Convex hulls and extreme points of families of starlike and close-to-convex mappings

The closed convex hull is obtained for the functions which are starlike of order a, A>fold symmetric, and real on (-1,1). The same result for the close-to-convex functions which are &-fold symmetric is obtained. Integral representations are given for the hulls of these and other families in terms of probability measures on suitable sets. These results are used to solve extremal problems.