Subordination and extreme-point theory

David Hallenbeck, Thomas MacGregor
1974 Pacific Journal of Mathematics  
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
more » ... tion 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/is an extreme point. If FeH p , 1 < p < oo, and φ is an inner function with φ(Q) = 0, then F(φ) is an extreme point.
doi:10.2140/pjm.1974.50.455 fatcat:7y6yfzxj2befxkhaysldgbj2za