Grüss Type k-Fractional Integral Operator Inequalities and Allied Results

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DOI: https://doi.org/10.28924/2291-8639-21-2023-133
Published: Dec 4, 2023
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Ghulam Farid, Sajid Mehmood, Laxmi Rathour, Mawahib Elamin, Huda Uones Mohamd Ahamd, Neama Yahia, Grüss Type k-Fractional Integral Operator Inequalities and Allied Results, Int. J. Anal. Appl., 21 (2023), 133.

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Ghulam Farid, Sajid Mehmood, Laxmi Rathour, Mawahib Elamin, Huda Uones Mohamd Ahamd, Neama Yahia

Abstract

This paper aims to derive fractional Grüss type integral inequalities for generalized k-fractional integral operators with Mittag-Leffler function in the kernel. Many new results can be deduced for several integral operators by giving specific values to the parameters involved in Mittag-Leffler function. Moreover, the results of this paper reproduce a lot of already published results.

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References

  1. M. Andrić, G. Farid, J. Pečarić, A Further Extension of Mittag-Leffler Function, Fract. Calc. Appl. Anal. 21 (2018), 1377–1395. https://doi.org/10.1515/fca-2018-0072.
  2. M. Andrić, G. Farid, S. Mehmood et al. Pólya-Szegö and Chebyshev Types Inequalities via an Extended Generalized Mittag-Leffler Function, Math. Inequal. Appl. 22 (2019), 1365–1377. https://doi.org/10.7153/mia-2019-22-94.
  3. M. Andrić, G. Farid, J. Pečarić, Analytical Inequalities for Fractional Calculus Operators and the Mittag-Leffler Function: Applications of Integral Operators Containing an Extended Generalized Mittag-Leffler Function in the Kernel, Element, Zagreb, 2021.
  4. H. Chen, U.N. Katugampola, Hermite-Hadamard and Hermite-Hadamard-Fejér Type Inequalities for Generalized Fractional Integrals, J. Math. Anal. Appl. 446 (2017), 1274–1291. https://doi.org/10.1016/j.jmaa.2016.09.018.
  5. G. Farid, A.U. Rehman, V.N. Mishra, et al. Fractional Integral Inequalities of Gruss Type via Generalized MittagLeffler Function, Int. J. Anal. Appl. 17 (2019), 548–558. https://doi.org/10.28924/2291-8639-17-2019-548.
  6. G. Farid, Bounds of Fractional Integral Operators Containing Mittag-Leffler Function, U.P.B. Sci. Bull., Ser. A, 81 (2019), 133–142.
  7. G. Grüss, Über 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.
  8. S. Habib, G. Farid, S. Mubeen, Gru¨ss 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.
  9. S. Habib, S. Mubeen, M.N. Naeem, Chebyshev Type Integral Inequalities for Generalized k-Fractional Conformable Integrals, J. Inequal. Spec. Funct. 9 (2018), 53–65.
  10. F. Jarad, E. Uğurlu, T. Abdeljawad, D. Baleanu, On a New Class of Fractional Operators, Adv. Differ. Equ. 2017 (2017), 247. https://doi.org/10.1186/s13662-017-1306-z.
  11. E. Kacar, Z. Kacar, H. Yildirim, Integral Inequalities for Riemann-Liouville Fractional Integrals of a Function with Respect to Another Function, Iran. J. Math. Sci. Inform. 13 (2018), 1–13.
  12. E. Kacar, H. Yildirim, Gruss Type Integral Inequalities for Generalized Riemann-Liouville Fractional Integrals, Int. J. Pure Appl. Math. 101 (2015), 55–70. https://doi.org/10.12732/ijpam.v101i1.6.
  13. T.U. Khan, M.A. Khan, Generalized Conformable Fractional Operators, J. Comput. Appl. Math. 346 (2019), 378–389. https://doi.org/10.1016/j.cam.2018.07.018.
  14. A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, NorthHolland Mathematics Studies, 204, Elsevier, Amsterdam, 2006.
  15. Y.C. Kwun, G. Farid, W. Nazeer, S. Ullah, S.M. Kang, Generalized Riemann-Liouville k -Fractional Integrals Associated With Ostrowski Type Inequalities and Error Bounds of Hadamard Inequalities, IEEE Access. 6 (2018), 64946–64953. https://doi.org/10.1109/access.2018.2878266.
