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Transactions of the American Mathematical Society

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

 

 

Extreme points in a class of polynomials having univalent sequential limits


Author: T. J. Suffridge
Journal: Trans. Amer. Math. Soc. 163 (1972), 225-237
MSC: Primary 30A06; Secondary 30A34
DOI: https://doi.org/10.1090/S0002-9947-1972-0294609-3
MathSciNet review: 0294609
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Abstract: This paper concerns a class $ {\mathcal{P}_n}$ (defined below) of polynomials of degree less than or equal to $ n$ having the properties: each polynomial which is univalent in the unit disk and of degree $ n$ or less is in $ {\mathcal{P}_n}$ and if $ \{ {P_{{n_k}}}\} _{k = 1}^\infty $ is a sequence of polynomials such that $ {P_{{n_k}}} \in {\mathcal{P}_{{n_k}}}$ and $ {\lim _{k \to \infty }}{P_{{n_k}}} = f$ (uniformly on compact subsets of the unit disk) then $ f$ is univalent. The approach is to study the extreme points in $ {\mathcal{P}_n}$ ( $ P \in {\mathcal{P}_n}$ is extreme if $ P$ is not a proper convex combination of two distinct elements of $ {\mathcal{P}_n}$). Theorem 3 shows that if $ P \in {\mathcal{P}_n}$ is extreme then $ ((n + 1)/n)P(z) - (1/n)zP'(z)$ is univalent and Theorem 6 gives a geometric condition on the image of the boundary of the disk under this mapping in order that $ P$ be extreme. Theorem 10 states that the collection of polynomials univalent in the unit disk and having the property $ P(z) = z + {a_2}{z^2} + \cdots + {a_n}{z^n},{a_n} = 1/n$, are dense in the class $ S$ of normalized univalent functions. These polynomials have the very striking geometric property that the tangent line to the curve $ P({e^{i\theta }})$, $ 0 \leqq \theta \leqq 2\pi $, turns at a constant rate (between cusps) as $ \theta $ varies.


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DOI: https://doi.org/10.1090/S0002-9947-1972-0294609-3
Keywords: Extreme point, convex hull, univalent polynomial
Article copyright: © Copyright 1972 American Mathematical Society