I'm trying to understand the reasoning given in Donald Marshall's complex analysis:
The Green's function for the unit disk $\mathbb{D}$ is given by $$ g_{\mathbb{D}}(z, a)=\log \left|\frac{1-\bar{a} z}{z-a}\right| . $$
If $g(z)=\log \frac{|1-\bar{a} z|}{|z-a|}$ then, by Lindelöf's maximum principle, each candidate subharmonic function $v$ in the Perron family $\mathcal{F}_a$ is bounded by $g$. Moreover, $\max (g-\varepsilon, 0) \in \mathcal{F}_a$, when $\varepsilon>0$. Letting $\varepsilon \rightarrow 0$, we conclude $g=g_{\mathbb{D}}(z, a)$. How are we exactly applying Lindelöf's maximum principle?
The definition of Green's function that I have is as follows, let $\mathcal{F}_{p_0}$ be the collection of subharmonic functions $v$ on $W \backslash p_0$ satisfying $$ v=0 \text { on } W \backslash K \text {, for some compact } K \subset W \text { with } K \neq W $$ and $$ \limsup _{p \rightarrow p_0}(v(p)+\log |z(p)|)<\infty . $$
Note that $v \in \mathcal{F}_{p_0}$ is not assumed to be subharmonic at $p_0$, and indeed it can tend to $+\infty$ as $p \rightarrow p_0$. Set $$ g_W\left(p, p_0\right)=\sup \left\{v(p): v \in \mathcal{F}_{p_0}\right\} . $$
The only result proved in the section before "stating" what the Green's function for $\mathbb{D}$ is the following lemma: Suppose $p_0 \in W$ and suppose $z: U \rightarrow \mathbb{D}$ is a coordinate function such that $z\left(p_0\right)=0$. If $g_W\left(p, p_0\right)$ exists, then $$ \begin{aligned} & g_W\left(p, p_0\right)>0 \text { for } p \in W \backslash\left\{p_0\right\}, \text { and } \\ & g_W\left(p, p_0\right)+\log |z(p)| \text { extends to be harmonic in } U . \end{aligned} $$