  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.
