The Symmetric Versions of Rouché's Theorem via ∂--Calculus. (4th February 2014)

• Record Type: Journal Article
• Authors: Mortini, Raymond; Rupp, Rudolf
• Other Names: Lindström Mikael Academic Editor.
• Is Part Of: Journal of complex analysis. Volume 2014(2014)
• Publication Date: 2014-02-04
• Subjects: Functions of complex variables -- Periodicals; Mathematical analysis -- Periodicals; 515.905
• DOI: 10.1155/2014/260953
• Languages: English
• ISSNs: 2314-4963

Abstract

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.