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Transactions of the American Mathematical Society

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Geometric properties of a class of support points of univalent functions


Author: Johnny E. Brown
Journal: Trans. Amer. Math. Soc. 256 (1979), 371-382
MSC: Primary 30C55
DOI: https://doi.org/10.1090/S0002-9947-1979-0546923-8
MathSciNet review: 546923
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Abstract: Let S denote the set of functions $ f(z)$ analytic and univalent in $ \vert z\vert\, < \,1$, normalized by $ f(0)\, = \,0$ and $ f'(0)\, = \,1$. A function f is a support point of S if there exists a continuous linear functional L, nonconstant on S, for which f maximizes Re $ \operatorname{Re} \,L(g)$, $ g \in S$. The support points corresponding to the point-evaluation functionals are determined explicitly and are shown to also be extreme points of S. New geometric properties of their omitte $ \operatorname {arcs}\,\Gamma $ are found. In particular, it is shown that for each such support point $ \Gamma $ lies entirely in a certain half-strip, $ \Gamma $ has monotonic argument, and the angle between radius and tangent vectors increases from zero at infinity to a finite maximum value at the tip of the $ \operatorname{arc}\,\Gamma $. Numerical calculations appear to indicate that the known bound $ \pi /4$ for the angle between radius and tangent vectors is actually best possible.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1979-0546923-8
Keywords: Support points, extreme points, univalent functions
Article copyright: © Copyright 1979 American Mathematical Society

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