On the inequality of I. Schur

[摘要] Denote by pi(n) the set of all real algebraic polynomials of degree at most n. The classical inequality of I. Schur asserts that the transformed Chebyshev polynomial (T) over bar(n)(x) = T-n(x cos(pi/2n)) has the greatest uniform norm of its first derivative on [-1,1] among all f is an element of L-n, where L-n = {f:f is an element of pi(n), f(-1) = f(1) = 0, parallel to f parallel to less than or equal to 1}. Here we extend this result to the kth derivative by proving the inequality parallel to f((k))parallel to less than or equal to parallel to (T) over bar(n)((k))parallel to (k = 1,...,n) for all f is an element of L-n. For k greater than or equal to 2 we prove the same inequality in the larger class D-n = {f:f is an element of pi n, f(-1) = f(1) = 0, \f(cox(j pi/n)/cox(pi/2n))\ less than or equal to 1, j = 1, ...,n - 1}. This extension is in the spirit of the refinement of the Markov inequality found by Duffin and Schaeffer. (C) 1997 Academic Press.

[发布日期] 1997-12-15