Grüss-type fractional inequality via Caputo-Fabrizio integral operator

Asha B. Nale; Satish K. Panchal; Vaijanath L. Chinchane

doi:10.2478/ausm-2022-0018

Abstract

In this article, the main objective is to establish the Grüss-type fractional integral inequalities by employing the Caputo-Fabrizio fractional integral.

References

[1] T. A. Aljaaidi and D. B. Pachpatte,Some Grüss-type inequalities using generalized Katugampola fractional integral, AIMS Mathematics, 2 (5), (2020), 1011–1024. doi: 10.3934/math.2020070.10.3934/math.2020070
Search in Google Scholar Back to article
[2] D. Baleanu, J. A. T. Machado and C. J. Luo, Fractional Dynamic and Control, Springer, 2012, pp 159–171.10.1007/978-1-4614-0457-6
Search in Google Scholar Back to article
[3] S. Belarbi and Z. Dahmani,On some new fractional integral inequality, J. Inequal. Pure and Appl. Math., 10 (3) (2009), Art.86, 5 pp.
Search in Google Scholar Back to article
[4] P. L. Butzer, A. A. Kilbas and J. J. Trujillo, Fractional Calculus in the Mellin setting and Hadardmard-type fractional integrals, J. Math. Anal. Appl., (2002), 1–27.10.1016/S0022-247X(02)00001-X
Search in Google Scholar Back to article
[5] M. Caputo and M. Fabrizio, A new defination of fractional derivative without singular kernel, Progr., Fract., Differ., Appl., 1 (2) (2015), 73–85.10.18576/pfda/020101
Search in Google Scholar Back to article
[6] M. Caputo and M. Fabrizio, Applications of new time and spatial fractional derivative with exponential kernels, Progr., Fract., Differ., Appl., 2 (2) (2016), 7–8.10.18576/pfda/020101
Search in Google Scholar Back to article
[7] V. L. Chinchane and D. B. Pachpatte, A note on some integral inequalities via Hadamard integral, J. Fractional Calculus Appl., 4 (11) (2013), 1–5.10.1155/2014/527910
Search in Google Scholar Back to article
[8] V. L. Chinchane and D. B. Pachpatte, On some integral inequalities using Hadamard fractional integral, Malaya J. Math., 1 (1) (2012), 62–66.10.26637/mjm0101/008
Search in Google Scholar Back to article
[9] V. L. Chinchane and D. B. Pachpatte, Some new integral inequalities using Hadamard fractional integral operator, Adv. Inequal. Appl., 2014, 2014:12.10.1155/2014/527910
Search in Google Scholar Back to article
[10] V. L. Chinchane and D. B. Pachpatte, On some Grüss-type fractional inequalities using Saigo fractional integral operator, Journal of Mathematics, vol. 2014, Article ID 527910, 9 pages, 2014. https://doi.org/10.1155/2014/527910.10.1155/2014/527910
Search in Google Scholar Back to article
[11] Z. Dahmani, On Minkowski and Hermit-Hadamard integral inequalities via fractional integration, Ann. Funct. Anal., 1 (1) (2010), 51–58.10.15352/afa/1399900993
Search in Google Scholar Back to article
[12] Z. Dahmani, New inequalities in fractional integrals, Int. J. Nonlinear Sci., 9, 4(2010), 493–497.
Search in Google Scholar Back to article
[13] Z. Dahmani, The Riemann-Liouville operator to generate some new inequalities, Int. J. Nonlinear Sci., 12 (4) (2011), 452-455.
Search in Google Scholar Back to article
[14] Z. Dahmani, Some results associate with fractional integrals involving the extended Chebyshev, Acta Univ. Apulensis Math. Inform., 27 (2011), 217–224.
Search in Google Scholar Back to article
[15] Z. Dahmani and A. Benzidane, New inequalities using Q-fractional theory, Bull. Math. Anal. Appl., 4 (1) (2012), 190–196.
Search in Google Scholar Back to article
[16] Z. Dahmani, L. Tabharit and S. Taf, New generalisation of Grüss inequality using Riemann-Liouville fractional integrals, Bull. Math. Anal. Appl., 2 (3)(2012), 92–99.
Search in Google Scholar Back to article
[17] Z. Denton and A. S. Vatsala, Fractional integral inequalities and application, Comput. Math. Appl., 59 (2010), 1087–1094.10.1016/j.camwa.2009.05.012
Search in Google Scholar Back to article
[18] S. S. Dragomir, Some integral inequalities of Grüss type, Indian J. Pur. Appl. Math., 31 (4) (2002), 397–415.
Search in Google Scholar Back to article
[19] S. S. Dragomir, A companion of the Grüss inequality and applications, Appl. Math. Lett., 17 (2004), no. 4, 429–435.10.1016/S0893-9659(04)90085-7
Search in Google Scholar Back to article
[20] S. S. Dragomir and S. Wang, An inequality of Ostrowski-Grüss type and its applications to the estimation of error bounds for some special means and for home numerical quadrature rules, Comput. Math. Appl., 33 (1997), no. 11, 15–20.10.1016/S0898-1221(97)00084-9
Search in Google Scholar Back to article
