# Question on Rouche's Theorem - Boundary Included?

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.