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If $U$ is a polydisc in $\mathbb C^n$, that is, $U=\{z \in \mathbb C^n:|z_i|<1\}$, can we find a biholomorphic map from $U$ to $\mathbb C^n$?

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Your edit completely changed the question... :-/ – Mariano Suárez-Alvarez Feb 15 '13 at 22:16
The best would be for you to undo that edit and ask a separate question. – Mariano Suárez-Alvarez Feb 15 '13 at 22:17
up vote 2 down vote accepted

In an open set biholomorphic to $\mathbb C^n$ there are no bounded holomorphic functions which are not constant.

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In the case $n = 1$, the unit disc is not biholomophic to $\mathbb C$.

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