I have been asked to prove Poincaré's theorem using Schwarz Lemma. The statements are as follows:
Schwarz Lemma. Let $B^n_1(\mathbf{0}) := \{(z_1, \ldots, z_n)\in \Bbb C^n: |z_1|^2 + \ldots + |z_n|^2 < 1\}$ and $f: B^n_1(\mathbf{0}) \to \Bbb C$ be holomorphic with $f(\mathbf{0}) = 0$. Let $|f(z)| \le M$ for all $z\in B^n_1(\mathbf{0})$ for some $M > 0$, i.e. $f$ is bounded. Then, $|f(z)| \le M\|z\|$ for all $z\in B^n_1(\mathbf{0})$ and $\|f'(\mathbf{0})\|\le M$.
With $B^n_1(\mathbf{0})$ (the unit ball in $\Bbb C^n$) as above, let $P^n_1(\mathbf{0}):= \{(z_1, \ldots, z_n)\in \Bbb C^n: |z_i| < 1 \text{ for all }1\le i\le n\}$ the unit polydisc in $\Bbb C^n$.
Poincaré's Theorem. If $n \ge 2$, there does not exist a biholomorphism between $B^n_1(\mathbf{0})$ and $P^n_1(\mathbf{0})$.
My work. Suppose, for a contradiction, that $n\ge 2$ and there exists a biholomorphism $f: B^n_1(\mathbf{0}) \to P^n_1(\mathbf{0})$, that is, $f,f^{-1}$ are holomorphic and bijective. Let $f = (f_1, f_2, \ldots, f_n)$ be the coordinate-wise representation of $f$. Also, denote the derivative map of $f$ at $\mathbf{0}$ by $f'(\mathbf{0}): \Bbb C^n\to \Bbb C^n$. Without loss of generality, we can assume $f(\mathbf{0}) = \mathbf{0}$. If $f(\mathbf{0}) \ne \mathbf{0}$, then we can consider $\widetilde{f} = g \circ f: B^n_1(\mathbf{0}) \to P^n_1(\mathbf{0})$ where $g$ is an automorphism of the polydisc, sending $f(\mathbf{0})$ to $\mathbf{0}$.
Using Schwarz Lemma as stated above, I have proved that if $f(\mathbf{0}) = \mathbf{0}$ then $f'(\mathbf{0})$ maps $B^n_1(\mathbf{0})$ into $P^n_1(\mathbf{0})$. If we can show (how?) that the image of $B^n_1(\mathbf{0})$ under $f'(\mathbf{0})$ is $P^n_1(\mathbf{0})$, then, $f'(\mathbf{0})$ maps $\partial B^n_1(\mathbf{0})$ into $\partial P^n_1(\mathbf{0})$. Also, $f'(\mathbf{0})$ is invertible (use chain rule). Knowing that linear maps send spheres to ellipsoids, $f'(\mathbf{0})(\partial B^n_1(\mathbf{0})) \subset \partial P^n_1(\mathbf{0})$ is a contradiction.
Question. Could I get some help with showing that $f'(\mathbf{0})$ maps $B^n_1(\mathbf{0})$ onto $P^n_1(\mathbf{0})$? If this is not true, how else can I prove Poincaré's Theorem using Schwarz Lemma? Thanks!