On reverse Markov–Nikol’skii inequalities for polynomials with restricted zeros

Abstract

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 Turán 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 Turán type inequalities are also discussed.