We study extreme points of normalized families of analytic functions with values in a convex set. Suppose F is univalent and maps the unit disc onto a convex domain, other than a strip, a wedge, or a half plane, whose boundary contains a line segment. We prove that the set of extreme points of the family of functions subordinate to F is strictly contained between the minimal (compositions of F with inner functions vanishing at the origin) and maximal (compositions of F with extreme points of the family of bounded functions vanishing at the origin) sets. This provides the first example of a convex domain with this property.

Let F be a univalent function mapping the unit disc onto a convex domain D. Let f be subordinate to F. We show that if f is an extreme point of the subordination family to F then the closure of the boundary values ${\rm f}(\theta)$ of f $({\rm f}(\theta) = {\lim\atop{\rm r}\to 1}\ {\rm f(re}\sp{{\rm i}\theta})\ {\rm a.e.})$ has a nonempty intersection with the set of extreme points of the boundary of D. We demonstrate other necessary conditions in special cases. We also completely determine extreme points of the family of functions subordinate to a univalent function mapping the unit disc onto a convex domain whose boundary consists of finitely many arcs with positive curvature.

Next, we determine closed convex hulls and their extreme points for several subordination families with multivalent classes as majorants. These are used to determine precise coefficient bounds and to solve integral mean problems. We also determine support points of these subordination families. These results were known previously only in case of univalent majorants. Finally, we determine the support points of the set of functions subordinate to a function F which lies in the closed convex hull of the normalized convex functions.