Let $f$ be a non-constant entire function with $f(z) \neq 0$ for all $z \in \mathbb{C}$. Consider the set $U = \{z : |f(z)| < 1\}$. Show that all connected components of $U$ are unbounded.
Let $f$ be a non-constant entire function with $f(z) \neq 0$ for all $z \in \mathbb{C}$. Consider the set $U = \{z : |f(z)| < 1\}$. Show that all connected components of $U$ are unbounded.