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JOURNALS // Matematicheskii Sbornik // Archive

Mat. Sb., 1994 Volume 185, Number 8, Pages 63–80 (Mi sm918)

This article is cited in 20 papers

Estimates of the distances from the poles of logarithmic derivatives of polynomials to lines and circles

V. I. Danchenko


Abstract: Estimates are obtained for the distances $d(Q,\Gamma)$ from poles of the logarithmic derivative $\theta_Q=Q'/Q$ of a polynomial $Q$ to lines $\Gamma$ of the extended complex plane in dependence on the degree $\deg Q$ of the polynomial $Q$ and the norm of $\theta_Q$ in a certain metric on $\Gamma$. The smallest deviations are defined to be
$$ d_n(\Gamma )=\inf \{d(Q,\Gamma ):\|\theta _Q\|_{C(\Gamma )}\leqslant 1, \deg Q\le n\},\qquad n=1,2,\dotsc . $$
In this case if $\Gamma_1$ is the real axis, then $d_n(\Gamma_1)\asymp\ln\ln n/\ln n$, and if $\Gamma_2$ is the unit circle $\vert z\vert=1$, then $d_n(\Gamma_2)\asymp\ln n/n$. When the derivative $\theta'_Q$ is normalized in the metric of $C(\Gamma_1)$, $d_n'(\Gamma_1)\asymp\ln n/\sqrt{n}$ for the corresponding smallest deviation. When $\theta_Q$ is normalized in the metric of $L_p(\Gamma_1)$, $1<p<\infty$, the corresponding smallest deviations do not decrease to zero as $n$ increases, and are bounded below by the quantity $1/p(\sin\pi/p)^{p/(p-1)}$.

UDC: 517.53

MSC: 30C10, 30C15

Received: 28.09.1993


 English version:
Russian Academy of Sciences. Sbornik. Mathematics, 1995, 82:2, 425–440

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