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.

It's been a while since I looked at several complex variables, but doesn't this require $U$ to be pseudoconvex? — Harald Hanche-Olsen, Jun 03 '15 at 17:46

score 2 · Accepted Answer · answered Jun 03 '15 at 18:55

2

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.

answered Jun 03 '15 at 18:55

mrf

43,639

It's false when $n=1$ as well. For example in the case where $U$ is an annulus. – zhw. Jun 03 '15 at 19:27
@zhw. Yes. but at least for $n=1$ it's possible for some $U$. – mrf Jun 03 '15 at 19:52

Finding an entire function $f$

1 Answers1