State the argument principle.

Let $U \subset \mathbb{C}$ be an open set and $f: U \rightarrow \mathbb{C}$ a holomorphic injective function. Show that $f^{\prime}(z) \neq 0$ for each $z$ in $U$ and that $f(U)$ is open.

Stating clearly any theorems that you require, show that for each $a \in U$ and a sufficiently small $r>0$,

$$g(w)=\frac{1}{2 \pi i} \int_{|z-a|=r} \frac{z f^{\prime}(z)}{f(z)-w} d z$$

defines a holomorphic function on some open disc $D$ about $f(a)$.

Show that $g$ is the inverse for the restriction of $f$ to $g(D)$.