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

Let ${\mathrm{\Pi }}_n$ 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 Turán type reverse Markov-Nikol’skii inequality: $\mathrm{\parallel }{P}^{\mathrm{\prime }}{\mathrm{\parallel }}_{L_p[-1,1]}>c{(\sqrt{n})}^{1-1\mathrm{/}p+1\mathrm{/}q}\mathrm{\parallel }P{\mathrm{\parallel }}_{L_q[-1,1]}$, $P\in {\mathrm{\Pi }}_n$, where $0<p\le q\le \mathrm{\infty }$, $1-1\mathrm{/}p+1\mathrm{/}q\ge 0$ ($c>0$ is a constant independent of $P$ and $n$). We show that Zhou’s estimate remains true in the case $p=\mathrm{\infty }$, $q>1$. Some of related Turán type inequalities are also discussed.