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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

A Schwarz lemma for multivalued functions and
distortion theorems for Bloch functions
with branch points


Authors: Ian Graham and David Minda
Journal: Trans. Amer. Math. Soc. 351 (1999), 4741-4752
MSC (1991): Primary 30C80, 30C45
Published electronically: August 25, 1999
MathSciNet review: 1694292
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Abstract | References | Similar Articles | Additional Information

Abstract: We give a version of the Schwarz lemma for multivalued mappings between hyperbolic plane regions. As in the original work of Nehari on this subject, the derivative must remain bounded near the branch points. Our version of the distance-decreasing principle represents a considerable strengthening of previous results. We apply it to the study of Bloch functions with branch points of specified order. We obtain upper and lower estimates for $|f'|$, an upper estimate for $|f|$, and a lower estimate for the radius of the largest schlicht disk in the image of $f$ centered at $f(0)$. We also obtain some results requiring estimates of second order derivatives of $f$.


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Additional Information

Ian Graham
Affiliation: Department of Mathematics, University of Toronto, Toronto, Ontario M5S 3G3, Canada
Email: graham@math.utoronto.ca

David Minda
Affiliation: Department of Mathematical Sciences, University of Cincinnati, Cincinnati, Ohio 45221-0025
Email: david.minda@math.uc.edu

DOI: https://doi.org/10.1090/S0002-9947-99-02540-4
Received by editor(s): February 6, 1997
Received by editor(s) in revised form: October 12, 1998
Published electronically: August 25, 1999
Additional Notes: The research of the first author was partially supported by NSERC
The research of the second author was partially supported by a National Science Foundation Grant
Article copyright: © Copyright 1999 American Mathematical Society