I'm to solve an exercise which supposes that $ f: U \rightarrow \mathbb{C}$ is holomorphic on an open subset U $\subset \mathbb{C}$ and that $f'(a) \neq 0$ for some $a \in U$.
I'm asked to show that $f$ is injective on some disk around $a$, and then having done so that it's inverse $g$ on such a disk is given by
$$ g(w) = \frac{1}{2 \pi i}\int_{\gamma} \frac{zf'(z)}{f(z) -w}dz$$
where $\gamma (t) = a + re^{it}$, $t\in [0, 2\pi]$ is the path around the boundary of that disk.
I've managed to show that it's injective in the disk, but I have no idea how to show the inverse is of that form.
The section of the textbook I'm using this is in has covered the argument principle, Rouche's theorem, the open mapping theorem, and the inverse function theorem, but I can't see any obvious way to apply any of them to show $f$ is injective on some disk.
I thought of using the inverse function theorem now that I've established $f$ is injective, but this just tells me that $g'(w) = \frac{1}{f'(g(w))}$, and I can't see how to make that of the desired form.
I'd really appreciate any help you could offer, thanks in advance.