Lower bounds for derivatives of polynomials and Remez type inequalities

Abstract:

P. Turán [Über die Ableitung von Polynomen, Comositio Math. 7 (1939), 89–95] proved that if all the zeros of a polynomial $p$ lie in the unit interval $I \overset {\text {def}}{=} [-1,1]$, then $\|p’\|_{L^{\infty }(I)}\ge {\sqrt {\deg (p)}}/{6}\; \|p\|_{L^{\infty }(I)}\;$. Our goal is to study the feasibility of $\lim _{{n\to \infty } }{\|p_{n}’\|_{X}} / {\|p_{n}\|_{Y}} =\infty$ for sequences of polynomials $\{p_{n}\}_{n\in \mathbb N }$ whose zeros satisfy certain conditions, and to obtain lower bounds for derivatives of (generalized) polynomials and Remez type inequalities for generalized polynomials in various spaces.