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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Subordination families and extreme points


Authors: Yusuf Abu-Muhanna and D. J. Hallenbeck
Journal: Trans. Amer. Math. Soc. 308 (1988), 83-89
MSC: Primary 30C80
DOI: https://doi.org/10.1090/S0002-9947-1988-0946431-1
MathSciNet review: 946431
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Abstract: Let $ s(F)$ denote the set of functions subordinate to a univalent function $ F$ in $ \Delta $ the unit disk. Let $ {B_0}$ denote the set of functions $ \phi (z)$ analytic in $ \Delta $ satisfying $ \vert\phi (z)\vert < 1$ and $ \phi (0) = 0$. We prove that if $ f = F \circ \phi $ is an extreme point of $ s(F)$, then $ \phi $ is an extreme point of $ {B_0}$. Let $ D = F(s)$ and $ \lambda (w,\,\partial D)$ denote the distance between $ w$ and $ \partial D$ (boundary of $ D$). We also prove that if $ \phi $ is an extreme point of $ {B_0}$ and $ \vert\phi ({e^{it}})\vert < 1$ for almost all $ t$, then $ \int_0^{2\pi } {\log \lambda (F(\phi ({e^{it}}){e^{i\theta }}),\,\partial D)\,dt = - \infty } $ for almost all $ \theta $.


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DOI: https://doi.org/10.1090/S0002-9947-1988-0946431-1
Keywords: Extreme point, Nevanlinna class, subordination, univalent function
Article copyright: © Copyright 1988 American Mathematical Society