On the components of the lemniscate containing no critical points of a polynomial other than its zeros

Let $P$ be a complex polynomial of degree $n$ and let $E$ be a connected component of the set $\{z\colon|P(z)|\leq1\}$ containing no critical points of $P$ other than its zeros. We prove the inequality $|(z-a)P'(z)/P(z)|\leq n$ for all $z\in E\setminus\{a\}$, where $a$ is the zero of the polynomial $P$ lying in $E$. Equality is attained for $P(z)=cz^n$ and any $z$, $c\neq0$. Bibl. 4 titles.