On the Maximum and Minimum Modulus of Rational Functions

Lubinsky, D. S.

Canadian journal of mathematics, Tome 52 (2000) no. 4, pp. 815-832

Résumé

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.

Détail

DOI : 10.4153/CJM-2000-035-3

Mots-clés : 30E10, 30C15, 31A15, 41A21

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