  We consider the class univalent log-harmonic mappings on the unit disk.
Firstly, we obtain necessary and sufficient conditions for a complex-valued
continuous function to be starlike or convex in the unit disk. Then we present
a general idea, for example, to construct log-harmonic Koebe mapping,
log-harmonic right half-plane mapping and log-harmonic two-slits mapping and
then we show precise ranges of these mappings. Moreover, coefficient estimates
for univalent log-harmonic starlike mappings are obtained. Growth and
distortion theorems for certain special subclass of log-harmonic mappings are
studied. Finally, we propose two conjectures, namely, log-harmonic coefficient
and log-harmonic covering conjectures.
