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# On the Maximum and Minimum Modulus of Rational Functions

Published:2000-08-01
Printed: Aug 2000
• D. S. Lubinsky
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## Abstract

We show that if $m$, $n\geq 0$, $\lambda >1$, and $R$ is a rational function with numerator, denominator of degree $\leq m$, $n$, respectively, then there exists a set $\mathcal{S}\subset [0,1]$ of linear measure $\geq \frac{1}{4}\exp (-\frac{13}{\log \lambda })$ such that for $r\in \mathcal{S}$, $\max_{|z| =r}| R(z)| / \min_{|z| =r} | R(z) |\leq \lambda ^{m+n}.$ 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 Pad\'{e} approximants for functions meromorphic in the unit ball.
 MSC Classifications: 30E10 - Approximation in the complex domain 30C15 - Zeros of polynomials, rational functions, and other analytic functions (e.g. zeros of functions with bounded Dirichlet integral) {For algebraic theory, see 12D10; for real methods, see 26C10} 31A15 - Potentials and capacity, harmonic measure, extremal length [See also 30C85] 41A21 - Pade approximation