On Bernstein's inequality for polynomials

TL;DR

This paper explores Bernstein's inequality for polynomials, providing multiple proofs and extending the inequality to $L^p$-norms, enhancing understanding of polynomial derivative bounds.

Contribution

The paper introduces new approaches and extensions of Bernstein's inequality, especially for $L^p$-norms, broadening its applicability and theoretical understanding.

Findings

Multiple proofs of Bernstein's inequality presented

Extensions to $L^p$-norms analyzed

New variants of the inequality proposed

Abstract

Bernstein's classical inequality asserts that given a trigonometric polynomial $T$ of degree $n\geq1$, the sup-norm of the derivative of $T$ does not exceed $n$ times the sup-norm of $T$. We present various approaches to prove this inequality and some of its natural extensions/variants, especially when it comes to replacing the sup-norm with the $L^p-norm$.