Branch point area methods in conformal mapping

Abstract

The classical estimate of Bieberbach that ⋎a ₂⋎≤2 for a given univalent function ϕ(z)=z+a ₂ z ²+… in the classS leads to the best possible pointwise estimates of the ratio ϕ"(z)/ϕ'(z) for ϕ∈S, first obtained by Kœbe and Bieberbach. For the corresponding class Σ of univalent functions in the exterior disk, Goluzin found in 1943 by variational methods the corresponding best possible pointwise estimates of ϕ"(z)/ϕ'(z) for ψ∈Σ. It was perhaps surprising that this time, the expressions involve elliptic integrals. Here, we obtain an area-type theorem which has Goluzin's pointwise estimate as a corollary. This shows that Goluzin's estimate, like the Kœbe-Bieberbach estimate, is firmly rooted in areabased 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.