ON EXTREME-POINTS OF SUBORDINATION FAMILIES WITH A CONVEX MAJORANT

Let Δ={z∈C||z|<1}. Let B0 denote the set of functions φ analytic in Δ and satisfying |φ(z)| < 1, φ(0) = 0. Suppose F is analytic and univalent in Δ and maps it onto a convex domain D other than a strip, a wedge or a half plane. Let s(F) = {Foφ|φε{lunate}B0}. Let Es(F) denote the set of extreme points of s(F). D. J. Hallenbeck and T. H. MacGregor asked in [3] whether the set Es(F) ever lies strictly between the minimal ({Foφ|φε{lunate}B0, limr→1|φ(reiθ)| = 1 a.e.}) and the maxima ({Fcφ|φ is an extreme point of B0}) sets. We prove that this always happens when the boundary of D contains a line segment. Let D = F(Δ) be any convex domain, and let f be subordinate to F. We show that if f is an extreme point of s(F) then the closure of the boundary values f(θ) of f(f(θ) = limr→1f(reiθ a.e.) has a nonempty intersection with the set of extreme points of the boundary of D. We also completely determine extreme points of s(F) when the boundary of D consists of finitely many arcs wih positive curvature. © 1990.