Bernstein-type inequalities (Q1759355)

Let \(E\subset [-\pi,\pi]\) be compact and symmetric with respect to the origin and let \(\Gamma_E = \{ e^{it}\,| \;t\in E \}\). Denote by \(\omega_{\Gamma_E}\) the equilibrium density of \(\Gamma_E\) on the unit circle \({\mathbb T}\). It is proved that if \(\theta\in E\) is an inner point of \(E\) then for any trigonometric polynomial \(T_n\) of degree at most \(n\) we have \[ |T'_n(\theta)| \leq n2\pi\omega_{\Gamma_E}(e^{i\theta})\|T_n\|_E. \eqno (1) \] This Bernstein-type inequality is sharp. In particular, if \(E=[-\beta, -\alpha]\cap [\alpha, \beta]\) with some \(0\leq \alpha <\beta \leq \pi\), then \[ \omega_{\Gamma_E}(e^{i\theta})=\frac{1}{2\pi}\frac{|\sin\theta|}{\sqrt{|\cos\theta - \cos\alpha||\cos\theta - \cos\beta|}}. \] So for \(E=[-\beta, \beta]\), he gets from (1) the Videnskii inequality. Also, it is shown that the original Bernstein inequality implies its Szegő variant as well as both Videnskii's inequality and its half-integer variant.