Support points of the set of univalent functions

Abstract:

Let S be the usual set of analytic, univalent, normalized functions on the unit disk $\Delta$. Let $f \in S$. Then f is a support point of S, if there exists a continuous linear functional J on the space of analytic functions on $\Delta$, J nonconstant on S, such that $\operatorname {Re} J(f) = \max \{ \operatorname {Re} J(g):g \in S\}$. Theorem. Let f be a support point of S. Then f is analytic in the closed unit disk except for a pole of order two at one point of the unit circle. The complement of $f(\Delta )$ is a single arc $\Gamma$, regular and analytic everywhere, tending to $\infty$ in such a way that the angle between $\Gamma$ and the radial direction is always less than $\pi /4$. Near $\infty ,\Gamma$ can be described in the form $\sum _{n = - 1}^\infty {d_n}{t^n}(0 < t < \delta ,{d_{ - 1}} \ne 0)$. In particular, $\Gamma$ is asymptotic to a line ${d_{ - 1}}{t^{ - 1}} + {d_0}$.