Support points of the set of univalent functions
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- by Louis Brickman and Donald Wilken
- Proc. Amer. Math. Soc. 42 (1974), 523-528
- DOI: https://doi.org/10.1090/S0002-9939-1974-0328057-1
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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 ${d_{ - 1}}{t^{ - 1}} + {d_0}$.References
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Bibliographic Information
- © Copyright 1974 American Mathematical Society
- 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