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.
$q$-Mittag-Leffler functions, univalent, starlike and convex functions, radius of starlikeness and convexity, Laguerre-P\'olya class of entire functions
"Radii of starlikeness and convexity of $q$-Mittag-Leffler functions,"
Turkish Journal of Mathematics: Vol. 43:
5, Article 43.
Available at: https://journals.tubitak.gov.tr/math/vol43/iss5/43