Biholomorphic Equivalence in $\mathbb{C}^n$

Question

I am stuck at the following problem :

For $n \geq 2$, are the sets $\Delta_n$ and $H_n$ biholomorphically equivalent ?

Here, $ \Delta_n = \{ ( z_1, z_2, \ldots z_n) : |z_i| < 1, i = 1, 2, \ldots, n\} $ and $ H_n := \{ (z_1, z_2, \ldots, z_n) : \Im (z_1) > 0 \}$.

I was trying to see the case $n=2$ first. But I am not even sure if the statement is true or false. Any help or comments are very much appreciated.

Answer 1

Let $\varphi: H_n \to \Delta_n$ be holomorphic. Then $z \mapsto \varphi(i, z, 0, \dotsc, 0)$ is entire on $\mathbb{C}$ and bounded in each of its coordinates.