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Chebyshev type inequality containing a fractional integral operator with a multi-index Mittag-Leffler function as a kernel

  • Kamlesh Jangid , S. D. Purohit , Kottakkaran Sooppy Nisar and Serkan Araci ORCID logo EMAIL logo
From the journal Analysis

Abstract

In this paper, we derive certain Chebyshev type integral inequalities connected with a fractional integral operator in terms of the generalized Mittag-Leffler multi-index function as a kernel. Our key findings are general in nature and, as a special case, can give rise to integral inequalities of the Chebyshev form involving fractional integral operators present in the literature.

MSC 2010: 33B15; 33B99; 33E12

References

[1] S. Belarbi and Z. Dahmani, On some new fractional integral inequalities, JIPAM. J. Inequal. Pure Appl. Math. 10 (2009), no. 3, Article ID 86. Search in Google Scholar

[2] P. L. Chebyshev, Sur les expressions approximatives des integrales definies par les autres prises entre les me mes limites, Proc. Math. Soc. Charkov. 2 (1882), 93–98. Search in Google Scholar

[3] C. Fox, The asymptotic expansion of generalized hypergeometric functions, Proc. Lond. Math. Soc. (2) 27 (1928), no. 5, 389–400. 10.1112/plms/s2-27.1.389Search in Google Scholar

[4] K. Jangid, S. D. Purohit and R. Agarwal, On Grüss type inequality involving a fractional integral operator with a multi-index Mittag-Leffler function as a kernel, communicated. Search in Google Scholar

[5] S. L. Kalla and A. Rao, On Grüss type inequality for a hypergeometric fractional integral, Matematiche (Catania) 66 (2011), no. 1, 57–64. Search in Google Scholar

[6] A. A. Kilbas, M. Saigo and R. K. Saxena, Generalized mittag-leffler function and generalized fractional calculus operators, Int. Trans. Special Func. 15 (2004), 31–49. 10.1080/10652460310001600717Search in Google Scholar

[7] V. S. Kiryakova, Multiple (multiindex) Mittag-Leffler functions and relations to generalized fractional calculus, J. Comput. Appl. Math. 118 (2000), 241–259. 10.1016/S0377-0427(00)00292-2Search in Google Scholar

[8] A. M. Mishra, D. Kumar and S. D. Purohit, Unified integral inequalities comprising pathway operators, AIMS Math. 5 (2020), no. 1, 399–407. 10.3934/math.2020027Search in Google Scholar

[9] G. M. Mittag-Leffler, Sur la nouvelle fonction Eα(x), C. R. Acad. Sci. Paris 137 (1903), 554–558. Search in Google Scholar

[10] T. R. Prabhakar, A singular integral equation with a generalized Mittag-Leffler function in the kernel, Yokohama Math. J. 19 (1971), 7–15. Search in Google Scholar

[11] S. Purohit, N. Jolly, M. K. Bansal, J. Singh and D. Kumar, Chebyshev type inequalities involving the fractional integral operator containing multi-index Mittag-Leffler function in the kernel, Appl. Appl. Math. (2020), no. 6, 29–38. Search in Google Scholar

[12] S. D. Purohit and R. K. Raina, Chebyshev type inequalities for the Saigo fractional integrals and their 𝑞-analogues, J. Math. Inequal. 7 (2013), no. 2, 239–249. 10.7153/jmi-07-22Search in Google Scholar

[13] T. O. Salim and A. W. Faraj, A generalization of Mittag-Leffler function and integral operator associated with fractional calculus, J. Fract. Calc. Appl. 3 (2012), no. 5, 1–13. Search in Google Scholar

[14] R. K. Saxena and K. Nishimoto, Further results on generalized Mittag-Leffler functions of fractional calculus, J. Fract. Calc. 39 (2010), 29–41. Search in Google Scholar

[15] R. K. Saxena and K. Nishimoto, 𝑁-fractional calculus of generalized Mittag-Leffler functions, J. Fract. Calc. 37 (2010), 43–52. Search in Google Scholar

[16] A. K. Shukla and J. C. Prajapati, On a generalization of Mittag-Leffler function and its properties, J. Math. Anal. Appl. 336 (2007), no. 2, 797–811. 10.1016/j.jmaa.2007.03.018Search in Google Scholar

[17] H. M. Srivastava, M. Bansal and P. Harjule, A study of fractional integral operators involving a certain generalized multi-index Mittag-Leffler function, Math. Methods Appl. Sci. 41 (2018), no. 16, 6108–6121. 10.1002/mma.5122Search in Google Scholar

[18] H. M. Srivastava and v. Tomovski, Fractional calculus with an integral operator containing a generalized Mittag-Leffler function in the kernel, Appl. Math. Comput. 211 (2009), no. 1, 198–210. 10.1016/j.amc.2009.01.055Search in Google Scholar

[19] W. T. Sulaiman, Some new fractional integral inequalities, J. Math. Anal. 2 (2011), no. 2, 23–28. Search in Google Scholar

[20] A. Wiman, Über den Fundamentalsatz in der Teorie der Funktionen Ea(x), Acta Math. 29 (1905), no. 1, 191–201. 10.1007/BF02403202Search in Google Scholar

Received: 2020-11-03
Accepted: 2020-12-09
Published Online: 2021-01-13
Published in Print: 2021-02-01

© 2021 Walter de Gruyter GmbH, Berlin/Boston

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