  In this paper we deal with the radii of starlikeness and convexity of the
$q-$Mittag--Leffler function for three different kinds of normalization by
making use of their Hadamard factorization in such a way that the resulting
functions are analytic in the unit disk of the complex plane. By applying
Euler-Rayleigh inequalities for the first positive zeros of these functions
tight lower and upper bounds for the radii of starlikeness of these functions
are obtained. The Laguerre-P\'olya class of real entire functions plays a
pivotal role in this investigation.
