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 p lie in the unit interval I = def [ − 1 , 1 ] I \overset {\text {def}}{=} [-1,1] , then ‖ p ′ ‖ L ∞ ( I ) ≥ deg ⁡ ( p ) / 6 ‖ p ‖ L ∞ ( I ) \|p’\|_{L^{\infty }(I)}\ge {\sqrt {\deg (p)}}/{6}\; \|p\|_{L^{\infty }(I)}\; . Our goal is to study the feasibility of lim n → ∞ ‖ p n ′ ‖ X / ‖ p n ‖ Y = ∞ \lim _{{n\to \infty } }{\|p_{n}’\|_{X}} / {\|p_{n}\|_{Y}} =\infty for sequences of polynomials { p n } n ∈ N \{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.