Monday Analysis Seminar: On a result by Clunie and SheilSmall 2
 Date

20111028 10:30 
20111028 12:00
 Place
 Science Building #3210

Speaker/Organizer
 Kenichi Sakan (Osaka City University)

 In 1984 J. Clunie and T. SheilSmall proved that for any complexvalued and sensepreserving 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.