Denote the class of Schlicht functions (injective, holomorphic on the unit disk, with $f(0)=0$ and $f'(0)=1$) by $\mathcal{S}$. And let $\gamma :[0,\infty )\rightarrow \mathbb{C}$ be a simple curve (continuous and injective) and such that $\gamma (0)=0$ and $\lim _{t\to \infty}\gamma (t)=\infty$.
I have to show that, for each such $\gamma$, there is a unique $t>0$ so that $\mathbb{C} \backslash \gamma \left( [t,\infty )\right)$ is the image of some $f\in \mathcal{S}$.
Here's my idea so far:
For each $t$, via the Riemann Maping Theorem, I can construct a bijective, holomorphic function $f_t:\mathbb{C} \backslash \gamma \left( [t,\infty )\right) \rightarrow \mathbb{D}$ ($\mathbb{D}$ is the unit disk) such that $f_t(0)=0$. Furthermore, such an $f_t$ is unique up to a choice of the argument of f_t'(0). If I could somehow show that there must be some $t>0$ such that \left| f_t'(0)\right| =1 I would be done, but I am stuck as how to do this.
Anyone have any ideas? In the same direction or not, all suggestions are welcome.