\begin{definition}[Definition:Barrier]
A complex function $\varphi \in \map \C {\overline \Omega}$ is a '''barrier''' for $\Omega$ at $z \in \partial \Omega$ {{iff}}:
:$\varphi$ is subharmonic
:$\map \varphi z = 0$
:$\varphi < 0$ on $\partial \Omega \setminus \set z$
{{explain|Define $\map \C {\overline \Omega}$}}
{{explain|Define $\Omega$}}
{{explain|Define $\partial \Omega$}}
{{explain|What is the domain and range of $\varphi$?}}
{{explain|If $\varphi$ is a complex function, its range is also complex, and so cannot be ordered, so the meaning of $\varphi < 0$ needs to be explained carefully}}
\end{definition}