The following is an old qualifying exam problem I cannot solve:
Let $f$ be a meromorphic function (quotient of two holomorphic functions) on an open neighborhood of the closed unit disk. Suppose that the imaginary part of $f$ does not have any zeros on the unit circle, then the number of zeros of $f$ in the unit disk equals the number of poles of $f$ in the unit disk.
It seems this problem begs Rouche's theorem, but I cannot seem to apply it correctly.