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 (defined below) of polynomials of degree less than or equal to having the properties: each polynomial which is univalent in the unit disk and of degree or less is in and if is a sequence of polynomials such that and (uniformly on compact subsets of the unit disk) then is univalent. The approach is to study the extreme points in ( is extreme if is not a proper convex combination of two distinct elements of ). Theorem 3 shows that if is extreme then is univalent and Theorem 6 gives a geometric condition on the image of the boundary of the disk under this mapping in order that be extreme. Theorem 10 states that the collection of polynomials univalent in the unit disk and having the property , are dense in the class of normalized univalent functions. These polynomials have the very striking geometric property that the tangent line to the curve , , turns at a constant rate (between cusps) as varies.

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Additional Information

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