Vol. 50, No. 2, 1974

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Subordination and extreme-point theory

David James Hallenbeck and Thomas Harold MacGregor

Vol. 50 (1974), No. 2, 455–468

This paper examines the set of extreme points of the convex hull of families of analytic functions defined through subordination. The set of extreme points is determined for the class of functions each of which is subordinate to some starlike, univalent mapping of the open unit disk. This set is also determined for the family defined by subordination to some convex mapping, and a partial determination is obtained for subordination to some close-to-convex mapping. This information is used to solve extremal problems over such families. Results are also presented about the extreme points for the functions which are subordinate to a given analytic function F. For example, if f(z) = F(xz) and |x| = 1 then f is an extreme point. If F Hp,1 < p < , and ϕ is an inner function with ϕ(0) = 0, then F(ϕ) is an extreme point.

Mathematical Subject Classification
Primary: 30A32
Received: 28 September 1972
Revised: 15 October 1973
Published: 1 February 1974
David James Hallenbeck
Thomas Harold MacGregor