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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?

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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.

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