  16. S. Mehmood, G. Farid, K.A. Khan, M. Yussouf, New Fractional Hadamard and Fejér-Hadamard Inequalities Associated With Exponentially (h, m)-Convex Function, Eng. Appl. Sci. Lett. 3 (2020), 9–18.
  17. S. Mubeen, S. Iqbal, Gru¨ss Type Integral Inequalities for Generalized Riemann-Liouville k-Fractional Integrals, J. Inequal. Appl. 2016 (2016), 109. https://doi.org/10.1186/s13660-016-1052-x.
  18. S. Mubeen, G. M. Habibullah, k-Fractional Integrals and Applications, Int. J. Contemp. Math. Sci. 7 (2012), 89–94.
  19. D.S. Mitrinović, J.E. Pečarić, A.M. Fink, Classical and New Inequalities in Analysis, Kluwer Academic, Dordrecht, (1993).
  20. S.D. Purohit, N. Jolly, M.K. Bansal, et al. Chebyshev Type Inequalities Involving the Fractional Integral Operator Containing Multi-Index Mittag-Leffler Function in the Kernel, Appl. Appl. Math. 15 (2020), 29–38.
  21. T.R. Prabhakar, A Singular Integral Equation With a Generalized Mittag-Leffler Function in the Kernel, Yokohama Math. J. 19 (1971), 7–15.
  22. S. Rashid, F. Jarad, M.A. Noor, K.I. Noor, D. Baleanu, J.B. Liu, On Grüss Inequalities Within Generalized KFractional Integrals, Adv. Differ. Equ. 2020 (2020), 203. https://doi.org/10.1186/s13662-020-02644-7.
  23. S. Rashid, F. Safdar, A.O. Akdemir, M.A. Noor, K.I. Noor, Some new fractional integral inequalities for exponentially m-convex functions via extended generalized Mittag-Leffler function, J. Inequal. Appl. 2019 (2019), 299. https://doi.org/10.1186/s13660-019-2248-7.
  24. G. Rahman, K. Sooppy Nisar, 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.
  25. G. Rahman, D. Baleanu, M. Al Qurashi, S.D. Purohit, S. Mubeen, M. Arshad, The Extended Mittag-Leffler Function via Fractional Calculus, J. Nonlinear Sci. Appl. 10 (2017), 4244–4253. https://doi.org/10.22436/jnsa.010.08.19.
  26. T.O. Salim, A.W. Faraj, A Generalization of Mittag-Leffler Function and Integral Operator Associated With Integral Calculus, J. Frac. Calc. Appl. 3 (2012), 1–13.
  27. H.M. Srivastava, Z. Tomovski, Fractional Calculus With an Integral Operator Containing a Generalized Mittag–leffler Function in the Kernel, Appl. Math. Comput. 211 (2009), 198–210. https://doi.org/10.1016/j.amc.2009.01.055.
  28. M.Z. Sarikaya, M. Dahmani, M.E. Kiris, F. Ahmad, (k, s)-Riemann-Liouville Fractional Integral and Applications, Hacettepe J. Math. Stat. 45 (2016), 77–89.
  29. J. Tariboon, S.K. Ntouyas, W. Sudsutad, Some New Riemann-Liouville Fractional Integral Inequalities, Int. J. Math. Math. Sci. 2014 (2014), 869434. https://doi.org/10.1155/2014/869434.
  30. Z. Zhang, G. Farid, S. Mehmood, C.Y. Jung, T. Yan, Generalized k-Fractional Chebyshev-Type Inequalities via Mittag-Leffler Functions, Axioms. 11 (2022), 82. https://doi.org/10.3390/axioms11020082.
  31. Z. Zhang, G. Farid, S. Mehmood, K. Nonlaopon, T. Yan, Generalized k-Fractional Integral Operators Associated with Pólya-Szegö and Chebyshev Types Inequalities, Fractal Fract. 6 (2022), 90. https://doi.org/10.3390/fractalfract6020090.
  32. G. Farid, S. Mehmood, L. Rathour, L. N. Mishra, V. N. Mishra, Fractional Hadamard and Fejér-Hadamard Inequalities Associated With exp. (α, h − m)-Convexity, Dyn. Contin. Discr. Impuls. Syst. Ser. A: Math. Anal. 30 (2023), 353–367.
  33. G. Farid, S.B. Akbar, L. Rathour, L.N. Mishra, V.N. Mishra, Riemann-Liouville Fractional Versions of Hadamard inequality for Strongly m-Convex Functions, Int. J. Anal. Appl. 20 (2022), 5. https://doi.org/10.28924/2291-8639-20-2022-5.
  34. S.K. Paul, L.N. Mishra, V.N. Mishra, D. Baleanu, An effective method for solving nonlinear integral equations involving the Riemann-Liouville fractional operator, AIMS Math. 8 (2023), 17448–17469. https://doi.org/10.3934/math.2023891.