If is the unit disk, a bounded component of then (use the compactness of , now use that .
If I have a function f non-vanishing, entire and non-constant, how can I demonstrate that each component of the set {z in ; |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.