# Arestov's theorems on Bernstein's inequality

## TL;DR

This paper presents a straightforward and partly novel proof of Arestov's extension of Bernstein's inequality in Lp spaces, utilizing an approach related to Mahler's inequality for polynomials.

## Contribution

It introduces a simplified proof of Arestov's theorem, extending Boyd's approach from algebraic to trigonometric polynomials.

## Key findings

- Proof of Arestov's extension of Bernstein's inequality in Lp for all p ≥ 0
- Extension of Boyd's approach to trigonometric polynomials
- Simplification of existing proof techniques

## Abstract

We give a simple, elementary, and at least partially new proof of Arestov's famous extension of Bernstein's inequality in $L_p$ to all $p \geq 0$. Our crucial observation is that Boyd's approach to prove Mahler's inequality for algebraic polynomials $P_n \in {\mathcal P}_n^c$ can be extended to all trigonometric polynomials $T_n \in {\mathcal T}_n^c$.