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Positive Superharmonic Functions and the Hölder Continuity of Conformal Mappings
Author(s) -
Anderson J. M.,
Hinkkanen A.
Publication year - 1989
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/s2-39.2.256
Subject(s) - subharmonic function , conformal map , unit disk , exponent , hölder condition , connection (principal bundle) , boundary (topology) , mathematics , mathematical analysis , domain (mathematical analysis) , plane (geometry) , function (biology) , zero (linguistics) , pure mathematics , mathematical physics , physics , geometry , linguistics , evolutionary biology , biology , philosophy
We study the rate at which a positive superharmonic function u can tend to zero at a boundary point z 0 of a plane domain G . In particular, if G is a quasidisk, and α > 0 is given, we show that the condition that lim inf u ( z )/dist ( z , ∂ G ) 1/α > 0 as z 0 in G for any such u is related to the condition that the conformal map f of the unit disk onto G witha f (1) = z o is Hölder continuous with exponent α at the point 1. This leads us to consider the problem of finding the best exponent α for which f is Hölder continuous. The answer depends on how we characterize quasidisks or quasicircles. In this connection we give a negative answer to a question of Näkki and Palka.