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

 

Small values of polynomials
and potentials with $L_p$ normalization


Author: D. S. Lubinsky
Journal: Proc. Amer. Math. Soc. 127 (1999), 529-536
MSC (1991): Primary 30C10, 31A15; Secondary 41A17, 41A44, 30C85
MathSciNet review: 1468199
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Abstract | References | Similar Articles | Additional Information

Abstract: For a polynomial $P$ of degree $\leq n$, normalized by the condition

\begin{displaymath}\frac 1{2\pi }\int _0^{2\pi }\mid P(re^{i\theta })\mid ^pd\theta =1, \end{displaymath}

we show that $E(P;r;\varepsilon ):=\{z:\mid z\mid \leq r,\mid P(z)\mid \leq \varepsilon ^n\}$ has $cap$ at most $r\varepsilon \kappa _{np}$, where $\kappa _{np}\leq 2$ is explicitly given and sharp for each $n,r$. Similar estimates are given for other normalizations, such as $p=0$, and for planar measure, and for generalized polynomials and potentials, thereby extending work of Cuyt, Driver and the author for $p=\infty $. The relation to Remez inequalities is briefly discussed.


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Additional Information

D. S. Lubinsky
Affiliation: Department of Mathematics, Witwatersrand University, Wits 2050, South Africa
Email: 036dsl@cosmos.wits.ac.za

DOI: http://dx.doi.org/10.1090/S0002-9939-99-04549-9
PII: S 0002-9939(99)04549-9
Keywords: Cartan's lemma, capacity, polynomials, $L_p$ norm
Received by editor(s): July 17, 1996
Received by editor(s) in revised form: June 2, 1997
Communicated by: J. Marshall Ash
Article copyright: © Copyright 1999 American Mathematical Society