Markov-type inequalities for polynomials with restricted zeros
Let \(D^{\alpha}\) be a lens shaped, symmetric with respect to the real axis region in \(\mathbb C\), bounded by two circular arcs joining 1 and -1 and meeting each other at an inner angle \(\alpha\pi\) in \(\pm 1\). The author shows the following version of Markov's inequality for polynomials with restrictions on the distribution of the zeros: For every \(0\leq\alpha<1\) there exists a constant \(c(\alpha)\) depending only on \(\alpha\) such that \[ \|p^\prime\|\leq c(\alpha)n^{2-\alpha}\|p\| \] for polynomials \(p\) of degree \(n\) nonvanishing on \(D^\alpha\). Here \(\|\cdot\|=\sup|\cdot|([-1,1])\). Applying this result to the polynomial \(q(z)=\sup_{z\in D^\alpha}|p(z)|+p(z)\) yields, in particular, the well-known Szegö estimate \[ |p^\prime(\pm 1)|\leq c(\alpha)n^{2-\alpha}\sup_{z\in D^\alpha}|p(z)|. \]
scientific article; zbMATH DE number 1389114