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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
MathSciNet review: 1307558
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Abstract: We construct domains $U$ in the plane such that if $G(re^{i\theta })$ is the Green's function of $U$ with pole at zero, while $\tilde G(r e^{i\theta })$ is the symmetric non-increasing rearrangement of $G(re^{i\theta })$ for each fixed $r$ and $G^{*}$ is the Green's function of the circular symmetrization $U^{*}$, again with pole at zero, then there are positive numbers $r$ and $\varepsilon $ such that

\begin{equation*}G^{*}(r e^{i\theta }) < \tilde G(r e^{i\theta }), \end{equation*}

whenever $0<|\pi -\theta |<\varepsilon $. One of our constructions will have $U$ 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).

References [Enhancements On Off] (What's this?)

  • 1. Albert Baernstein II, Integral means, univalent functions and circular symmetrization, Acta Math. 133 (1974), 139–169. MR 417406,
  • 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

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