An Extension of a Theorem of Gehring and Pommerenke
نویسنده
چکیده
Gehring and Pommerenke have shown that if the Schwarzian derivative S f of an analytic function f in the unit disk D satisfies ISf(z)] ~_ 2(1 I z [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. Received December 26, 1993
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