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On the Maximum and Minimum Modulus of Rational Functions

Published online by Cambridge University Press:  20 November 2018

D. S. Lubinsky
D. S. Lubinsky*
Affiliation:
John Knopfmacher Centre, Department of Mathematics, Witwatersrand University, Wits 2050, South Africa email: [email protected]
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Abstract

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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.

Keywords

30E1030C1531A1541A21
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 52 , Issue 4 , 01 August 2000 , pp. 815 - 832
Copyright
Copyright © Canadian Mathematical Society 2000

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