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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, San Diego, 1980. MR 81g:33001
  • [2] W. K. Hayman, Subharmonic Functions, Vol. 2, Academic Press, London, 1989. MR 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] E. B. Saff and V. Totik, Logarithmic Potential with External Fields, Springer, Berlin, 1997. MR 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

Received by editor(s): February 10, 2000
Published electronically: June 13, 2001
Communicated by: Jonathan M. Borwein
Article copyright: © Copyright 2001 American Mathematical Society

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