# Math Help - unbounded component

1. ## unbounded component

If I have a function f non-vanishing, entire and non-constant, how can I demonstrate that each component of the set {z in $\mathbb{C}$; |f(z)|<1} is unbounded? If I assume that there is at least one bounded component of that set, how can I get a contradiction? I suppose that I should use Liouville's Theorem at somewhere...

Thanks.

2. ## Re: unbounded component

If $D$ is the unit disk, $U$ a bounded component of $f^{-1}(D)$ then $f(U)=D$ (use the compactness of $\overline{U})$, now use that $0\notin f(U)$.