- The condition of Rouche is exactly the "strict triangle inequality". The triangle inequality always holds on the complex plane (as a normed space). So we only need to check the function values on the boundary, if they were always different, then 2 functions have the same zeros situation on interior.
- With the preceding discussion, "different function values on boundary" is equivalent to "different function arguments on boundary", because the only way triangle inequality becomes equality is that 0, f(z) and g(z) are on the same line. Hence, f(z) and g(z) would wind origin with the same number of circles when z goes through boundary. That's the reason why f and g would have the same number of zeros considering multiplicity.
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[CA] 1962 Strong Version of Rouché's Theorem
CA Originally published 2020-01-21
Rouche's Theorem used to be comparing the distance of 2 function images on the complex plane with the modulus of one function on the boundary points, and if the former was strictly smaller, then we can say, 2 functions have the same number of zeros counting multiplicity.
The stronger version, a.k.a symmetric version of Rouche's theorem, est. by Theodor E in 1962, saying that if the distance of these 2 boundary images was strictly less than the sum of the modulus of functions, then we can imply that they have the same number of zeros counting multiplicity. There are 2 points we need to mension for this strong version: