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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Support points of subordination families


Author: D. J. Hallenbeck
Journal: Proc. Amer. Math. Soc. 103 (1988), 414-416
MSC: Primary 30C80
MathSciNet review: 943058
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Abstract: Let $ s(F)$ denote the set of functions subordinate to a function $ F$ analytic in the unit disc $ \Delta $. Let $ Hs(F)$ denote the closed convex hull of $ s(F)$ and supp $ s(F)$ the set of support points of $ s(F)$. We prove the following

Theorem. Let $ F$ be analytic in $ \Delta $ and satisfy

(1) $ Hs(F) = \{ \int_{\partial \Delta } {F(xz)d\mu (x):\mu \;{\text{a}}\;{\text{probablity}}\;{\text{measure}}\;{\text{on}}\;\partial \Delta } \} $ and

(2) $ F(z) = G(z)/{(z - {x_0})^\alpha }$ where $ G$ is analytic in $ \Delta $, continuous in $ \bar \Delta $, $ G({x_0}) \ne 0$ and $ \alpha > 1$. Then supp $ s(F) = \left\{ {F(xz):\vert x\vert = 1} \right\}$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1988-0943058-8
PII: S 0002-9939(1988)0943058-8
Keywords: Support points, subordination
Article copyright: © Copyright 1988 American Mathematical Society