Finding an entire function $f$

Question

Let $U\subset\mathbb{C}^n$ be a bounded domain. Give an example of an entire function $f:\mathbb{C}^n\longrightarrow\mathbb{C}$ such that: $$f[U]\subset D(0,1)$$ $$f[ext(U)]\subset ext[{D(0,1)}]$$

$D(0,1)=\{z\in\mathbb{C}:|z|<1\}$

$ext(U)=\overline{U}^c$

Any hint would be appreciated.

Answer 1

This is impossible. The assumptions imply that $f^{-1}(c) \subset U$ for all $c$ with $|c| < 1$, and the level set of an entire function (of several variables) can never be compact.