= Markov inequality for polynomial derivatives
{c}
{title2=$\|p'\|_{[-1,1]}\le n^2\|p\|_{[-1,1]}$}
= Markov polynomial inequality
{c}
{synonym}
Apply the <Bernstein inequality for algebraic polynomials> to the central interval $|x|\le\cos(\pi/(2n))$, using $\sin(\pi/(2n))\ge1/n$. On the remaining regions apply the <Chebyshev nodal derivative comparison> to $p'$. Since $T_n'(\cos\theta)=n\sin(n\theta)/\sin\theta$, the identity expressing $\sin(n\theta)/\sin\theta$ as a sum of $n$ unit complex phases gives $|T_n'|\le n^2$. The endpoint limit is $T_n'(1)=n^2$, so the constant is sharp.
Back to article page