[21] G. Farid, A. U. Rehman, V. N. Mishra, S. Mehmood, Fractional integral inequalities of Gruss type via generalized Mittag-Leffler function, Int. J. Anal. Appl., 17 (4) (2019), 548–558.
Search in Google Scholar Back to article
[22] N. Elezovic, L. Marangunic and J. Pecaric, Some improvements of Grüss type inequality, J. Math. Inequal., 1 (2007), no. 3, 425–436.10.7153/jmi-01-36
Search in Google Scholar Back to article
[23] A. A. George, Fractional Differentiation Inequalities, Springer Publishing Company, New York, 2009.
Search in Google Scholar Back to article
[24] G. Grüss, Uber das maximum des absoluten Betrages von 1b-a∫abf(x)g(x)dx-1(b-a)2∫abf(x)dx∫abg(x)dx {1 \over {{\rm{b}} - {\rm{a}}}}\int_{\rm{a}}^{\rm{b}} {{\rm{f}}\left( {\rm{x}} \right){\rm{g}}\left( {\rm{x}} \right){\rm{dx}} - {1 \over {{{\left( {{\rm{b}} - {\rm{a}}} \right)}^2}}}} \int_{\rm{a}}^{\rm{b}} {{\rm{f}}\left( {\rm{x}} \right){\rm{dx}}\int_{\rm{a}}^{\rm{b}} {{\rm{g}}\left( {\rm{x}} \right){\rm{dx}}} } , Math. Z., 39 (1935), 215–226.10.1007/BF01201355
Back to article
[25] A. M. Mercer, P. Mercer, New proofs of the Grüss inequality, Aust. J. Math. Anal. Appl., 1 (2) (2004), Art. 12.
Search in Google Scholar Back to article
[26] A. M. Mercer, An improvement of the Grüss inequality, JIPAM. J. Inequal. Pure Appl. Math., 6 (2005), no. 4, Article 93.
Search in Google Scholar Back to article
[27] Yu Miao, Fei Han and Jianyong Mu, A new Ostrowski-Grüss type inequality, Kragujevac Journal of Mathematics, 37 (2) (2013), Pages 307–317.
Search in Google Scholar Back to article
[28] G. A. M. Nchama, A. L. Mecias and M. R. Richard, The Caputo-Fabrizio fractional integral to generate some new inequalities, Inf. Sci. Lett., 2 (8), (2019), Pages 73–80.10.18576/isl/080205
Search in Google Scholar Back to article
[29] B. G. Pachpatte, On multidimensional Grüss type inetegral inequalities, J. Inequal. Pure and Appl. Math., Vol 03, Iss. 2, Art. 27, (2002), 1–7.
Search in Google Scholar Back to article
[30] B. G. Pachpatte, A note on Chebyshev-Grüss inequalities for differential equations, Tamsui Oxf. J. Math. Sci., 22 (1), (2006), 29–36.
Search in Google Scholar Back to article
[31] G. H. Peng and Y. Miao, A note on Grüss type inequality, Appl. Math. Sci., (Ruse) 3 (2009), no. 8, 399–402.
Search in Google Scholar Back to article
[32] I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives to method of their solution and some of their application, Acadamic Press, San Diego CA, 1999.
Search in Google Scholar Back to article
[33] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Application of Fractional Differential Equations, Elsevier, Amersterdam, 2006.
Search in Google Scholar Back to article
[34] G. Rahman, K. S. Nisar, S. Rashid and T. Abdeljawad, Certain Grüss-type inequalities via tempered fractional integrals concerning another function, J. Inequal. Appl., 2020, 147 (2020). https://doi.org/10.1186/s13660-020-02420-x.10.1186/s13660-020-02420-x
Search in Google Scholar Back to article
[35] S. Rashid, F. Jarad and M. A. Noor, Grüss-type integrals inequalities via generalized proportional fractional operators, RACSAM, 114, 93 (2020). https://doi.org/10.1007/s13398-020-00823-5.10.1007/s13398-020-00823-5
Search in Google Scholar Back to article
[36] E. Set, M. Tomar and M. Z. Sarikaya, On generalized Grüss type inequalities for k-fractional integrals, Applied Mathematics and Computation, 269 (2015), 29–34. DOI: 10.1016/j.amc.2015.07.026.10.1016/j.amc.2015.07.026
Search in Google Scholar Back to article
[37] S. G. Somko, A. A. Kilbas and O. I. Marichev, Fractional Integral and Derivative Theory and Application, Gordon and Breach, Yverdon, Switzerland, 1993.
Search in Google Scholar Back to article
[38] W. Yang, Some new fractional quantum integral inequalities, Appl. Math. Latt., 25, 6 (2012), 963–969.10.1016/j.aml.2011.11.005
Search in Google Scholar Back to article
[39] X. Wang, M. S. Saleem, K. N. Aslam, X. Wu and T. Zhou, On Caputo–Fabrizio fractional integral inequalities of Hermite–Hadamard type for modified -convex Functions, Journal of Mathematics, vol. 2020, Article ID 8829140, 17 pages, 2020. https://doi.org/10.1155/2020/8829140.10.1155/2020/8829140
Search in Google Scholar Back to article
[40] C. Zhu, W. Yang and Q. Zhao, Some new fractional q-integral Grüss-type inequalities and other inequalities, J. Inequal. Appl., 2012:299.10.1186/1029-242X-2012-299
Search in Google Scholar Back to article

Grüss-type fractional inequality via Caputo-Fabrizio integral operator

Abstract

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