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Fractional integral inequalities of Grüss type via generalized Mittag-Leffler function. (English) Zbl 1438.26035

Summary: We use generalized fractional integral operator containing the generalized Mittag-Leffler function to establish some new integral inequalities of Grüss type. A cluster of fractional integral inequalities have been identified by setting particular values to parameters involved in the Mittag-Leffler special function. Presented results contain several fractional integral inequalities which reflects their importance.

MSC:

26D10 Inequalities involving derivatives and differential and integral operators
26A33 Fractional derivatives and integrals
33E12 Mittag-Leffler functions and generalizations
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[1] A. Akkurt, Z. Ka¸car and H. Yildirim, Generalized fractional integral inequalities for continuous random variables, J. Probab. Stat. 2015(2015), Art. ID 958980. Google Scholar · Zbl 1426.60022
[2] A. Akkurt, S. Kilin¸c and H. Yildirim, Gr¨uss type inequalities involving the generalized Gauss hypergeometric functions, Int. J. Pure Appl. Math. 106(4) 2016, 1103-1114. Google Scholar
[3] M. Andri´c, G. Farid and J. Peˇcari´c, A further extension of Mittag-Leffler function, Fract. Calc. Appl. Anal., 21(5) (2018), 1377-1395. Google Scholar · Zbl 1426.33051
[4] A. S. Balakishiyev, E. A. Gadjieva, F. G¨urb¨uz and A. Serbetci, Boundedness of some sublinear operators and their commutators on generalized local Morrey spaces, Complex Var. Elliptic Equ. 63(11) (2018), 1620-1641. Google Scholar · Zbl 1395.42027
[5] G. Gr¨uss, Uber das maximum des absolten Betrages von ¨ 1 b−a R b a f(x)g(x)dx − 1 (b−a) 2 R b a f(x)dx R b a f(x)dx, Math. Z. 39 (1935), 215-226. Google Scholar · Zbl 0010.01602
[6] F. G¨urb¨uz, Some estimates for generalized commutators of rough fractional maximal and integral operators on generalized weighted Morrey spaces, Canad. Math. Bull. 60(1) (2017), 131-145. Google Scholar · Zbl 1361.42014
[7] E. Ka¸car and H. Yildirim, Gr¨uss type integral inequalities for generalized Riemann-Liouville fractional integrals, Int. J. Pure Appl. Math. 101(1) (2015), 55-70. Google Scholar
[8] V. N. Mishra, K. Khatri, L. N. Mishra, Deepmala, Inverse result in simultaneous approximation by Baskakov-DurrmeyerStancu operators, J. Inequal. Appl. 2013 (2013), Art. ID 586. Google Scholar · Zbl 1295.41013
[9] X. Li, R. N. Mohapatra and R. S. Rodriguez, Gr¨uss-type inequalities, J. Math. Anal. Appl. 267 (2002), 434-443. Google Scholar · Zbl 1007.26016
[10] T. R. Prabhakar, A singular integral equation with a generalized Mittag-Leffler function in the kernel, Yokohama Math. J. 19 (1971), 7-15. Google Scholar · Zbl 0221.45003
[11] G. Rahman, D. Baleanu, M. A. Qurashi, S. D. Purohit, S. Mubeen and M. Arshad, The extended Mittag-Leffler function via fractional calculus, J. Nonlinear Sci. Appl. 10 (2017), 4244-4253. Google Scholar · Zbl 1412.33039
[12] T. O. Salim and A. W. Faraj, A generalization of Mittag-Leffler function and integral operator associated with integral calculus, J. Frac. Calc. Appl., 3(5) (2012), 1-13. Google Scholar · Zbl 1488.33067
[13] A. K. Shukla and J. C. Prajapati, On a generalization of Mittag-Leffler function and its properties, J. Math. Anal. Appl. 336 (2007), 797-811. Google Scholar · Zbl 1122.33017
[14] H. M. Srivastava and Z. Tomovski, Fractional calculus with an integral operator containing generalized Mittag-Leffler function in the kernal, Appl. Math. Comput. 211(1) (2009), 198-210. Google Scholar · Zbl 1432.30022
[15] G. Wang, P. Agarwal and M. Chand, Certain Gr¨uss type inequalities involving the generalized fractional integral operator, J. Inequal. Appl. 2014 (2014), Art. ID 147. Google Scholar · Zbl 1308.26025
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