Radii of starlikeness and convexity of generalized Mittag-Leffler functions

Arpad Baricz, Anuja Prajapati


In this paper our aim is to find the radii of starlikeness and convexity of the generalized Mittag-Leffler function for three different kinds of normalization by using their Hadamard factorization in such a way that the resulting functions are analytic in the unit disk of the complex plane. The characterization of entire functions from Laguerre-P\'{o}lya class and a result of H. Kumar and M.A. Pathan on the reality of the zeros of generalized Mittag-Leffler functions, which origins goes back to Dzhrbashyan, Ostrovski\u{i} and Peresyolkova, play important roles in this paper. Moreover, the interlacing properties of the zeros of Mittag-Leffler function and its derivative is also useful in the proof of the main results. By using the Euler-Rayleigh inequalities for the real zeros of the generalized Mittag-Leffler function, we obtain some tight lower and upper bounds for the radii of starlikeness and convexity of order zero.


Generalized Mittag-Leffler functions; starlike and convex functions; radius of starlikeness; radius of convexity; entire functions; real zeros; Euler-Rayleigh inequalities; Weierstrassian decomposition.

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ISSN: 1331-0623 (Print), 1848-8013 (Online)