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

Let 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\'an type reverse Markov-Nikol'skii inequality: \|P'\|₋䂹-₁, ₁>c\, (n) ^1-1/p+1/q\, \|P\|₋ₐ-₁, ₁, P ₙ, where 00 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\'an type inequalities are also discussed.