Uniform convergence of the Bieberbach polynomials in closed smooth domains of bounded boundary rotation.
Let \(\pi_n(z)\), \(n=1,2,...\) be the Bieberbach polynomials for \((G,z_0)\), where \(G\) is a finite simply connected domain in \(\mathbb{C}\) bounded by a rectifiable Jordan curve \(L\) and \(z_0\in G\). The paper deals with the uniform convergence on \(\overline{G}\) of the sequence of Bieberbach polynomials. The following definition, given here in a short version (the details can be seen in the paper), is crucial and it is added to the condition of \(L\) being smooth . ``The domain \(G\) is of bounded boundary rotation (BBR) if the tangent direction angle of \(L\) has bounded variation.'' The main result of the present paper is the following. Theorem. If \(G\) is a finite smooth domain of BBR, then for every \(\varepsilon >0\) there exists a constant \(c=c(\varepsilon)\) such that \[ \| \phi_0-\pi_n\| _{\overline{G}}:=\max_{z\in \overline{G}}| \phi_0(z)-\pi_n(z)| \leq \frac{c}{n^{1-\varepsilon}},\quad n\geq 1.\tag{1} \] Estimate (1) is an improvement of that obtained by Mergelyan in 1951 when \(L\) is a smooth Jordan curve.
scientific article; zbMATH DE number 2026874
scientific article