Geometric properties of a class of support points of univalent functions
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- by Johnny E. Brown PDF
- Trans. Amer. Math. Soc. 256 (1979), 371-382 Request permission
Abstract:
Let S denote the set of functions $f(z)$ analytic and univalent in $|z| < 1$, 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.References
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Additional Information
- © Copyright 1979 American Mathematical Society
- 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