Let $F(x,y)$ be a holomorphic function of two variables without zeroes on $|x|=1$, $\phi(x)$ be a holomorphic function on a disk. Consider function $\mbox{Ф}$ that sends point $y_0$ to $\sum\phi(\alpha_i)$, where $\alpha_i$ are zeroes of $F(x,y_0)$ in $|x|\leq1$ with multiplicities. Prove that $\mbox{Ф}$ is holomorphic.
It looks like the proof is somehow related to Rouche's theorem but I can't see it.
Hint: use the Residue Theorem to express $\Phi(y)$ as the contour integral of a meromorphic function around the unit circle.