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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Sharp estimates for the maximum over minimum modulus of rational functions


Author: D. S. Lubinsky
Journal: Proc. Amer. Math. Soc. 129 (2001), 3519-3529
MSC (2000): Primary 41A17, 41A20
Published electronically: June 13, 2001
MathSciNet review: 1860483
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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 $r\in {\mathcal{S}}$,

\begin{displaymath}\max _{\left \vert z\right \vert =r}\left \vert R\left ( z\ri... ...left \vert R\left ( z\right ) \right \vert \leq \lambda ^{m+n}.\end{displaymath}

In an earlier paper, we showed that

\begin{displaymath}meas\left ( {\mathcal{S}}\right ) \geq \frac{1}{4}\exp \left ( -\frac{13}{\log \lambda }\right ) ,\end{displaymath}

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 $\lambda \rightarrow 1+$,

\begin{displaymath}meas\left ( {\mathcal{S}}\right ) \geq 4\exp \left ( -\frac{\pi ^{2}}{2\log \lambda }\right ) \big ( 1+o(1) \big ) .\end{displaymath}


References [Enhancements On Off] (What's this?)

  • [1] I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London-Toronto, Ont., 1980. Corrected and enlarged edition edited by Alan Jeffrey; Incorporating the fourth edition edited by Yu. V. Geronimus [Yu. V. Geronimus]\ and M. Yu. Tseytlin [M. Yu. Tseĭtlin]; Translated from the Russian. MR 582453 (81g:33001)
  • [2] W. K. Hayman, Subharmonic functions. Vol. 2, London Mathematical Society Monographs, vol. 20, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London, 1989. MR 1049148 (91f:31001)
  • [3] D. S. Lubinsky, On the Maximum and Minimum Modulus of Rational Functions, Canad. J. Math., 52(2000), 815-832. CMP 2000:15
  • [4] Edward B. Saff and Vilmos Totik, Logarithmic potentials with external fields, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 316, Springer-Verlag, Berlin, 1997. Appendix B by Thomas Bloom. MR 1485778 (99h:31001)

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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
Email: 036dsl@cosmos.wits.ac.za

DOI: http://dx.doi.org/10.1090/S0002-9939-01-06268-2
PII: S 0002-9939(01)06268-2
Received by editor(s): February 10, 2000
Published electronically: June 13, 2001
Communicated by: Jonathan M. Borwein
Article copyright: © Copyright 2001 American Mathematical Society