Is a holomorphic mapping from open unit disk to itself always biholomorphic (bijective)

Question

I am learning to understand the behavior of holomorphic mappings from open unit disk to itself. And I found a related question said that a holomorphic function is always conformal, while another related question said that conformal mapping from unit disk to itself is always bijective.

Now after the reading I'm a bit confused about the relationship among holomorphic mappings, conformal mappings and bijective holomorphic mappings from open unit disk to itself… Are they all equivalent, or does there exist a holomorphic mapping from open unit disk to itself that is not bijective?

Answer 1

If $f(z)=\frac{z^2}2$, then $f$ is neither an injective nor a surjective map from the open unit disk into itself.