Sharp estimates for the maximum over minimum modulus of rational functions
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- by D. S. Lubinsky PDF
- Proc. Amer. Math. Soc. 129 (2001), 3519-3529 Request permission
Abstract:
Let $m, n\geq 0, \lambda >1$, and $R$ be a rational function with numerator, denominator of degree $\leq m,n$, respectively. In several applications, one needs to know the size of the set ${\mathcal {S}}\subset \left [ 0,1\right ]$ such that for $r\in {\mathcal {S}}$, \[ \max _{\left | z\right | =r}\left | R\left ( z\right ) \right | /\min _{\left | z\right | =r}\left | R\left ( z\right ) \right | \leq \lambda ^{m+n}.\] In an earlier paper, we showed that \[ meas\left ( {\mathcal {S}}\right ) \geq \frac {1}{4}\exp \left ( -\frac {13}{\log \lambda }\right ) ,\] where $meas$ denotes linear Lebesgue measure. Here we obtain, for each $\lambda$, the sharp version of this inequality in terms of condenser capacity. In particular, we show that as $\lambda \rightarrow 1+$, \[ meas\left ( {\mathcal {S}}\right ) \geq 4\exp \left ( -\frac {\pi ^{2}}{2\log \lambda }\right ) \big ( 1+o(1) \big ) .\]References
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Additional Information
- D. S. Lubinsky
- Affiliation: The John Knopfmacher Centre for Applicable Analysis and Number Theory, Department of Mathematics, Witwatersrand University, Wits 2050, South Africa
- Address at time of publication: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332-0160
- MR Author ID: 116460
- ORCID: 0000-0002-0473-4242
- Email: 036dsl@cosmos.wits.ac.za
- Received by editor(s): February 10, 2000
- Published electronically: June 13, 2001
- Communicated by: Jonathan M. Borwein
- © Copyright 2001 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 129 (2001), 3519-3529
- MSC (2000): Primary 41A17, 41A20
- DOI: https://doi.org/10.1090/S0002-9939-01-06268-2
- MathSciNet review: 1860483