On polynomials with a prescribed zero on a quasicircle
Let \(G\) be a finite domain in the complex plane. Fix a point \(z_0\in G\) and a point \(z_1\) on the boundary \(\partial G\). Set \(\alpha_n(G,z_0,z_1):=1-\sup|P_n(z_0)|\), where the supremum is taken over the set of all polynomials of degree \(\leq n\) which vanish at \(z_1\). When \(G={\mathbf D}\) is the unit circle, one has \(c_1n^{-1}\leq \alpha_n(\mathbf{D},z_0,z_1)\leq c_2n^{-1}\), where \(c_1\) and \(c_2\) are absolute constants (this inequality is due to Halász). The authors prove an asymptotically sharp result which extends the result above to quasicircles.
scientific article; zbMATH DE number 5305536