The Symmetric Versions of Rouché's Theorem via ∂–Calculus. (4th February 2014) - Mortini, Raymond; Rupp, Rudolf
Abstract : 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 compactaK . 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.