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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