On the Inequality of I. Schur*

Denote by pn the set of all real algebraic polynomials of degree at most n. The classical inequality of I. Schur asserts that the transformed Chebyshev polynomial Tn x.sTn xcos pr2n.. has the greatest uniform norm of its first derivative on wy1, 1xamong all fgSn, where Sn[ f: fgpn, f y1.sf 1.s0, 5f5F14. Here we extend this result to the kth derivative by proving the inequality f k. FTn k. ks1, . . . , n.for all fgSn. For kG2 we prove the same inequality in the larger class cos jprn. Dn[ f:fgpn,f y1.sf 1.s0, f F1, js1, . . . , ny1 . / 5 cos pr2n. This extension is in the spirit of the refinement of the Markov inequality found by Duffin and Schaeffer.