Existence of bounded analytic function on unbounded domain? 
Given any proper open connected unbounded set $U$  in $\mathbb C$.Does there always exist a non constant bounded analytic function $ f\colon U \to \mathbb C$ ?

Edit: $U$ is any arbitrary domain. I don't have idea to do it. Please help.

Take $f(z) = {1 \over z} $ on $U=\{z \mid |z|>1 \}$.
This example can be extended to any $U$ such that $U^c$ contains an open set.
