Let $D= \{ z\in \mathbb{C}: |z|<1\}$ and $T= \{ z\in \mathbb{C} : |z|=1\}$.
Let $f: \Omega \rightarrow \mathbb{C} $ be analytic on an open set $\Omega \supset D \cup T$, and suppose that $f(z)\ne0$, for all $z\in T$. Let $N_D(f)$ be the number of zeroes of $f$ in $D$, counted with multiplicities. Indicate a closed path $\gamma:[0,1] \rightarrow \mathbb{C}$ for which $N_D(f)$ is equal to the winding number of $\gamma$ around $0$.
I know that the winding number is $n(f,\gamma) = \int_\gamma \frac{f'(z)}{f(z)}dz$.
Also, the questions wants $N_D(f)$ equal to the winding number of $\gamma$ around $0$. So I'm suspecting that we will need Rouche's theorem?
But I'm not sure how to continue.
Any help will be appreciated!