How to show that there a set that has no barrier function? I mean that how to show by example that existence of barrier function of any set $U\subset \mathbb{C}$ is dependent of its set.
Definition (barrier function):
Let $U \subset \mathbb{C}$ and $z_0 \in \partial U$. Let $b:U \leftarrow \mathbb{R}$, then $b$ is barrier function of $U$ in $z_0$, if
(i) $b$ is continuous in $U$
(ii) b is subharmonic in closure of $U$ ($\Delta b \leq 0$)
(iii) $b(z) \leq 0$, $\forall z \in \mathbb{C}$
(iv) $b(z)=0$ if and only if $z=z_0$.