Pointwise estimates for the derivative of algebraic polynomials

TL;DR

This paper establishes a sufficient condition on polynomial coefficients ensuring the pointwise Bernstein inequality holds for all points within the unit disk, advancing understanding of polynomial derivative bounds.

Contribution

It provides a new criterion on polynomial coefficients that guarantees the Bernstein inequality holds pointwise in the unit disk.

Findings

Derived a sufficient condition on coefficients for the Bernstein inequality

Extended the inequality to hold pointwise for all points in the unit disk

Clarified the relationship between polynomial coefficients and derivative bounds

Abstract

We give the sufficient condition on coefficients $a_k$ of an algebraic polynomial $P(z)=\sum_{k=0}^{n}a_kz^k$, $a_n\not=0,$ for the pointwise Bernstein inequality $|P'(z)|\le n|P(z)|$ to be true for all $z\in\overline{\mathbb D}:=\{w\in\mathbb C : |w|\le 1\}$.