An extension of an inequality of I. Schur

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)