A proof of Sethares' conjecture (Q1777546)

A criteria for extremality of Teichmüller mappings is constructed. Recall that a quasiconformal self mapping of the unit disc is Teichmüller if its complex dilatation satisfies that \(\frac{\mu_f}{| \mu_f| }=\frac{\overline{\varphi(z)}}{| \varphi(z)| }\) where \(\varphi\) is holomorphic. Sethares proved that \(f\) is extremal if \(\varphi\) has either a double pole or no pole of order exceeding two. In the present paper it is proved that this condition is also sufficient and a more general criteria for extremality is presented.