Bernstein's inequality for algebraic polynomials on circular arcs (Q1943999)

For \(0 < \omega \leq \pi\), let \(K_{\omega}=\{e^{i\theta} \, |\;\theta\in [-\omega, \omega]\}\) be the circular arc on the unit circle of central angle \(2\omega\) and with midpoint at 1. If \(P_n\) is a polynomial of degree at most \(n\), then \[ |P'_n(e^{i\theta})| \leq \frac{n}{2}\left(1 + \frac{\sqrt{2}\cos(\theta/2)}{\sqrt{\cos\theta - \cos\omega}} \right)\|P_n\|_{K_{\omega}}, \quad \theta\in (-\omega, \omega). \eqno (1) \] For even \(n\), inequality (1) is an easy consequence of the classical Videnskii inequality on trigonometric polynomials, and for odd \(n\), it also follows from a related inequality of Videnskii for a trigonometric expression in which the frequencies of cosine and sine are an integer plus one half. It is shown that this Bernstein-type inequality is sharp. Moreover, the authors give an alternative proof for (1) using a result of P. Borwein and T. Erdélyi.