On the dilatation of univalent planar harmonic mappings
HTML articles powered by AMS MathViewer
- by Allen Weitsman PDF
- Proc. Amer. Math. Soc. 126 (1998), 447-452 Request permission
Abstract:
It is shown that if $f$ is a univalent harmonic mapping of the unit disk onto a domain having a smooth boundary arc which is convex with respect to the domain, and if the dilatation has modulus 1 on the arc, then the arc must be a line segment.References
- Yusuf Abu-Muhanna and Abdallah Lyzzaik, The boundary behaviour of harmonic univalent maps, Pacific J. Math. 141 (1990), no. 1, 1–20. MR 1028262, DOI 10.2140/pjm.1990.141.1
- Robert Finn, New estimates for equations of minimal surface type, Arch. Rational Mech. Anal. 14 (1963), 337–375. MR 157096, DOI 10.1007/BF00250712
- P. Hebroni, Sur les inverses des éléments dérivables dans un anneau abstrait, C. R. Acad. Sci. Paris 209 (1939), 285–287 (French). MR 14
- W. Hengartner and G. Schober, On the boundary behavior of orientation-preserving harmonic mappings, Complex Variables Theory Appl. 5 (1986), no. 2-4, 197–208. MR 846488, DOI 10.1080/17476938608814140
- W. Hengartner and G. Schober, Harmonic mappings with given dilatation, J. London Math. Soc. (2) 33 (1986), no. 3, 473–483. MR 850963, DOI 10.1112/jlms/s2-33.3.473
- R. Laugesen, Planar harmonic maps with inner and Blaschke dilatations, to appear.
Additional Information
- Allen Weitsman
- Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
- Email: weits@math.purdue.edu
- Received by editor(s): November 18, 1995
- Received by editor(s) in revised form: May 6, 1996, and August 6, 1996
- Communicated by: Albert Baernstein II
- © Copyright 1998 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 126 (1998), 447-452
- MSC (1991): Primary 30C62, 31A05, 31A20, 49Q05
- DOI: https://doi.org/10.1090/S0002-9939-98-04076-3
- MathSciNet review: 1415341