Branch point area methods in conformal mapping

The classical estimate of Bieberbach -- that $|a_2|\le2$ for a given univalent function $ϕ(z)=z+a_2z^2+...$ in the class $S$ -- leads to best possible pointwise estimates of the ratio $ϕ''(z)/ϕ'(z)$ for $ϕ\in S$, first obtained by Kœbe and Bieberbach. For the corresponding class $Σ$ of univalent functions in the exterior disk, Goluzin found in 1943 -- by extremality methods -- the corresponding best possible pointwise estimates of $ψ''(z)/ψ'(z)$ for $ψ\inΣ$. 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 Kœbe-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.

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