Pointwise estimates for the derivative of algebraic polynomials

Abstract

We give the sufficient condition on coefficients $a_k$ of an algebraic polynomial $P(z)={\sum }_{k=0}^n{a}_k{z}^k$, $a_n\ne 0,$ for the pointwise Bernstein inequality $|P^{\prime }(z)|\le n|P(z)|$ to be true for all $z\in \overline{\mathbb{D}}:=\{w\in \mathbb{C}:|w|\le 1\}$.