Schwarzian Derivatives and Uniform Local Univalence

Details Schwarzian Derivatives and Uniform Local Univalence Schwarzian Derivatives and Uniform Local Univalence

2007-06-28

arxiv.org/abs/0706.4296v2

Mathematics

Complex Variables

Scientific paper

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.

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