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If the criteria is satisfied for Rouche's theorem to be applied to two complex functions i.e. it is such that $|f(z)|<|g(z)|$ on the contour and both functions are analytic on the contour. Does this mean that $|g(z)|$ and $|f(z) + g(z)|$ have the same number of zeros inside and on the contour?

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Yes, because if $|f| < |g|$ then neither $g$ nor $f+g$ is zero.

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