Computational Methods and Function Theory 8 (2008), No. 1, 21--34
Copyright Heldermann Verlag 2008
Martin Chuaqui
mchuaqui@mat.puc.cl , P. Universidad Católica de Chile, Facultad de Matemáticas, Casilla 306, Santiago 22, Chile.
Peter Duren
duren@umich.edu , University of Michigan, Department of Mathematics, Ann Arbor, Michigan 48109--1043, U.S.A.
Brad Osgood
osgood@ee.stanford.edu , Stanford University, Department of Electrical Engineering, Stanford, California 94305, U.S.A.
Quantitative estimates are obtained for the (finite) valence of functions analytic in the unit disk with Schwarzian derivative that is bounded or of slow growth. A harmonic mapping is shown to be uniformly locally univalent with respect to the hyperbolic metric if and only if it has finite Schwarzian norm, thus generalizing a result of B. Schwarz for analytic functions. A numerical bound is obtained for the Schwarzian norms of univalent harmonic mappings.
Keywords: Analytic function, valence, harmonic mapping, Schwarzian derivative, uniform local univalence, Schwarzian norm, minimal surface, harmonic lift.
MSC 2000: Primary 30C99; Secondary 31A05, 30C55.
[FullText-pdf (281 K)] [FullText-ps (471 K)]