Subordination, Extreme points and support points

Abstract

Let s(F) denote the set of functions subordinate to a function F analytic in the open unit disk Δ. Let be the set of functions f analytic in Δ such that

where ∞s(f)denotes the closed convex hull of s(f). Let denote those functions analytic in Δ such that the set of support points of s( f) is . In this paper we investigate the problems of describing functions in and Recently, Abu-Muhanna showed that univalent functions, with convex and satisfying a smoothness condition at ∞ must be in . We show that : is closed in the space of analytic functions. Consequently any F univalent with convex is in Moreover if such an F is not a half-plane mapping it belongs to

AMS No:

• 30C80
• 30C55