A Note on the Maximum Number of Zeros of r(z)-z

An important theorem of Khavinson and Neumann (Proc. Am. Math. Soc. 134: 1077–1085, 2006 ) states that the complex harmonic function r ( z ) - z ¯ , where r is a rational function of degree n ≥ 2 , has at most 5 ( n - 1 ) zeros. In this note, we resolve a slight inaccuracy in their proof and in addi...

Full description

Saved in:
Bibliographic Details
Published in:Computational methods and function theory 2015-09, Vol.15 (3), p.439-448
Main Authors: Luce, Robert, Sète, Olivier, Liesen, Jörg
Format: Article
Language:English
Subjects:
Analysis
Computational Mathematics and Numerical Analysis
Functions of a Complex Variable
Mathematics
Mathematics and Statistics
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:An important theorem of Khavinson and Neumann (Proc. Am. Math. Soc. 134: 1077–1085, 2006 ) states that the complex harmonic function r ( z ) - z ¯ , where r is a rational function of degree n ≥ 2 , has at most 5 ( n - 1 ) zeros. In this note, we resolve a slight inaccuracy in their proof and in addition we show that for certain functions of the form r ( z ) - z ¯ no more than 5 ( n - 1 ) - 1 zeros can occur. Moreover, we show that r ( z ) - z ¯ is regular, if it has the maximal number of zeros.
ISSN:1617-9447
2195-3724
DOI:10.1007/s40315-015-0110-6