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A generalized Schwarz lemma at the boundary

Author: Dov Chelst
Journal: Proc. Amer. Math. Soc. 129 (2001), 3275-3278
MSC (2000): Primary 30C80
Published electronically: June 6, 2001
MathSciNet review: 1845002
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Abstract: Let $\phi$ be an analytic function mapping the unit disc $\mathbb {D}$ to itself. We generalize a boundary version of Schwarz’s lemma proven by D. Burns and S. Krantz and provide sufficient conditions on the local behavior of $\phi$ near a finite set of boundary points that requires $\phi$ to be a finite Blaschke product. Afterwards, we supply several counterexamples to illustrate that these conditions may also be necessary.

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Additional Information

Dov Chelst
Affiliation: Department of Mathematics, Hill Center, Rutgers, The State University of New Jersey, 110 Frelinghuysen Rd., Piscataway, New Jersey 08854-8019

Keywords: Schwarz’s lemma, Schur functions, bounded analytic functions, Blaschke product
Received by editor(s): March 10, 2000
Published electronically: June 6, 2001
Additional Notes: The author would like to thank Dr. R.B. Burckel for referring him to the article by Krantz and Burns and to also thank Drs. X. Huang, S. Goldstein and B. Walsh for their advice on this article’s contents.
Communicated by: Juha M. Heinonen
Article copyright: © Copyright 2001 American Mathematical Society