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Support points of the set of univalent functions


Authors: Louis Brickman and Donald Wilken
Journal: Proc. Amer. Math. Soc. 42 (1974), 523-528
MSC: Primary 30A36
DOI: https://doi.org/10.1090/S0002-9939-1974-0328057-1
MathSciNet review: 0328057
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Abstract: Let S be the usual set of analytic, univalent, normalized functions on the unit disk $ \Delta $. Let $ f \in S$. Then f is a support point of S, if there exists a continuous linear functional J on the space of analytic functions on $ \Delta $, J nonconstant on S, such that $ \operatorname{Re} J(f) = \max \{ \operatorname{Re} J(g):g \in S\} $. Theorem. Let f be a support point of S. Then f is analytic in the closed unit disk except for a pole of order two at one point of the unit circle. The complement of $ f(\Delta )$ is a single arc $ \Gamma $, regular and analytic everywhere, tending to $ \infty $ in such a way that the angle between $ \Gamma $ and the radial direction is always less than $ \pi /4$. Near $ \infty ,\Gamma $ can be described in the form $ \sum _{n = - 1}^\infty {d_n}{t^n}(0 < t < \delta ,{d_{ - 1}} \ne 0)$. In particular, $ \Gamma $ is asymptotic to a line $ {d_{ - 1}}{t^{ - 1}} + {d_0}$.


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DOI: https://doi.org/10.1090/S0002-9939-1974-0328057-1
Article copyright: © Copyright 1974 American Mathematical Society

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