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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



The access theorem for subharmonic functions

Authors: R. Hornblower and E. S. Thomas
Journal: Trans. Amer. Math. Soc. 172 (1972), 287-297
MSC: Primary 31A20
MathSciNet review: 0308420
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Abstract: A chain from a point $ {z_0}$ of the open unit disk $ \Delta $ to the boundary of $ \Delta $ is a set $ \Gamma = \cup \{ {\gamma _n}\vert n = 0,1,2, \cdots \} $ where the $ {\gamma _n}$ are compact, connected subsets of $ \Delta ,{z_0}$ is in $ {\gamma _0},{\gamma _n}$ meets $ {\gamma _{n + 1}}$ and the $ {\gamma _n}$ approach the boundary of $ \Delta $. The following ``Access Theorem'' is proved: If $ u$ is subharmonic in $ \Delta ,{z_0}$ is a point of $ \Delta $ and $ M < u({z_0})$, then there is a chain from $ {z_0}$ to the boundary of $ \Delta $ on which $ u \geq M$ and on which $ u$ tends to a limit. A refinement, in which the chain is a polygonal arc, is established, and an example is constructed to show that the theorem fails if $ M = u({z_0})$ even for bounded, continuous subharmonic functions.

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Keywords: Subharmonic function, harmonic function, Mac Lane classes, Milloux-Schmidt inequality, conformal map
Article copyright: © Copyright 1972 American Mathematical Society

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