Geodesic
Existence Criterion for Estimates of Derivatives of Rational Functions
Matematičeskie zametki, Tome 78 (2005) no. 4, pp. 493-502.

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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 $\mathrm{const}\cdot s!\,\omega(K,z_0,s)$. 
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V. I. Danchenko. Existence Criterion for Estimates of Derivatives of Rational Functions. Matematičeskie zametki, Tome 78 (2005) no. 4, pp. 493-502. http://geodesic.mathdoc.fr/item/MZM_2005_78_4_a1/

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