Arestov's theorems on Bernstein's inequality

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 $p\ge 0$. 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 $T_n\in {\mathcal{T}}_n^c$.