On convolution, convex, and starlike mappings

"Let $C$ and $S^*$ stand for the classes of convex and starlike mapping in $\D$, and let $\cc$, $\cs$ denote the closures of the respective convex hulls. We derive characterizations for when the convolution of mappings in $\cc$ is convex, as well as when the convolution of mappings in $\cs$ is starlike. Several characterizations in terms of convolution are given for convexity within $\cc$ and for starlikeness within $\cs$. We also obtain a correspondence via convolution between $C$ and $S^*$, as well as correspondences between the subclasses of convex and starlike mappings that have $n$-fold symmetry."