My textbook seems to use the following deduction.
Let $R$ be a Riemann surface, let $B$ be a coordinate disk of $R$, and let $u$ be a subharmonic function on $R \setminus B$ such that $u=0$ off a compact subset of $R$, and $u \leq 0$ on $\partial B$. Then $u \leq 0$ on all $R \setminus B$.
I do not see how can this be formally justified. It seems morally to be the maximum principle, which is formulated as follows:
The maximum principle. Let $R$ be a Riemann surface, $K$ a compact subset of $R$. If a function $u$ is subharmonic on $R$ and $u \leq 0$ on $R \setminus K$, then $u \leq 0$ on all $R$.
So in our case we may regard $R$ minus the closed coordinate disk $\bar B$ as a Riemann surface by itself, and then we want $u$ to be $\leq 0$ off a compact subset of $R \setminus \bar B$. But near the coordinate disk it is not clear that we can really find a "separation" on which $u \leq 0$ to guarantee that such a compact set exists.
Is the claim of the textbook true, then? If so, how can it be justified? (Maybe assuming that $R$ is hyperbolic helps?)