  If $\Omega$ is a simply connected domain in $\overline{{\mathbb C}}$ then,
according to the Ahlfors-Gehring theorem, $\Omega$ is a quasidisk if and only
if there exists a sufficient condition for the univalence of holomorphic
functions in $\Omega$ in relation to the growth of their Schwarzian derivative.
We extend this theorem to harmonic mappings by proving a univalence criterion
on quasidisks. We also show that the mappings satisfying this criterion admit a
homeomorphic extension to $\overline{{\mathbb C}}$ and, under the additional
assumption of quasiconformality in $\Omega$, they admit a quasiconformal
extension to $\overline{{\mathbb C}}$.
  The Ahlfors-Gehring theorem has been extended to finitely connected domains
$\Omega$ by Osgood, Beardon and Gehring, who showed that a Schwarzian criterion
for univalence holds in $\Omega$ if and only if the components of
$\partial\Omega$ are either points or quasicircles. We generalize this theorem
to harmonic mappings.

Complex Variables