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

2024/05/29 by М. А. Комаров, Komarov, Mikhail A.
Computer Science · Mathematics · #41A17 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical functions and polynomials #Matrix Theory and Algorithms #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.2405.19158

openalex publication_date 2024/05/29 · openalex created_date 2024/05/31 · openalex updated_date 2026/07/28

Abstract

Let Π_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: ‖P'‖_Lp[-1,1]>c (√(n))^1-1/p+1/q ‖P‖_Lq[-1,1], P∈ Π_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 Turán type inequalities are also discussed.

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