Geometric properties of a class of support points of univalent functions

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.