Consider the region $1<|z|<3$. Using Rouche's theorem, it is possible to show that zero roots lie inside $|z|=1$ and three roots lie inside $|z|=3$.
My question is, does Rouche's theorem include the boundary of the region, e.g. if three roots lie inside $|z|=3$, is it possible a root exists on the boundary of this circle?
The way Rouche's theorem is usually applied, you have a function $f+g$ and you show that $|f| < |g|$ on the boundary of a circle. Since this is a strict inequality, it follows $f+z \neq 0$ on the boundary. So no zeros can occur on the boundary.