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 Mat. Zametki, 2005, Volume 78, Issue 4, Pages 493–502 (Mi mz2608)

Existence Criterion for Estimates of Derivatives of Rational Functions

V. I. Danchenko

Abstract: Suppose that $K$ is a compact set in the open complex plane. In this paper, we prove an existence criterion for an estimate of Markov–Bernstein type for derivatives of a rational function $R(z)$ at any fixed point $z_0\in K$. We prove that, for a fixed integer $s$, the estimate of the form $|R^{(s)}(z_0)|\le C(K,z_0,s)n\|R\|_{C(K)}$, where $R$ is an arbitrary rational function of degree $n$ without poles on $K$ and $C$ is a bounded function depending on three arguments $K$, $z_0$, and $s$, holds if and only if the supremum $\omega(K,z_0,s)=\sup\{\operatorname{dist}(z,K)/|z-z_0|^{s+1}\}$ over $z$ in the complement of $K$ is finite. Under this assumption, $C$ is less than or equal to $\mathrm{const}\cdot s! \omega(K,z_0,s)$.

DOI: https://doi.org/10.4213/mzm2608

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English version:
Mathematical Notes, 2005, 78:4, 456–465

Bibliographic databases:

UDC: 517.53
Revised: 12.10.2004

Citation: V. I. Danchenko, “Existence Criterion for Estimates of Derivatives of Rational Functions”, Mat. Zametki, 78:4 (2005), 493–502; Math. Notes, 78:4 (2005), 456–465

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/mz2608
• https://doi.org/10.4213/mzm2608
• http://mi.mathnet.ru/eng/mz/v78/i4/p493

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This publication is cited in the following articles:
1. V. I. Danchenko, “Integral Estimates of Lengths of Level Lines of Rational Functions and Zolotarev's Problem”, Math. Notes, 94:3 (2013), 314–319
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