Let $f : G \to C$ be analytic and suppose that $G$ is bounded. Fix $z_0\in \partial G$ and suppose that $\limsup_{z→w} | f(z)| ≤ M$ for $w \in\partial G$, $w\neq z_0$. Show that if $\lim_{z→z_0} |z − z_0|^\alpha | f(z)| = 0$ for every $\alpha > 0$, then $| f(z)| ≤ M$ for every $z \in \partial G$. (Hint: If $a < G$, consider $\varphi(z) = (z − z_0)(z − a)^{−1}.$)
Firstly, let's remind Phragmén-Lindelöf Theorem:
Let $G$ be a simply connected region and let $f$ be an analytic function on $G$. Suppose there is an analytic function $\varphi: G \to \mathbb{C}$ which never vanishes and is bounded on $G$. If $M$ is a constant and $\partial_\infty G=A\cup B$ such that: $$a) \text{ For every }a\in A, \limsup_{z\to a}|f(z)|\leq M; $$ $$b) \text{ For every }b\in B \text{ and }\eta>0, \limsup_{z\to b}|f(z)||\varphi(z)|^\eta \leq M; $$ then $|f(z)|\leq M$ for all $z\in G$.
In the hypothesis of Phragmén-Lindelöf's Theorem, $G$ is simply connected, so we can use that, as $\varphi \neq 0$ in $G$, it exists an holomorphic determination of the logarithm.
In this case, we can suppose that $G$ is open because if $f$ is analytic on $G$ is analytic in an open set $\Omega$ such that $G\subset \Omega$ too.
As $G$ is bounded, let's fix $a\in \mathbb{C}\backslash \overline{G}$. The Möbius transformation $$\varphi(z) = \frac{z − z_0}{z − a}$$ verifies that $\varphi(z_0)=0$, $\varphi(a)=\infty$ and $\varphi(\infty)=1$. This transforms the line that pass through $z_0$ and $a$ into another line.
That's my progress at the moment, because I don't know how to define a logarithm with that. I think, once I've done that, I could follow the same proof as the Phragmén-Lindelöf Theorem to finish my exercise.
Any help would be appreciate.