I have a holomorphic function $f: G \rightarrow \mathbb{C} $.
G is a domain and $ G \subset \mathbb{C} $.
Furthermore $ V, V^{*}$ are domains, too, with continous differentiable closed boundary curves $ \Gamma, \Gamma^{*}$ and $\bar{V} \subset G$
I want to show: If $f: \Gamma \rightarrow \Gamma^*$ is bijective, then $f : V \rightarrow V^* $bijectiv. For a hint I can use the following theorem, applied on $h(z)=f(z)-w, w \notin \Gamma^*$:
$f: G \rightarrow \mathbb{C} $, G domain, $G \subset \mathbb{C}$. A is the set of roots and poles of f in G. Futhermore V is a domain, too, and $ \bar{V} \subset G$ , $\delta V =:\Gamma $ a curve in $ G \setminus A$. The roots and poles of f are $ n_1,...,n_l$ and $ d_1,...,d_l$. Then:
$ \frac{1}{2 \pi i} \int_{\gamma } \frac{f'(z)}{f(z)} dz= (n_1,...,n_l)-( d_1,...,d_l)$ Can somebody help me?