An extension of a theorem of Gehring and Pommerenke

Abstract

Gehring and Pommerenke have shown that if the Schwarzian derivative Sf of an analytic function f in the unit disk D satisfies ISf(z)] ~_ 2(1 -Iz[2) -2, then f(D) is a Jordan domain except when f(D) is the image under a M6bius transformation of an infinite parallel strip. The condition ISf(z)l <_ 2(1 - lzl2) -2 is the classical sufficient condition for univalence of Nehari. In this paper we show that the same type of phenomenon established by Gehring and Pommerenke holds for a wider class of univalence criteria of the form [Sf(z)[ ~_ p([z[) also introduced by Nehari. These include [Sf(z)[ <_ lr2/2 and [Sf(z)[ (_ 4(1 -[z12) -1. We also obtain results on H61der continuity and quasiconformal extensions.