This is the integral: $$\int_{\gamma}z^2\frac{f'(z)}{f(z)}dz$$ we know that:
$\gamma$ is a simple closed curve; $f$ is analytical in $R$ (the region bounded by the curve) and in all the points of the curve; $f$ has only one zero of order $n$ in $R$.
What I attempted to do is the following:
First I tried with integration by parts, and since $\gamma$ is closed, i have $\int_{\gamma}z\log(f(z))dz$, but I have no idea how to further procede since complex logarithm isn't continuous as far as I know.
Then I tried with $f(z)=(z-z_0)^ng(z)$ where $g(z_0)\neq 0$ and then applied Cauchy's formula, but that didn't lead me to anywhere either. Any ideas?