The access theorem for subharmonic functions

Abstract:

A chain from a point ${z_0}$ of the open unit disk $\Delta$ to the boundary of $\Delta$ is a set $\Gamma = \cup \{ {\gamma _n}|n = 0,1,2, \cdots \}$ 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.