Monday Analysis Seminar: On a result by Clunie and Sheil-Small

2011-10-27 14:45 - 2011-10-27 16:15
Science Building #3-202
Ken-ichi Sakan (Osaka City University)
In 1984 J. Clunie and T. Sheil-Small proved that for any complex-valued and sense-preserving injective harmonic mapping $F$ in the unit disk $\Bbb D$,  if $F(\Bbb D)$ is a convex domain, then the inequality $|G(z_2)-G(z_1)|<|H(z_2)-H(z_1)|$ holds  for all distinct points $z_1,z_2\in\Bbb D$. Here $H$ and $G$ are holomorphic mappings in $\Bbb D$ determined by $F=H+\overline G$, up to a constant function.
We extend this inequality by replacing the unit disk  by an arbitrary nonempty domain $\Omega$ in $\Bbb C$, and improve it in the case where  $\Omega = \Bbb D$,  F(\Omega)$  is a convex domain and $F$ is quasiconformal.   As an application we show a result on characterizations  of quasiconformality of $F$ in the case where $F(\Omega)$ is a convex domain.

In the introductory part of this talk,  together with an explanation of the subject,  we give a brief review of investigations and some important observations  on quasiconformality of harmonic mappings.