doi.org/10.3390/math11081953 doi: 10.3390/math11081953
[10]	G. Grüss, Über das Maximum des absoluten betrages von, Math. Z., 39 (1935), 215–226.
[11]	D. S. Mitrinovic, J. E. Pečaric, A. M. Fink, Classical and new inequalities in analysis, Dordrecht, The Netherlands, 1993.
[12]	Z. Dahmani, L. Tabharit, S. Taf, New generalizations of Grüss inequality using Riemann-Liouville fractional integrals, Bull. Math. Anal. Appl., 2 (2010), 93–99.
[13]	J. Tariboon, S. K. Ntouyas, W. Sudsutad, Some new Riemann-Liouville fractional integral inequalities, Int. J. Math. Sci., 2014 (2014), 1–6. https://doi.org/10.1155/2014/869434 doi: 10.1155/2014/869434
[14]	M. Z. Sarikaya, H. Yaldiz, N. Basak. New fractional inequalities of Ostrowski-Grüss type, Lobachevskii J. Math., 69 (2014), 227–235. https://doi.org/10.1134/S1995080213040124 doi: 10.1134/S1995080213040124
[15]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Amsterdam: Elsevier, 2006.
[16]	E. Kacar, Z. Kacar, H. Yildirim, Integral inequalities for h(x)-Riemann-Liouville fractional integrals, Iran. J. Math. Sci. Inform., 13 (2018), 1–13. https://doi.org/10.7508/ijmsi.2018.1.001 doi: 10.7508/ijmsi.2018.1.001
[17]	J. V. Sousa, D. S. Oliveira, E. K. Oliveira, Grüss-type inequalities by means of generalized fractional integrals, Bull. Braz. Math. Soc. Ser., 50 (2019), 1029–1047. https://doi.org/10.1007/s00574-019-00138-z doi: 10.1007/s00574-019-00138-z
[18]	T. A. Aljaaidi, D. B. Pachpatte, Some Grüss type inequalities using Katugampola fractional integral, AIMS Math., 5 (2020), 1011–1024. https://doi.org/10.3934/math.2020070 doi: 10.3934/math.2020070
[19]	W. Sudsutad, S. K. Ntouyas, J. Tariboon, Fractional integral inequalities via Hadamard's fractional integral, Abst. Appl. Anal., 2014 (2014), 1–11. https://doi.org/10.1155/2014/563096 doi: 10.1155/2014/563096
[20]	S. Mubeen, G. M. Habibullah, $k$-fractional integrals and application, Int. J. Contemp. Math. Sci., 7 (2012), 89–94.
[21]	S. K. Ntouyas, J. Tariboon, M. Tomar, Some new integral inequalities for $k$-fractional integrals, Malaya J. Math., 4 (2016), 100–110. https://doi.org/10.26637/mjm401/013 doi: 10.26637/mjm401/013
[22]	E. Set, M. Tomar, M. Z. Sarikaya, On generalized Grüss type inequalities for $k$-fractional integrals, Appl. Math. Comput., 269 (2015), 29–34. https://doi.org/10.1016/j.amc.2015.07.026 doi: 10.1016/j.amc.2015.07.026
[23]	S. Rashid, F. Jarad, M. A. Noor, K. I. Noor, D. Baleanu, J. B. Liu, On Grüss inequalities within generalized $k$-fractional integrals, Adv. Differ. Equ., 203 (2020). https://doi.org/10.1186/s13662-020-02644-7 doi: 10.1186/s13662-020-02644-7
[24]	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., 647 (2020). https://doi.org/10.1186/s13662-020-03108-8 doi: 10.1186/s13662-020-03108-8
[25]	C. Li, W. Deng, L. Zhao, Well-posedness and numerical algorithm for the tempered fractional ordinary differential equations, Discrete Contin. Dyn. Syst. Ser. B., 24 (2019), 1989–2015. https://doi.org/10.3934/dcdsb.2019022 doi: 10.3934/dcdsb.2019022
[26]	H. M. Fahad, A. Fernandez, M. U. Rehman, M. Siddiqi, Tempered and Hadamard type fractional calculus with respect to functions, Mediterr. J. Math., 18 (2021), 143. https://doi.org/10.1007/s00009-021-01783-9 doi: 10.1007/s00009-021-01783-9
[27]	G. Rahman, K. S. Nisar, S. Rashid, T. Abdeljawad, Certain Grüss-type inequalities via tempered fractional integrals concerning another function, J. Inequal. Appl., 147 (2020). https://doi.org/10.1186/s13660-020-02420-x doi: 10.1186/s13660-020-02420-x
[28]	S. K. Yildirim, H. Yildirim, Grüss type integral inequalities for generalized $\eta$-conformable fractional integrals, Turk. J. Math. Comput. Sci., 14 (2022), 201–211.
[29]	S. Habib, G. Farid, S. Mubeen, Grüss type integral inequalities for a new class of $k$-fractional integrals, Int. J. Nonlinear Anal. Appl., 12 (2021), 541–554. https://doi.org/10.22075/IJNAA.2021.4836 doi: 10.22075/IJNAA.2021.4836
[30]	G. Rahman, N. K. Sooppy, F. Qi, Some new inequalities of the Grüss type for conformable fractional integrals, AIMS Math., 3 (2018), 575–583. https://doi.org/10.3934/Math.2018.4.575 doi: 10.3934/Math.2018.4.575
[31]	S. Rashid, F. Jarad, M. A. Noor, Grüss-type integrals inequalities via generalized proportional fractional operators, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math., 114 (2020), 93.
