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Bloch constants for planar harmonic mappings
Author(s):
Huaihui
Chen;
P.
M.
Gauthier;
W.
Hengartner
Journal:
Proc. Amer. Math. Soc.
128
(2000),
3231-3240.
MSC (2000):
Primary 30C99;
Secondary 30C62
Posted:
March 2, 2000
MathSciNet review:
1707142
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Abstract:
We give a lower estimate for the Bloch constant for planar harmonic mappings which are quasiregular and for those which are open. The latter includes the classical Bloch theorem for holomorphic functions as a special case. Also, for bounded planar harmonic mappings, we obtain results similar to a theorem of Landau on bounded holomorphic functions.
References:
-
- 1.
- S. Bochner, Bloch's theorem for real variables, Bull. Amer. Math. Soc. 52 (1946), 715-719. MR 8,204a
- 2.
- D. Bshouty and W. Hengartner, Univalent harmonic mappings in the plane, Ann. Univ. Mariae Curie-Sklodowska Sect. A 48 (1994), 12-42. MR 96m:30025
- 3.
- H. Chen and P. M. Gauthier, On Bloch's constant, J. Anal. Math. 69 (1996), 275-291. MR 97j:30002
- 4.
- E. Heinz, On one-to-one harmonic mappings, Pacific J. Math. 9 (1959), 101-105. MR 21:3683
- 5.
- H. Lewy, On the non-vanishing of the Jacobian in certain one-to-one mappings, Bull. Amer. Math. Soc. 42 (1936), 689-692.
- 6.
- S. Stoïlow, Principes topologiques de la théorie des fonctions analytiques, 1938. Gauthier-Villars.
- 7.
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Additional Information:
Huaihui
Chen
Affiliation:
Department of Mathematics, Nanjing Normal University, Nanjing, Jiangsu 210097, People's Republic of China
Email:
hhchen@njnu.edu.cn
P.
M.
Gauthier
Affiliation:
Département de mathématiques et de statistique, Université de Montréal, Montréal, Québec, H3C 3J7, Canada
Email:
gauthier@dms.umontreal.ca
W.
Hengartner
Affiliation:
Département de mathématiques, Université Laval, Québec, G1K 7P4, Canada
Email:
walheng@mat.ulaval.ca
DOI:
10.1090/S0002-9939-00-05590-8
PII:
S 0002-9939(00)05590-8
Keywords:
Bloch constant,
harmonic mappings
Received by editor(s):
December 14, 1998
Posted:
March 2, 2000
Additional Notes:
This research was supported in part by NSFC(China), NSERC(Canada) and FCAR(Québec).
Communicated by:
Albert Baernstein II
Copyright of article:
Copyright
2000,
American Mathematical Society
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