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St. Petersburg Mathematical Journal

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Bernstein-type inequalities for the derivatives of rational functions in $L_{p}$-spaces, $0<p<1$, on Lavrent'ev curves

Author: A. A. Pekarskii
Translated by: S. V. Kislyakov
Original publication: Algebra i Analiz, tom 16 (2004), nomer 3.
Journal: St. Petersburg Math. J. 16 (2005), 541-560
MSC (2000): Primary 41A17, 41A20, 41A25, 41A27, 30D55
Published electronically: May 2, 2005
MathSciNet review: 2083568
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Abstract: Let $S$ be a simple or a closed Lavrent'ev curve on the complex plane, let $0<p<1$with $1/p \not\in \mathbb{N} $, and let $s\in\mathbb{N} $. It is shown that for an arbitrary rational function $r$ of degree $n$such that $\vert r\vert^{p}$ is integrable on $S$ the following inequality is fulfilled:

\begin{displaymath}\bigg(\int_{S}\vert r^{(s)}(z)\vert^{\sigma} \,\vert dz\vert\... ...igg(\int_{S} \vert r(z)\vert^{p} \,\vert dz\vert\bigg)^{1/p} , \end{displaymath}

where $1/\sigma=s+1/p$, and $c>0$ depends only on $S, p$, and $s$.

Earlier (in 1995) this result was obtained by the author and Stahl for the segment and the circle. The inequality is used to deduce an inverse rational approximation theorem in the Smirnov class $E_{p}$. Other rational approximation problems in $L_{p}$ and $E_{p}$ are also treated.

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Additional Information

A. A. Pekarskii
Affiliation: Belorussian State Technological University, Ul. Sverdlova 13a, Minsk 220630, Belorussia

Keywords: Rational function, Bernstein-type inequalities, Smirnov space
Received by editor(s): September 1, 2003
Published electronically: May 2, 2005
Additional Notes: Supported by the Russian–Belorussian Foundation for Basic Research (grant no. F02R-057).
Article copyright: © Copyright 2005 American Mathematical Society

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