On reverse Markov-Nikol'skii inequalities for polynomials with restricted zeros

Let $\Pi_n$ be the class of algebraic polynomials $P$ of degree $n$, all of whose zeros lie on the segment $[-1,1]$. In 1995, S.P. Zhou has proved the following Tur\'{a}n type reverse Markov-Nikol'skii inequality: $\|P'\|_{L_p[-1,1]}>c\, {(\sqrt{n})}^{1-1/p+1/q}\, \|P\|_{L_q[-1,1]}$, $P\in \Pi_n$, where $0<p\le q\le \infty$, $1-1/p+1/q\ge 0$ ($c>0$ is a constant independent of $P$ and $n$). We show that Zhou's estimate remains true in the case $p=\infty$, $q>1$. Some of related Tur\'{a}n type inequalities are also discussed.