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

V. I. Danchenko

Sbornik. Mathematics, Tome 82 (1995) no. 2, pp. 425-440

Résumé

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 $L_p(\Gamma_1)$, $1$, the corresponding smallest deviations do not decrease to zero as $n$ increases, and are bounded below by the quantity $1/p(\sin\pi/p)^{p/(p-1)}$.