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

Let Πₙ 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'\|Lₚ[-1,1]>c\, (√n)1-1/p+1/q\, \|P\|Lq[-1,1], P∈ Πₙ, where 0<p≤ q≤ ∞, 1-1/p+1/q≥ 0 ($c>0$ is a constant independent of P and n). We show that Zhou's estimate remains true in the case p=∞, $q>1$. Some of related Tur\'{a}n type inequalities are also discussed.