Table of Contents
Journal of Complex Analysis
Volume 2014, Article ID 260953, 9 pages
Research Article

The Symmetric Versions of Rouché’s Theorem via -Calculus

1Département de Mathématiques, Institut Élie Cartan de Lorraine, UMR 7502, Université de Lorraine, Ile du Saulcy, 57045 Metz, France
2Fakultät für Angewandte Mathematik, Physik und Allgemeinwissenschaften, TH-Nürnberg, Kesslerplatz 12, 90489 Nürnberg, Germany

Received 22 August 2013; Accepted 28 October 2013; Published 4 February 2014

Academic Editor: Mikael Lindström

Copyright © 2014 Raymond Mortini and Rudolf Rupp. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Let be a pair of holomorphic functions. In this expositional paper we apply the -calculus to prove the symmetric version “ on ” as well as the homotopic version of Rouché's theorem for arbitrary planar compacta . 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, , equality is allowed.