I'm given that $\displaystyle \frac{1}{2 \pi i} \int_{\partial D(0,3)} \frac{f'(z)}{f(z)}\,dz = 2$ and also that $\displaystyle \frac{1}{2 \pi i} \int_{\partial D(0,3)} z \frac{f'(z)}{f(z)}\,dz = 2$. I know that $\displaystyle f$ is analytic on the curve and on the interior region as well, and that $\displaystyle f(z)\neq 0, \forall z \in \partial D(0,3)$.

The actual question of the problem aside, what does the second equation really tell me? I know that the first says that the winding number of $\displaystyle f(D(0,3))$ about the origin is 2, and that because $\displaystyle f$ is holomorphic in the disc there are no poles and so $\displaystyle f$ must have two zeroes in $\displaystyle D(0,3)$.

Eventually I'm trying to locate those zeroes, and there's a third equation as well that I imagine is supposed to help me along, but what more information can I glean from that second equation?

Thanks.