An extension of the Landau-Kolmogorov inequality. Solution of a problem of Erdös

For any fixed finite interval [a, b] on the real line, an arbitrary natural numberr and σ>0, we describe the extremal function to the problem\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\left\| {f^{(k)} } \right\|L_p \left[ {a,b} \right]^{ \to \sup } \left( {1 \leqslant k \leqslant r - 1, 1 \leqslant p< \infty } \right)$$ \end{document} over all functionsf ∈W∞r such that |f(r)(x)| ≤σ, |f(x)|≤1 on (−∞, ∞). Similarly, we solve the problem, raised by Paul Erdös, of characterizing the trigonometric polynomial of fixed uniform norm whose graph has maximal arc length over [a, b].