An extension of a theorem of Gehring and Pommerenke |
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Authors: | M. Chuaqui B. Osgood |
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Affiliation: | (1) Department of Mathematics, Pontifica Universidad Católica, Casilla 6177, Santiago, Chile;(2) Department of Mathematics, Stanford University, 94305-2125 Stanford, CA, USA;(3) Department of Mathematics, Institute des Hautes Etudes Sciences, 35 Route de Chartres, 91440 Bures-sur-Yvette, France |
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Abstract: | Gehring and Pommerenke have shown that if the Schwarzian derivativeSf of an analytic functionf in the unit diskD satisfies |Sf(z)|≤, 2(1 - |z|2)–2 thenf(D) is a Jordan domain except whenf(D) is the image under a Möbius transformation of an infinite parallel strip. The condition |Sf(z)|≤ 2(1 - |z|2)–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)|≤π 2/2 and|Sf((z)|≤4(1-|z| 2)–1. We also obtain results on Hölder continuity and quasiconformal extensions. |
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