  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.
