Let $f$ be analytic in $\overline{B}(0;1)$ 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 $g: B(0,\rho) \to \mathbb{C}$ by
$$ g(w) = \frac{1}{2\pi i}\int_{\gamma}\frac{zf'(z)}{f(z) - w}dz$$
where $\gamma$ is the circle $|z| = R$. Show that $g$ is analytic is analytic and discuss the properties of $g$.
This is similar to the Cauchy Formula but is in the section of The argument principle and Rouche's Theorem (Conway's Complex Analysis Book) and I really don't have a clue how to prove this. Can anyone help me?
Maybe should I consider the power series of the analytic function $f$ ?