For the statement of Rouché's theorem, I've always seen that both $f$ and $g$ have to be holomorphic on and inside a simple closed curve $ C $. However, I am solving a problem which seems to suggest that I should use Rouché's theorem even though I only know that $ f $ is holomorphic in the unit disk $ D $ and continuous in $ \bar{D} $. I also check Wikipedia's page on Rouché's theorem which says that $ f $ and $ g $ only need to be holomorphic inside the region, not on the boundary. Is this sufficient?
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Yes that also looks strange to me. Rouché’s theorem hypothesis is to have a simply connected open subset $U\subseteq \mathbb C$ and a compact $K \subset U$ whose boundary is a closed simple curve positively oriented.
mathcounterexamples.net
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Yes, it is sufficient since by continuity the inequality assumption of Rouché's theorem extends to some neighborhood inside the boundary curve, and thus inside the holomorphic domain. In other words, shift the curve along some inside normal vector field a little bit, which is possible because of the compactness of the curve, to get a situation that is conform with the version of the theorem as you know it.
Lutz Lehmann
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