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

 

A note on extreme points of subordination classes


Author: D. J. Hallenbeck
Journal: Proc. Amer. Math. Soc. 103 (1988), 1167-1170
MSC: Primary 30C45; Secondary 30C80
MathSciNet review: 955001
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Abstract: Let $ s(F)$ denote the set of functions subordinate to a univalent function $ F$ in $ \Delta $ in the unit disc. Let $ B$ denote the set of functions $ \phi (z)$ analytic in $ \Delta $ satisfying $ \vert{\phi (x)}\vert < 1$ and $ \phi (0) = 0$. Let $ D = F(\Delta )$ and $ \lambda (w,\partial D)$ denote the distance between $ w$ and $ \partial D$ (boundary of $ D$). We prove that if $ \phi $ is an extreme point of $ B$ then $ \int_0^{2\pi } {\log \lambda (F(\phi ({e^{it}})),\partial D)dt = - \infty } $. As a corollary we prove that if $ F \circ \phi $ is an extreme point of $ s(F)$ then $ \int_0^{2\pi } {\log \lambda (F(\phi ({e^{it}})),\partial D)dt = - \infty } $.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1988-0955001-6
PII: S 0002-9939(1988)0955001-6
Keywords: Extreme point, subordination, univalent function
Article copyright: © Copyright 1988 American Mathematical Society