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 & aacute;n type reverse Markov-Nikol'skii inequality:parallel to P 'parallel to L-p[-1,L-1] > c(root n)(1-1/p+1/q)parallel to P parallel to(Lq[-1,1]),P is an element of Pi(n), where 0 < p <= q <= infinity, 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 = infinity,q > 1. Some of related Tur & aacute;n type inequalities are also discussed.(c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies