The Symmetric Versions of Rouché’s Theorem via ∂--Calculus

الملخص EN

Let (f,g) be a pair of holomorphic functions.

In this expositional paper we apply the ∂--calculus to prove the symmetric version “|f+g|<|f|+|g| on ∂K” as well as the homotopic version of Rouché's theorem for arbitrary planar compacta K.

Using Eilenberg's representation theorem we also give a converse to the homotopic version.

Then we derive two analogs of Rouché's theorem for continuous-holomorphic pairs (a symmetric and a nonsymmetric one).

One of the rarely presented properties of the non-symmetric version is that in the fundamental boundary hypothesis, |f+g|≤|g|, equality is allowed.