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Journal of the London Mathematical Society 1989 s2-39(2):256-270; doi:10.1112/jlms/s2-39.2.256
© 1989 by London Mathematical Society
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© Oxford University Press

Positive Superharmonic Functions and the Hölder Continuity of Conformal Mappings

J. M. Anderson and A. Hinkkanen

Department of Mathematics, University College Gower Street, London WC1E 6BT
Department of Mathematics, University of Michigan Ann Arbor, Michigan 48109-1003, USA

We study the rate at which a positive superharmonic function u can tend to zero at a boundary point z0 of a plane domain G. In particular, if G is a quasidisk, and {alpha} > 0 is given, we show that the condition that lim inf u(z)/dist (z, {partial}G)1/{alpha} > 0 as z0 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) = zo is Hölder continuous with exponent {alpha} at the point 1. This leads us to consider the problem of finding the best exponent {alpha} 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.

Current address: Department of Mathematics, The University of Texas at Austin, Austin, Texas 78712, USA


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