Fractional Integral Inequalities of Gruss Type via Generalized Mittag-Leffler Function

Farid, G.; Rehman, A. U.; Mishra, Vishnu Narayan; Mehmood, S.

doi:10.28924/2291-8639-17-2019-548

Research Article

Fractional Integral Inequalities of Gruss Type via Generalized Mittag-Leffler Function

G. Farid
A. U. Rehman
Vishnu Narayan Mishra¹*
S. Mehmood

Department of Mathematics, Indira Gandhi National Tribal University, Lalpur, Amarkantak, Anuppur, Madhya Pradesh 484 887, India.

* Corresponding author

Abstract

We use generalized fractional integral operator containing the generalized Mittag-Leffler function to establish some new integral inequalities of Gr ¨uss 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.

How to Cite

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 (2019), 548–558. https://doi.org/10.28924/2291-8639-17-2019-548

BibTeX

@article{Farid2019,
  author  = {Farid, G. and Rehman, A. U. and Mishra, Vishnu Narayan and Mehmood, S.},
  title   = {Fractional Integral Inequalities of Gruss Type via Generalized Mittag-Leffler Function},
  journal = {Int. J. Anal. Appl.},
  volume  = {17},
  year    = {2019},
  pages   = {548--558},
  doi     = {10.28924/2291-8639-17-2019-548}
}

References

[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[9] X. Li, R. N. Mohapatra and R. S. Rodriguez, Gr ¨uss-type inequalities, J. Math. Anal. Appl. 267 (2002), 434-443.
[10] T. R. Prabhakar, A singular integral equation with a generalized Mittag-Leffler function in the kernel, Yokohama Math. J. 19 (1971), 7-15.
[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.
[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.
[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.
[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.
[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.