Research article

On Bounds of fractional integral operators containing Mittag-Leffler functions for generalized exponentially convex functions

  • Received: 14 January 2021 Accepted: 29 March 2021 Published: 14 April 2021
  • MSC : 26A51, 26A33, 33E12

  • Recently, a generalization of convex function called exponentially $ (\alpha, h-m) $-convex function has been introduced. This generalization of convexity is used to obtain upper bounds of fractional integral operators involving Mittag-Leffler (ML) functions. Moreover, the upper bounds of left and right integrals lead to their boundedness and continuity. A modulus inequality is established for differentiable functions. The Hadamard type inequality is proved which shows upper and lower bounds of sum of left and right sided fractional integral operators.

    Citation: Maryam Saddiqa, Ghulam Farid, Saleem Ullah, Chahn Yong Jung, Soo Hak Shim. On Bounds of fractional integral operators containing Mittag-Leffler functions for generalized exponentially convex functions[J]. AIMS Mathematics, 2021, 6(6): 6454-6468. doi: 10.3934/math.2021379

    Related Papers:

  • Recently, a generalization of convex function called exponentially $ (\alpha, h-m) $-convex function has been introduced. This generalization of convexity is used to obtain upper bounds of fractional integral operators involving Mittag-Leffler (ML) functions. Moreover, the upper bounds of left and right integrals lead to their boundedness and continuity. A modulus inequality is established for differentiable functions. The Hadamard type inequality is proved which shows upper and lower bounds of sum of left and right sided fractional integral operators.



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    [1] M. U. Awan, M. A. Noor, K. I. Noor, Hermite-Hadamard inequalities for exponentially convex functions, Appl. Math. Inf. Sci., 12 (2018), 405–409. doi: 10.18576/amis/120215
    [2] S. Varo$\hat{s}$anec, On $h$-convexity, J. Math. Anal. Appl., 326 (2007), 303–311.
    [3] M. E. Ozdemir, A. O. Akdemri, E. Set, On $(h-m)$-convexity and Hadamard-type inequalities, J. Math. Mech., 8 (2016), 51–58.
    [4] V. Mihesan, A generalization of the convexity, Seminar on Functional Equations, Approx. Convex., Cluj-Napoca, Romania, 1993.
    [5] G. A. Anastassiou, Generalized fractional Hermit-Hadamard inequalities involving $m$-convexity and $(s, m)$-convexity, Ser. Math. Inform., 28 (2013), 107–126.
    [6] G. Farid, A. U. Rehman, Q. U. Ain, $k$-fractional integral inequalities of hadamard type for $(h-m)$-convex functions, Comput. Meth. Diff. Equ., 8 (2020), 119–140.
    [7] W. He, G. Farid, K. Mahreen, M. Zahra, N. Chen, On an integral and consequent fractional integral operators via generalized convexity, AIMS Math., 5 (2020), 7631–7647. doi: 10.3934/math.2020488
    [8] G. Mittag-Leffler, Sur la nouvelle fonction $E_{\alpha}(x)$, C. R. Acad. Sci. Paris., 137 (1903), 554–558.
    [9] H. J. Haubold, A. M. Mathai, R. K. Saxena, Mittag-Leffler functions and their applications, J. Appl. Math., 2011 (2011), Article ID 298628.
    [10] M. Arshad, J. Choi, S. Mubeen, K. S. Nisar, A New Extension of Mittag-Leffler function, Commun. Korean Math. Soc., 33 (2018), 549–560.
    [11] G. Rahman, D. Baleanu, M. A. Qurashi, S. D. Purohit, The extended Mittag-Leffler function via fractional calculus, J. Nonlinear Sci. Appl., 10 (2017), 4244–4253. doi: 10.22436/jnsa.010.08.19
    [12] 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.
    [13] A. K. Shukla, J. C. Prajapati, On a generalization of Mittag-Leffler function and its properties, J. Math. Anal. Appl., 336 (2007), 797–811. doi: 10.1016/j.jmaa.2007.03.018
    [14] M. Andrić, G. Farid, J. Pečarić, A further extension of Mittag-Leffler function, Fract. Calc. Appl. Anal., 21 (2018), 1377–1395. doi: 10.1515/fca-2018-0072
    [15] T. R. Prabhakar, A singular integral equation with a generalized Mittag-Leffler function in the kernel, Yokohama Math. J., 19 (1971), 7–15.
    [16] H. M. Srivastava, Z. Tomovski, Fractional calculus with an integral operator containing generalized Mittag-Leffler function in the kernel, Appl. Math. Comput., 211 (2009), 198–210. doi: 10.1016/j.amc.2009.01.055
    [17] S. Ullah, G. Farid, K. A. Khan, A. Waheed, Generalized fractional inequalities for quasi-convex functions, Adv. Difference Equ., 2019 (2019), 2019:15.
    [18] G. Abbas, K. A. Khan, G. Farid, A. U. Rehman, generalization of some fractional integral inequalities via generalized Mittag-Leffler function, J. Inequal. Appl., 2017 (2017), 121. doi: 10.1186/s13660-017-1389-9
    [19] R. Gorenflo, A. A. Kilbas, F. Mainardi, S. V. Rogosin, Mittag-Leffler Function, Related Topics and Applications, Springer, Berlin, Germany, 2, 2014.
    [20] G. Farid, Bounds of fractional integral operators containing Mittag-Leffler function, U.P.B. Sci. Bull., 81 (2019), 133–142.
    [21] Z. Chen, G. Farid, A. U. Rehman, N. Latif, Estimations of fractional integral operators for convex functions and related results, Adv. Diff. Equ., 2020 (2020), 2020:163.
    [22] G. Farid, Bounds of Riemann-Liouville fractional integral operators, Comput. Meth. Diff. Equ., 9 (2021), 637–648.
    [23] G. Farid, Some Riemann-Liouville fractional integral inequalities for convex functions, J. Anal., 27 (2019), 1095–1102. doi: 10.1007/s41478-018-0079-4
    [24] L. Chen, G. Farid, S. I. Butt, S. B. Akbar, Boundedness of fractional integral operators containing Mittag-Leffler functions, Turkish J. Ineq., 4 (2020), 14–24.
    [25] G. Hong, G. Farid, W. Nazeer, S. B. Akbar, J. Pecaric, et al., Boundedness of fractional integral operators containing Mittag-Leffler functions via exponentially $s$-convex functions, J. Math., 2020 (2020), Article ID 3584105, 7.
    [26] G. Farid, S. B. Akbar, S. U. Rehman, J. Pecarić, Boundedness of fractional integral operators containing Mittag-Leffler functions via $(s, m)$-convexity, AIMS Math., 5 (2020), 966–978. doi: 10.3934/math.2020318
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