I am trying to check the conditions of the Phragmén Lindelöf Principle (see below) in order to use it on the following corollary:
Corollary Let $u$ be a subharmonic function on an unbounded proper domain subdomain $D$ of $\mathbb{C}$ such that \begin{eqnarray*} \limsup_{z \to \zeta} u(z) \leq 0 \quad \text{for} \quad \zeta \in \partial D \backslash \{\infty\} \quad \text{and} \quad \limsup_{z \to \infty} \frac{u(z)}{\log|z|} \leq 0. \end{eqnarray*} Then $u \leq 0$ on $D$.
Proof. Take $w \in \partial D$ and aplly Theorem 2.3.2 with $v(z)=\log |z-w|$. $\square$
This is the theorem that the corollary uses:
Theorem 2.3.2 (Phragmén Lindelöf Principle) Let $u$ be a subharmonic function on a unbounded domain D in $\mathbb{C}$ such that \begin{eqnarray*} \limsup_{z \to \zeta} u(z) \leq 0 \end{eqnarray*} for all $\zeta \in \partial D \backslash \{\infty\}$. Suppose also taht there exists a finite-valued superharmonic function $v$ on $D$ such that \begin{eqnarray*} \liminf_{z \to \infty} v(z) > 0 \quad \text{and} \quad \limsup_{z \to \infty} \frac{u(z)}{v(z)} \leq 0. \end{eqnarray*} Then $u \leq 0$ on $D$.
So we need $v$ to be finite-valued and superharmonic, which is the case since $z \neq w$ (right?). But I am having trouble understanding why the limit conditions are fulfilled: \begin{eqnarray*} \liminf_{z \to \infty} \log |z-w|>0 \\ \limsup _{z \to \infty} \frac{u(z)}{\log |z-w|} \leq 0 \end{eqnarray*} What if $w=\infty$? (I should mention that limits are taken with respect to $\infty$.) And why do we need the subdomain to be proper?
Thanks!
If $D \ne \Bbb C$ then $\partial D \setminus \{\infty\}$ is not empty, therefore you can chose a finite $w \in \partial D$ and apply the Phragmén-Lindelöf Principle with $v(z)=\log |z-w|$.
The corollary is wrong for $D = \Bbb C$: any positive constant function satisfies the hypotheses, but not the conclusion.