On the inequality of I. Schur
The authors extend to the \(k\)-th derivative the well-known Schur inequality by proving that for any polynomial \(f\) of degree at most \(n\) such that \(f(-1) = f(1)=0\) and \(|f|: =\sup |f|([-1,1])\leq 1\) one has \(|f^{(k)} |\leq|\overline T_n^{(k)} |\), where \(\overline T_n(x) =T_n(x \cos (\pi/2n))\) is a modified Chebyshev polynomial. Actually they show that the inequality also holds if the assumption that \(|f|\leq 1\) is replaced by a weaker one that \(|f(\cos (j\pi/n)/ \cos (\pi/2n)) |\leq 1\).
scientific article; zbMATH DE number 1115578