  In this paper our aim is to determine the radii of univalence, starlikeness
and convexity of the normalized regular Coulomb wave functions for two
different kinds of normalization. The key tools in the proof of our main
results are the Mittag-Leffler expansion for regular Coulomb wave functions,
and properties of zeros of the regular Coulomb wave functions and their
derivatives. Moreover, by using the technique of differential subordinations we
present some conditions on the parameters of the regular Coulomb wave function
in order to have a starlike normalized form. In addition, by using the
Euler-Rayleigh inequalities we obtain some tight bounds for the radii of
starlikeness of the normalized regular Coulomb wave functions. Some open
problems for the zeros of the regular Coulomb wave functions are also stated
which may be of interest for further research.
