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Positive Harmonic Functions and Complete Metrics

Part of: Geometric function theory Higher-dimensional theory Two-dimensional theory

Published online by Cambridge University Press:  20 November 2018

David A. Herron  and
Joel L. Schiff
David A. Herron
Affiliation:
Department of Mathematics University of Cincinnati Cincinnati, Ohio 45221-0025 U.S.A.
Joel L. Schiff
Affiliation:
Department of Mathematics University of Auckland Auckland New Zealand
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Abstract

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We introduce the class of Harnack domains in which a Harnack type inequality holds for positive harmonic functions with bounds given in terms of the distance to the domain's boundary. We give conditions connecting Harnack domains with several different complete metrics. We characterize the simply connected plane domains which are Harnack and discuss associated topics. We extend classical results to Harnack domains and give applications concerning the rate of growth of various functions defined in Harnack domains. We present a perhaps new characterization for quasidisks.

MSC classification

Secondary: 30C20: Conformal mappings of special domains 31A05: Harmonic, subharmonic, superharmonic functions 31A20: Boundary behavior (theorems of Fatou type, etc.) 31B05: Harmonic, subharmonic, superharmonic functions 31B25: Boundary behavior 30C55: General theory of univalent and multivalent functions
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 32 , Issue 3 , 01 September 1989 , pp. 286 - 297
Copyright
Copyright © Canadian Mathematical Society 1989

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