Analogs of Markov's inequality in normed spaces (Q1889665)

The Markov inequality states \[ | P^{(k)}_n(x)| \leq c_{n,k}\| P_n\| _{C([-1,1])}\quad \text{ for } -1\leq x \leq 1, 0\leq k \leq n \] for all polynomials \(P_n(x)\) of degree at most \(n\) in one real variable \(x\). The best constants \(c_{n,k}\) are explicitly known and they are attained for the Chebyshev polynomials \(T_n(x)=\cos(n\cdot \arccos x)\) at \(x=\pm1\). In the paper under review, some generalizations of these inequalities are given for real valued polynomials on Banach spaces. The interesting paper does not contain any proofs.