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

Abstract

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For any fixed finite interval [a, b] on the real line, an arbitrary natural numberr and [sigma]>0, we describe the extremal function to the problem... over all functionsf W^sub ∞^^sup r^ such that |f^sup (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].[PUBLICATION ABSTRACT]