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Sharp estimates for the maximum over minimum modulus of rational functions

Author(s): D. S. Lubinsky
Journal: Proc. Amer. Math. Soc. 129 (2001), 3519-3529.
MSC (2000): Primary 41A17, 41A20
Posted: June 13, 2001
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Abstract: Let $m,\,n\geq 0,\,\lambda >1$ , and 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

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:

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

DOI: 10.1090/S0002-9939-01-06268-2
PII: S 0002-9939(01)06268-2
Received by editor(s): February 10, 2000
Posted: June 13, 2001
Communicated by: Jonathan M. Borwein
Copyright of article: Copyright 2001, American Mathematical Society


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