Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Transactions of the American Mathematical Society
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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


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


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 31A20

Retrieve articles in all journals with MSC: 31A20


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1972-0308420-8
PII: S 0002-9947(1972)0308420-8
Keywords: Subharmonic function, harmonic function, Mac Lane classes, Milloux-Schmidt inequality, conformal map
Article copyright: © Copyright 1972 American Mathematical Society