Premium
An extension of an inequality of I. Schur
Author(s) -
Nikolov Geno
Publication year - 2005
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.200310302
Subject(s) - mathematics , combinatorics , schur polynomial , polynomial , extension (predicate logic) , norm (philosophy) , degree (music) , algebraic number , macdonald polynomials , difference polynomials , mathematical analysis , orthogonal polynomials , physics , computer science , political science , acoustics , law , programming language
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 ∈ n , whereHere 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)