On some geometric properties of the Le Roy-type Mittag-Leffler function

Abstract

In this paper, we consider the Le Roy-type Mittag-Leffler function. Our main focus is to establish some sufficient conditions so that the normalized Le-Roy type Mittag-Leffler function posses some geometric properties such as starlikeness, convexity, close-to-convexity (univalency) and uniformly convexity inside the unit disk. Using these results, geometric properties of the normalized Mittag-Leffler function are derived as application. Results obtained in this paper are new. Interesting consequences, corollaries and examples are provided to support that these results are better and improve several results available in the literature.

Keywords

• Mittag-Leffler function
• analytic function
• univalent
• starlike
• convex and close-to-convex functions

References

1. [1] M.A. Al-Bassam and Y.F. Luchko, On generalized fractional calculus and its application to the solution of integro-differential equations, J. Fract. Calc. 7, 69-88, 1995.
2. [2] D. Bansal and J.K. Prajapat, Certain geometric properties of the Mittag-Leffler functions, Complex Var. Elliptic Equ. 61 (3), 338-350, 2016.
3. [3] Á. Baricz, Generalized Bessel functions of the first kind, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2010.
4. [4] R.W. Conway and W.L. Maxwell, A queuing model with state dependent service rates J. Ind. Eng. 12, 132-136, 1962.
5. [5] P.L. Duren, Univalent functions, Grundlehren der Mathematischen Wissenschaften, 259, Springer-Verlag, New York, 1983.
6. [6] R. Garra and F. Polito, On some operators involving Hadamard derivatives, Integral Transforms Spec. Funct. 24 (10), 773-782, 2013.
7. [7] R. Garrappa, S. Rogosin and F. Mainardi, On a generalized three-parameter Wright function of Le Roy type, Fract. Calc. Appl. Anal. 20 (5), 1196-1215, 2017.
8. [8] S. Gerhold, Asymptotics for a variant of the Mittag-Leffler function, Integral Transforms Spec. Funct. 23 (6), 397-403, 2012.

9. [9] A.W. Goodman, On uniformly convex functions, Ann. Polon. Math. 56 (1), 87-92, 1991.
10. [10] A.W. Goodman, On uniformly starlike functions, J. Math. Anal. Appl. 155 (2), 364- 370, 1991.
11. [11] K. Górska, A. Horzela and R. Garrappa, Some results on the complete monotonicity of Mittag-Leffler functions of Le Roy type, Fract. Calc. Appl. Anal. 22 (5), 1284-1306, 2019.
12. [12] B.N. Gu and F. Qi, An extension of an inequality for ratios of gamma functions, J. Approx. Theory 163 (9), 1208-1216, 2011.
13. [13] T.H. MacGregor, The radius of univalence of certain analytic functions II, Proc Amer. Math. Soc. 14, 521-524, 1963.
14. [14] T.H. MacGregor, A class of univalent functions, Proc. Amer. Math. Soc. 15, 311-317, 1964.
15. [15] K. Mehrez, Some geometric properties of a class of functions related to the Fox-Wright functions, Banach J. Math. Anal. 14 (3), 1222-1240, 2020.
16. [16] K. Mehrez, S. Das and A. Kumar, Geometric properties of the products of modified Bessel functions of the first kind, Bull. Malays. Math. Sci. Soc. 44 (5), 2715-2733, 2021.
17. [17] M.G. Mittag-Leffler, Sur la nouvelle function e(x), Comptes Rendus hebdomadaires de Séances de l’Academié des Sciences, Paris 137, 554-558, 1903.
18. [18] M.G. Mittag-Leffler, Une généralisation de l’intégrale de Laplace-Abel, Comptes Rendus hebdomadaires de Séances de l’Academié des Sciences, Paris, 136, 537-539, 1903.
19. [19] P.T. Mocanu, Some starlike conditions for analytic functions, Rev. Roumaine. Math. Pures. Appl. 33, 117-124, 1988.
20. [20] S. Noreen, M. Raza, M.U. Din and S. Hussain, On Certain Geometric Properties of Normalized Mittag-Leffler Functions, U. P. B. Sci. Bull. Series A 81 (4), 167-174, 2019.
21. [21] S. Noreen, M. Raza, J.-L. Liu and M. Arif, Geometric Properties of Normalized Mittag-Leffler Functions, Symmetry 11 (1), 45, 2019.
22. [22] S. Ozaki, On the theory of multivalent functions, Sci. Rep. Tokyo Bunrika Daigaku A, 2, 167-188, 1935.
23. [23] T.K. Pogány, Integral form of Le Roy-type hypergeometric function, Integral Transforms Spec. Funct. 29 (7), 580-584, 2018.
24. [24] V. Ravichandran, On uniformly convex functions, Ganita 53 (2), 117-124, 2002.
25. [25] S.V. Rogosin, The role of the Mittag-Leffler function in fractional modeling, Mathematics 3, 368-381, 2015.
26. [26] F. Rønning, Uniformly convex functions and a corresponding class of starlike functions, Proc. Amer. Math. Soc. 118 (1), 189-196, 1993.
27. [27] É. Le Roy, Valeurs asymptotiques de certaines séries procédant suivant les puissances entiéreset positives d’une variable réelle, Bull des Sci Math. 24 (2), 245-268, 1900.
28. [28] T. Simon, Remark on a Mittag-Leffler function of Le Roy type, Integral Transforms Spec. Funct. 33 (2), 108-114, 2022.