Three counterexamples for a question concerning Green's functions and circular symmetrization
Author:
Alexander R. Pruss
Journal:
Proc. Amer. Math. Soc. 124 (1996), 1755-1761
MSC (1991):
Primary 31A15
DOI:
https://doi.org/10.1090/S0002-9939-96-03196-6
MathSciNet review:
1307558
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Abstract | References | Similar Articles | Additional Information
Abstract: We construct domains in the plane such that if
is the Green's function of
with pole at zero, while
is the symmetric non-increasing rearrangement of
for each fixed
and
is the Green's function of the circular symmetrization
, again with pole at zero, then there are positive numbers
and
such that
whenever . One of our constructions will have
simply connected. We also consider the case where the poles of the Green's functions do not lie at the origin. Our work provides a negative answer to a question of Hayman (1967).
- 1. Albert Baernstein II, Integral means, univalent functions and circular symmetrization, Acta Math. 133 (1974), 139–169. MR 417406, https://doi.org/10.1007/BF02392144
- 2. Arne Beurling, Études sur un problème de majoration, Thèse pour le doctorat, Almqvist & Wiksell, Uppsala, 1933.
- 3. W. K. Hayman, Research problems in function theory, The Athlone Press University of London, London, 1967. MR 0217268
- 4. Rolf Nevanlinna, Analytic functions, Translated from the second German edition by Phillip Emig. Die Grundlehren der mathematischen Wissenschaften, Band 162, Springer-Verlag, New York-Berlin, 1970. MR 0279280
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Additional Information
Alexander R. Pruss
Affiliation:
University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z2
Email:
pruss@math.ubc.ca
DOI:
https://doi.org/10.1090/S0002-9939-96-03196-6
Keywords:
Green's functions,
circular symmetrization
Received by editor(s):
September 30, 1994
Received by editor(s) in revised form:
November 16, 1994
Additional Notes:
The research was partially supported by Professor J. J. F. Fournier’s NSERC Grant #4822. This paper constitutes a portion of the author’s doctoral dissertation.
Communicated by:
Albert Baernstein II
Article copyright:
© Copyright 1996
American Mathematical Society