\(L_p\) Markov-Bernstein inequalities on arcs of the circle (Q5930032)

The paper is devoted to the proof of the following (very difficult!) result: \textbf{Theorem}: Let \(0<p<\infty\) and \(0\leq \alpha<\beta\leq 2\pi\). Then for all trigonometric polynomials \(s_n\) of degree \(\leq n\) the inequality NEWLINE\[NEWLINE \int_{\alpha}^{\beta} |s'_n(\theta)|^p\left[ \biggl|\sin\frac{\theta-\alpha}{2}\biggr|\biggl|\sin\frac{\theta-\beta}{2}\biggr|+ \left(\frac{\beta-\alpha}{n}\right)^2 \right]^{p/2} d\theta \leq Cn^p\int_{\alpha}^{\beta} |s_n (\theta)|^p d\theta, NEWLINE\]NEWLINE for a certain constant \(C\) which is independent of \(\alpha,\beta,n,s_n\). The result proves that a conjecture of T. Erdélyi is valid. The proof is quite technical. It uses complex analysis and meassure theory, and it is based on the proof of an analogous result for the algebraic polynomial setting. The paper should be very useful for people interested in inequalities in approximation theory, specially those working in the complex domain.