Factorizing holomorphic function

Question

Let $f:U\subset \mathbb{D} \rightarrow \mathbb{D}$ be a holomorphic and injective function where $0 \in U$ is simply connected and $\mathbb{D}$ represents the unit circle. For $f$ applies $f(0) = 0$ and $\mid f(z) \mid > \mid z \mid$ for all $z \in U \setminus \lbrace 0 \rbrace$.

How can one assume that one can write $f(z) = z \cdot g(z)$ with a holomorphic function $g$?

For seeing this it might not be needed to have all these requirements.

Answer 1

More generally,

If $f$ is holomorphic in a domain $U$ and $f(z_0)=0$ for a $z_0 \in U$, then $f(z)=(z-z_0)g(z)$, where $g$ is holomorphic in $U$.

This follows directly from the series expansion of $f$ around $z_0$, which is valid in all of $U$, assuming you know that holomorphic functions are analytic.