Constants in V. A. Markov's inequality in \(L^p\) norms (Q2344298)

For \(k\in\mathbb{N}\), \(k\geq 3\), \(n\in\mathbb{N}\) and \(1\leq p<\infty\) there is obtained an explicit constant \(C(k,n,p)\), for which the inequality \(\| P^{(k)}\|_p\leq C(k,n,p)\| P\|_p,\) holds for all polynomials \(P\) of degree at most \(n\) on the interval \([-1,1]\). This is an extension for \(p\in [1,\infty)\) of the classical inequality of V. A. Markov: \(\| P^{(k)}\|_{\infty}\leq \| T_n^{(k)}\|_{\infty}\| P\|_{\infty}\), where \(T_n\) are the Chebyshev polynomials. Moreover, there holds the relation \(\lim_{p\to\infty}C(k,n,p)=\| T_n^{(k)}\|_{\infty}\).