Some identities and inequalities for derivatives (Q1899359)

The authors reprove inequalities due to Bernstein and Brudnyi, respectively, and prove a few other new inequalities involving the derivatives of trigonometric or algebraic polynomials. These proofs are based on the following identity due to \textit{M. Riesz} [Jahresber. Dtsch. Math.-Ver. 23, 354-368 (1914; J.-buch F.d.M. 45, 405)]: If \(\phi_m\) is a trigonometric polynomial of degree \(m\) and \(t_k:= (2k- 1)\pi/2m\), then \[ \phi_m'(x)= {1\over 4m} \sum^{2m}_{k= 1} \phi_m(x+ t_k)(- 1)^{k+ 1} (\sin{1\over 2} t_k)^{- 2}. \] Setting \(\phi_m(x):= (1/ m)\sin mx\) and \(x:= 0\) yields \(1= {1\over 4m^2} \sum^{2m}_{k= 1} (\sin{1\over 2} t_k)^{- 2}\).