The maximal range problem for a quasidisk. (Q1399535)

Let \(G, D\) be simply connected domains in the complex plane \(\mathbb C\) with \(0\in G\cap D\). Let \(\mathcal P_n\) be the set of all holomorphic polynomials of degree \(\leq n\). Put \[ \mathbb P_n(G,D):= \{p\in \mathcal P_n; p(0)=0, \;p(G)\subset D\}, \] and define the maximal polynomial range by the formula \[ D_n(G):= \bigcup_{p\in \mathcal P_n(G,D)} p(G). \] For more details concerning this notion see \textit{V. V. Andrievskii} and \textit{St. Rusheweyh} [Complex polynomials and maximal range: background and applications, in: G.V. Milovanovic (Ed.), Recent Progress in Inequalities (dedicated to Prof. Dragoslav S. Mitinivic), Kluwer Academic Publishers, Dordrecht, Math. Appl. (Dordrecht) 430, 31--54 (1998; Zbl 0998.30002)]. Main result: Assume that \(G\) is a quasidisk and the boundary of \(D\) contains at least two points. Let \(\psi\) be the conformal map of the unit disk \(\mathbb D\) onto \(D\) such that \(\psi(0) = 0\), \(\psi'(0)>0\). Then there exists \(n_0\in \mathbb N\) dependent only on \(G\) such that \[ \psi((1-\alpha(1/n))D)\subset D_n(G), \quad n>n_0, \] where \(\alpha\) is a real-valued function dependent on geometric properties of \(G\) such that \(\alpha (\delta) >0\) for \(0<\delta <\pi\), and \(\alpha (\delta) \searrow 0\) as \(\delta \searrow 0\).