Support points of families of analytic functions described by subordination

Authors:
D. J. Hallenbeck and T. H. MacGregor

Journal:
Trans. Amer. Math. Soc. **278** (1983), 523-546

MSC:
Primary 30C45; Secondary 30C80

DOI:
https://doi.org/10.1090/S0002-9947-1983-0701509-8

MathSciNet review:
701509

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Abstract: We determine the set of support points for several families of functions analytic in the open unit disc and which are generally defined in terms of subordination. The families we study include the functions with a positive real part, the typically-real functions, and the functions which are subordinate to a given majorant. If the majorant is univalent then each support point has the form , where is a finite Blaschke product and . This completely characterizes the set of support points when is convex. The set of support points is found for some specific majorants, including where . Let and denote the set of normalized convex and starlike mappings, respectively. We find the support points of the families and defined by the property of being subordinate to some member of or , respectively.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1983-0701509-8

Keywords:
Analytic function,
continuous linear functional,
support point,
extreme point,
closed convex hull,
subordination,
univalent function,
convex mapping,
starlike mapping,
function with a positive real part,
bounded function,
Herglotz formula,
probability measure,
finite Blaschke product

Article copyright:
© Copyright 1983
American Mathematical Society