Lower bounds for the modulus of the logarithmic derivative of a polynomial

Estimates are given for the measure of a section of an arbitrary straight line of the set
$$ E_\delta=\{z:|P'(z)/(nP(z))|\le\delta\}\quad(\delta>0), $$
where $P(z)$ is a polynomial of degree $n$.
THEOREM. {\em Suppose $P(x)=(x-x_1)\dots(x-x_n)$ is a polynomial with real zeros. Then, for any $\delta>0$, on any interval $a\le x\le b$, containing all of the $x_k$ $(k=1,2,\dots,n)$, outside an exceptional set $E_\delta\subset[a,b]$ such that
$$ \operatorname{mes}E_\delta\le(\sqrt{1+\delta^2(b-a)^2}-1)/\delta, $$
we have the inequality}
$$ |P'(x)/(nP(x))|>\delta. $$

A similar estimate is given for polynomials whose roots lie either in $\operatorname{Im}z\ge0$ or in $\operatorname{Im}z\le0$.