Also, cannot pass through any zeros of f or g. Now for any function analytic in except for poles:
where is the winding number right? That's how many times the contour goes around the origin.
Now, by Rouche's theorem:
Since the contour never passes through any zero of either function, we can divide that expression by and end up with:
That means is inside the unit circle centered at one right? What then does that say about the expression: