Existence Criterion for Estimates of Derivatives of Rational Functions

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 $\mathrm{const}\cdot s!\,\omega(K,z_0,s)$.