Hull subordination and extremal problems for starlike and spirallike mappings

Author:
Thomas H. MacGregor

Journal:
Trans. Amer. Math. Soc. **183** (1973), 499-510

MSC:
Primary 30A32

DOI:
https://doi.org/10.1090/S0002-9947-1973-0338339-9

MathSciNet review:
0338339

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Abstract: Let be a compact subset of the family of functions analytic in , and let be a continuous linear operator of order zero on . We show that if the extreme points of the closed convex hull of is the set , then is hull subordinate to in . This generalizes results of R. M. Robinson corresponding to families of functions that are subordinate to or to . Families to which this theorem applies are discussed and we identify each such operator with a suitable sequence of complex numbers.

Suppose that is a nonconstant entire function and that . We show that the maximum of over the class of starlike functions of order *a* is attained only by the functions . A similar result is obtained for spirallike mappings. Both results generalize a theorem of G. M. Golusin corresponding to the family of starlike mappings.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1973-0338339-9

Keywords:
Subordination,
hull subordination,
continuous linear operator,
functions with positive real part,
Herglotz representation formula,
extreme points,
closed convex hull,
univalent functions,
starlike mapping,
starlike functions of order *a*,
spirallike functions,
extremal problems,
Kreĭn-Milman theorem,
probability measure,
convex mapping,
convex functions of order *a*,
continuous linear functional

Article copyright:
© Copyright 1973
American Mathematical Society