An Application of an Operator Associated with Generalized Mittag-Leffler Function

The main object of this paper is to give an application of an operator associated with generalized Mittag-Leffler function in the unit disk $% \mathcal{U}=\{z\in \mathbb{C}:\left\vert z\right\vert <1\}$ to the differential inequalities.
Keywords:

Analytic functions,

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  • [1] A. A. Attiya, Some Applications of Mittag-Leffler Function in the Unit Disk, Filomat 30:7 (2016), 2075–2081.
  • [2] D. Bansal, J. K. Prajapat, Certain geometric properties of the Mittag-Leffler functions, Complex Var. Elliptic Equ., 61(3)(2016), 338-350.
  • [3] B. A. Frasin, Starlikeness and convexity of integral operators involving Mittag-Leffler functions, TWMS Journal of Pure and Applied Mathematics, in press.
  • [4] M. Garg, P. Manohar and S.L. Kalla, A Mittag-Leffler-type function of two variables. Integral Transforms Spec. Funct. 24 (2013), no. 11, 934–944.
  • [5] V. Kiryakova, Generalized fractional calculus and applications. Pitman Research Notes in Mathematics Series, 301. Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1994.
  • [6] V. Kiryakova, Multiple (multiindex) Mittag-Leffler functions and relations to generalized fractional calculus. Higher transcendental functions and their applications, J. Comput. Appl. Math. 118 (2000), no. 1-2, 241–259.
  • [7] V. Kiryakova, The multi-index Mittag-Leffler functions as an important class of special functions of fractional calculus, Comput. Math. Appl. 59 (2010), no. 5, 1885–1895.
  • [8] F. Mainardia and R. Gorenflo, On Mittag-Leffler-type functions in fractional evolution processes. Higher transcendental functions and their applications. J. Comput. Appl. Math. 118 (2000), no. 1-2, 283–299.
  • [9] S. S. Miller and P.T. Mocanu, Second order differential inequalities in the complex plane, J. Math. Ana.Appl. 65(1978), 289-305.
  • [10] G. M. Mittag-Leffler, Sur la nouvelle fonction E(x), C. R. Acad. Sci., Paris, 137(1903), 554-558.
  • [11] H. M. Srivastava, B. A. Frasin and Virgil Pescar, Univalence of Integral Operators Involving Mittag-Leffler Functions, Appl. Math. Inf. Sci. 11, No. 3, 635-641 (2017).
  • [12] H.M. Srivastava and Z. Tomovski, Fractional calculus with an integral operator containing a generalized Mittag-Leffler function in the kernel, Appl. Math. Comp., 211(2009), 198-210.
  • [13] A. Wiman, Uber den Fundamental satz in der Theorie der Funcktionen E(x), Acta Math., 29(1905), 191-201.