Radii of starlikeness and convexity of generalized Mittag-Leffler functions

摘要

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 the Laguerre-Polya class and a result of H. Kumar and M. A. Pathan on the reality of the zeros of generalized Mittag-Leffler functions, whose origins go back to Dzhrbashyan, Ostrovskii and Peresyolkova, play an important role in this paper. Moreover, the interlacing properties of the zeros of the Mittag-Leffler function and its derivative are 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.