Chinese Annals of Mathematics, Series B

, Volume 38, Issue 2, pp 563–578 | Cite as

Bolzano’s theorems for holomorphic mappings

Article

Abstract

The existence of a zero for a holomorphic functions on a ball or on a rectangle under some sign conditions on the boundary generalizing Bolzano’s ones for real functions on an interval is deduced in a very simple way from Cauchy’s theorem for holomorphic functions. A more complicated proof, using Cauchy’s argument principle, provides uniqueness of the zero, when the sign conditions on the boundary are strict. Applications are given to corresponding Brouwer fixed point theorems for holomorphic functions. Extensions to holomorphic mappings from ℂ n to ℂ n are obtained using Brouwer degree.

Keywords

Holomorphic function Hadamard-Shih’s conditions Poincaré-Miranda’s conditions Bolzano’s theorem Cauchy’s theorem Brouwer fixed point theorem Brouwer degree 

2000 MR Subject Classification

30C15 30E20 55M20 

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Copyright information

© Fudan University and Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Institut de Recherche en Mathématique et PhysiqueUniversité Catholique de Louvain chemin du cyclotron2 1348, Louvain-la-NeuveBelgium

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