Article contents
On the Maximum and Minimum Modulus of Rational Functions
Published online by Cambridge University Press: 20 November 2018
Abstract
We show that if $m,n\ge 0,\lambda >1$, and
$R$ is a rational function with numerator, denominator of degree
$\le m,n$, respectively, then there exists a set
$S\subset [0,1]$ of linear measure
$\ge \,\frac{1}{4}\,\exp \left( -\frac{13}{\log \,\text{ }\!\!\lambda\!\!\text{ }} \right)$ such that for
$r\in S$,
1$$_{|z|=r\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,|z|=r}^{\max |R(z)|/min|R(z)|\le {{\lambda }^{m+n}}.}$$
Here, one may not replace $\frac{1}{4}\exp (-\frac{13}{\log \lambda })$ by
$\exp (-\frac{2-\varepsilon }{\log \lambda })$, for any
$\varepsilon >0$. As our motivating application, we prove a convergence result for diagonal Padé approximants for functions meromorphic in the unit ball.
- Type
- Research Article
- Information
- Copyright
- Copyright © Canadian Mathematical Society 2000
References
- 1
- Cited by