Is there a holomorphic and bijective function of the unit disc onto the complex plane?

Question

Is there a holomorphic and bijective function between the open unit ball of $\mathbb{C}$ and $\mathbb{C}$?

The usual homeomorphisms $\psi(z):=\frac{z}{1+|z|}$ and his composition with the conjugate map are not holomorphic on $\mathbb{C}$.

Answer 1

No, because its inverse would be a holomorphic map from $\mathbb C$ to the open unit disk on $\mathbb C$. Therefore, it would be bounded and then, by Liouville's theorem, it would be a constant map.