Let $D=D(0,1)$. Let $f\in \mathcal{O}( D), f(0)=0, f'(0)=1$
Let $N(\omega)= \int_{\partial D(0,\frac{1}{6})} \frac{f'(z)}{f(z) - \omega } dz$ defined for $w\in D(0,\frac{1}{12})$
I need to show that $f(D)$ contains $ D(0,\frac{1}{12})$.
What I did:
I showed that $N(w)$ is the index of $f(\partial D(0,\frac{1}{6})) $ related to $w$.
Remains to prove that $N(w) \neq 0$ so that any $w\in D(0,\frac{1}{12})$ is inside the loop $ f(\partial D(0,\frac{1}{6})) \subset f(D)$
So by connexity of the index function in connected spaces we should have $N(w)=N(0)$, remains to show that $0\in f(D(0,\frac{1}{6}))$ but I am not even sure if this is true.
Many thanks.