An extension of an inequality of I. Schur

✍ Scribed by Geno Nikolov

• Publisher: John Wiley and Sons
• Year: 2005
• Tongue: English
• Weight: 241 KB
• Volume: 278
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.200310302

✦ Synopsis

Abstract

Denote by π~n~ the set of all algebraic polynomials of degree at most n with complex coefficients. An inequality of I. Schur asserts that the first derivative of the transformed Tchebycheff polynomial $\overline {T}_n (x) = T_n (x , \rm {cos} , {{\pi} \over {2n}})$ has the greatest uniform norm in [−1, 1] among all f ∈ 𝒮~n~, where

equation image

Here we show that this extremal property of $\overline {T}_n$ persists in the wider class of polynomials f ∈ π~n~ which vanish at ±1, and for which there exist n − 1 points ${t_j}^{n-1}_{j=1}$ separating the zeros of $\overline {T}_n$ and such that $|f(t_j)| \le |\overline {T}_n (t_j)|$ for j = 1, …, n − 1. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)