On convolution, convex and starlike mappings

Abstract

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