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Schwarzian derivative criteria for valence of analytic and harmonic mappings

Published online by Cambridge University Press:  01 September 2007

MARTIN CHUAQUI ,
PETER DUREN  and
BRAD OSGOOD
MARTIN CHUAQUI
Affiliation:
Facultad de Matemáticas, P. Universidad Católica de Chile, Casilla 306, Santiago 22, Chile email: mchuaqui@mat.puc.cl
PETER DUREN
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109–1109, U.S.A. email: duren@umich.edu
BRAD OSGOOD
Affiliation:
Department of Electrical Engineering, Stanford University, Stanford, California 94305, U.S.A. email: osgood@ee.stanford.edu
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Abstract

For analytic functions in the unit disk, general bounds on the Schwarzian derivative in terms of Nehari functions are shown to imply uniform local univalence and in some cases finite and bounded valence. Similar results are obtained for the Weierstrass–Enneper lifts of planar harmonic mappings to their associated minimal surfaces. Finally, certain classes of harmonic mappings are shown to have finite Schwarzian norm.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 143 , Issue 2 , September 2007 , pp. 473 - 486
Copyright
Copyright © Cambridge Philosophical Society 2007

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References

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