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proof of Rouch\'e's theorem

Type of Math Object: 
Proof
Major Section: 
Reference
Groups audience: 

Mathematics Subject Classification

30E20 no label found

Comments

The proof contains the line

"Since C is compact, both |f| and |g| attain maxima and minima on C. Hence there exist positive real constants a,b such that
|f(z)| > a > b > |g(z)|
for all z in C."

I don't follow this. Why should the maximum of |g| be less than the minimum of |f|? We only know that |f(z)|>|g(z)| at each z, and there's no reason I can see why the maximum of |g| should be at the same point as the minimum of |f|.

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