Sharp Markov brothers type inequality in the spaces \(L_{p}\) and \(L_{1}\) on a closed interval
Let \(\mathcal{P}_n\) be the set of algebraic polynomials of degree exactly \(n\geq 2\) with real coefficients. An equality \(\|P^{(n-1)}\|_p\leq c_p(n)\|P\|_1\) with exact constant \(c_p(n),\;p\in [0,\infty],\) between the \(L_p\)-mean of the \((n-1)\)th derivative of an polynomial \(P\in \mathcal{P}_n\) and \(L_1\)-mean of this polynomial on the interval \([-1,1]\) is studied. Sharp constants and extremal polynomials are established. It is proved that the unique extremal polynomial, up to a multiplicative constant, for \(p\in [0,2n^2-1]\) is the Chebyshev polynomial of the second kind and for \(p\in (2n^2-1,\infty ]\) this is the Zolotareff polynomial in the metric \(L_1[-1,1]\).
scientific article; zbMATH DE number 6345607