A generalized Schwarz lemma at the boundary
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- by Dov Chelst
- Proc. Amer. Math. Soc. 129 (2001), 3275-3278
- DOI: https://doi.org/10.1090/S0002-9939-01-06144-5
- Published electronically: June 6, 2001
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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.References
- Daniel M. Burns and Steven G. Krantz, Rigidity of holomorphic mappings and a new Schwarz lemma at the boundary, J. Amer. Math. Soc. 7 (1994), no. 3, 661–676. MR 1242454, DOI 10.1090/S0894-0347-1994-1242454-2
- David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR 737190, DOI 10.1007/978-3-642-61798-0
Bibliographic Information
- Dov Chelst
- Affiliation: Department of Mathematics, Hill Center, Rutgers, The State University of New Jersey, 110 Frelinghuysen Rd., Piscataway, New Jersey 08854-8019
- Email: chelst@math.rutgers.edu
- 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
- © Copyright 2001 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 129 (2001), 3275-3278
- MSC (2000): Primary 30C80
- DOI: https://doi.org/10.1090/S0002-9939-01-06144-5
- MathSciNet review: 1845002