Extreme points in a class of polynomials having univalent sequential limits

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.