Exercise 3. Let $f$ be analytic in $\overline{B}(0; R)$ with $f(0)=0$, $f'(0) \neq 0$ and $f(z) \neq 0$ for $0<|z| \leq R$. Put $\rho=\min\{|f(z)|:|z|=R\}>0$. Define $g: B(0; \rho) \rightarrow \mathbb{C}$ by $$g(w) = \frac{1}{2\pi i} \int_\gamma \frac{z\,f'(z)}{f(z)-\omega} dz$$ where $\gamma$ is the circle $|z|=R$. Show that $g$ is analytic and discuss the properties of $g$.
Now, I want to apply Rouche's theorem in this question, so please i want some hints. I think $f$ maybe one to one since the kernel maybe the zero element.
I don't see any use for Rouché here. Just consider the poles of the integrand inside $\gamma$, and what their residues are.