[32]	T. A. Aljaaidi, D. B. Pachpatte, M. S. Abdo, T. Botmart, H. Ahmad, M. A. Almalahi, et al., $(k, \psi)$-Proportional fractional integral Pólya-Szegö and Grüss-type inequalities, Fractal Fract., 5 (2021), 172. https://doi.org/10.3390/fractalfract5040172 doi: 10.3390/fractalfract5040172
[33]	M. Z. Sarikaya, Z. Dahmani, M. E. Kiris, F. Ahmad, $(k, s)$-Riemann-Liouville fractional integral and applications, Hacet. J. Math. Stat., 45 (2016), 77–89.
[34]	E. Kacar, H. Yildirim, Grüss-type integrals inequalities for generalized Riemann-Liouville fractional integrals, Int. J. Pure. Appl. Math., 101 (2015), 55–70.
[35]	S. Mubeen, S. Iqbal, Grüss type integral inequalities for generalized Riemann-Liouville $k$-fractional integrals, J. Inequal. Appl., 109 (2016). https://doi.org/10.1186/s13660-016-1052-x doi: 10.1186/s13660-016-1052-x
[36]	M. Caputo, M. Fabrizio, A new definition of fractional derivative without singular kernel, Prog. Fract. Differ. Appl., 1 (2015), 73–85. https://dx.doi.org/10.12785/pfda/010201 doi: 10.12785/pfda/010201
[37]	A. B. Nale, S. K. Panchal, V. L. Chinchane, Grüss-type fractional inequality via Caputo-Fabrizio integral operator, Acta Univ. Sapien. Math., 14 (2022), 262–277. https://doi.org/10.2478/ausm-2022-0018 doi: 10.2478/ausm-2022-0018
[38]	M. Saigo, A remark on integral operators involving the Grüss hypergeometric functions, Kyushu Univ., 11 (1978), 135–143.
[39]	V. L. Chinchane, D. B. Pachpatte, On some Grüss-type fractional inequalities using Saigo fractional integral operator, J. Math., 2014 (2014), 1–9. https://doi.org/10.1155/2014/527910 doi: 10.1155/2014/527910
[40]	S. L. Kalla, A. Rao, On Grüss type inequality for a hypergeometric fractional integral, Le Mat., 66 (2011), 57–64. https://doi.org/10.4418/2011.66.1.5 doi: 10.4418/2011.66.1.5
[41]	G. Wang, P. Agarwal, M. Chand, Certain Grüss type inequalities involving the generalized fractional integral operator, J. Inequal. Appl., 147 (2014). https://doi.org/10.1186/1029-242X-2014-147 doi: 10.1186/1029-242X-2014-147
[42]	M. Saigo, N. Maeda, More generalization of fractional calculus, transform methods and special functions, Sci. Sofia, 1996,386–400.
[43]	S. Joshi, E. Mittal, R. M. Pandey, S. D. Purohit, Some Grüss type inequalities involving generalized fractional integral operator, Math. Inform. Phys., 12 (2019), 41–52.
[44]	V. Kac, P. Cheung, Quantum calculus, New York: Springer, 2002.
[45]	R. P. Agarwal, Certain fractional $q$-integrals and $q$-derivatives, Math. Proc. Cambridge Philos. Soc., 66 (1969) 365–370.
[46]	A. Secer, S. D. Purohit, K. A. Selvakumaran, M. Bayram, A generalized $q$-Grüss inequality involving the Riemann-Liouville fractional $q$-integrals, J. Appl. Math., 2014 (2014), 1–6. https://doi.org/10.1155/2014/914320 doi: 10.1155/2014/914320
[47]	J. Tariboon, S. K. Ntouyas, Quantum integral inequalities on finite intervals, J. Ineq. Appl., 121 (2014). https://doi.org/10.1186/1029-242X-2014-121 doi: 10.1186/1029-242X-2014-121
[48]	M. Bilal, A. Iqbal, S. Rastogi, Quantum symmetric analogue of various integral inequalities over finite intervals, J. Math. Inequal., 17 (2023), 615–627. https://doi.org/10.7153/jmi-2023-17-40 doi: 10.7153/jmi-2023-17-40
[49]	J. Tariboon, S. K. Ntouyas, P. Agarwal, New concepts of fractional quantum calculus and applications to impulsive fractional $q$-Difference equations, Adv. Differ. Equ., 18 (2015). https://doi.org/10.1186/s13662-014-0348-8 doi: 10.1186/s13662-014-0348-8
[50]	W. Sudsutad, S. K. Ntouyas, J. Tariboon, Integral inequalities via fractional quantum calculus, J. Ineq. Appl., 81 (2016). https://doi.org/10.1186/s13660-016-1024-1 doi: 10.1186/s13660-016-1024-1
[51]	S. Iqbal, M. Samraiz, G. Rahman, K. S. Nisar, T. Abdeljawad, Some new Grüss inequalities associated with generalized fractional derivative, AIMS Math., 8 (2022), 213–227. https://doi.org/10.3934/math.2023010 doi: 10.3934/math.2023010
[52]	S. Naz, M. N. Naeem, Y. M. Chu, Some $k$-fractional extension of Grüss-type inequalities via generalized Hilfer-Katugampola derivative, Adv. Differ. Equ., 29 (2021). https://doi.org/10.1186/s13662-020-03187-7 doi: 10.1186/s13662-020-03187-7

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)