Inequalities of Duffin-Schaeffer type (Q2753569)

Let \(T_n(x)=\cos n \arccos x\) and let the points \(t_\nu \in [-1,1]\), \(\nu=0,1,\ldots , n\) interlace with the zeros of \(T_n\). If \(f\) is a polynomial of degree at most \(n\) and \(|f(t_\nu)|\leq |T_n(t_\nu)|\) for each \(\nu\), then \(\|f'\|_{C[-1,1]} \leq n^2\). Moreover, \(\|f'\|= n^2\) if and only if \(f=\pm T_n\).NEWLINENEWLINENEWLINEIn the particular case of \(t_\nu= \cos(\nu\pi/n)\) this inequality was established in 1941 by Duffin and Schaeffer. Since \(|T_n( \cos(\nu\pi/n))|=1\), it contains the A.A. Markov inequality \(\|f'\|\leq n^2 \|f\|\).