Branch point area methods in conformal mapping

Abstract

The classical estimate of Bieberbach -- that $|a_2|\le2$ for a given univalent function $\phi(z)=z+a_2z^2+...$ in the class $S$ -- leads to best possible pointwise estimates of the ratio $\phi''(z)/\phi'(z)$ for $\phi\in S$, first obtained by K\oe{}be and Bieberbach. For the corresponding class $\Sigma$ of univalent functions in the exterior disk, Goluzin found in 1943 -- by extremality methods -- the corresponding best possible pointwise estimates of $\psi''(z)/\psi'(z)$ for $\psi\in\Sigma$. It was perhaps surprising that this time, the expressions involve elliptic integrals. Here, we obtain the area-type theorem which has Goluzin's pointwise estimate as a corollary. This shows that the K\oe{}be-Bieberbach estimate as well as that of Goluzin are both firmly rooted in the area-based methods. The appearance of elliptic integrals finds a natural explanation: they arise because a certain associated covering surface of the Riemann sphere is a torus.