The maximal range problem for a bounded domain
Let \(\Omega\subset\mathbb C\) be a bounded domain, \(0\in\Omega\), \(L:=\partial\Omega\), and let \(\mathcal P_n\) be the set of all complex polynomials of degree at most \(n\). Denote \(\mathcal P_n(\Omega):=\{p\in\mathcal P_n: p(0)=0, p(\mathbb D)\subset\Omega\}\), where \(\mathbb D\) is the unit disk, and \(\Omega_n:=\bigcup_{p\in\mathcal P_n(\Omega)}p(\mathbb D)\) is the maximal range of \(\mathcal P_n(\Omega)\). For a conformal mapping \(f\) of \(\mathbb D\) onto \(\Omega\) with \(f(0)=0\), denote \(f_s(z):=f((1-s)z)\), \(0\leq s<1\). Assume \(f\) is not a polynomial. For a simply connected domain \(\Omega\) denote by \(s(n,\Omega)\) the infimum of \(s\) such that there exists a univalent polynomial \(p\in\mathcal P_n(\Omega)\) with the property \(f_s\prec p\prec f\). An earlier result of the authors states that \[ s(n,\Omega)\leq\frac{c_0}{n}\tag{1} \] for all sufficiently large \(n\) and a universal constant \(c_0>1\). The authors prove Theorem 1 which shows in particular that (1) is sharp for a wide class of bounded domains. Theorem 2 of the article implies that (1) is of the right order if \(L\) has in at least one point something like an acute angle with respect to \(\Omega\). Let \(\Omega(t):=\{z\in\Omega: \text{dist}(z,L)>t\}\), \(t>0\), and \(\Omega^0(t)\) denote the simply connected component of \(\Omega(t)\ni0\) containing the origin. The discrepancy between the domain and its maximal ranges is described by Theorem 3 according to which, for simply connected \(\Omega\), there exists \(n_0=n_0(\Omega)\) and \(c_4=c_4(\Omega)>0\) such that \(\Omega_n\supset\Omega^0(c_4(\log n)^{-1/2})\) for \(n>n_0\). The better estimates for the closeness of \(\Omega_n\) to \(\Omega\) are given in Theorem 4 and Theorem 5 in terms of the Hausdorff distance. Theorem 5 relates to John domains satisfying the \(k\)-quasidisk condition for \(0\leq k<1\). Theorem 6 characterizes the closeness of a boundary point \(z_0\in L\) to \(p(\mathbb D)\), \(p\in\mathcal P_n(\Omega)\) univalent, in the case of Dini-smooth boundary arcs joining at \(z_0\) and forming an inner angle \(\alpha\pi\), \(0<\alpha<1\), and it shows that under some geometrical assumptions on \(\Omega\) the estimate of the Hausdorff distance from \(L\) to \(\partial\Omega_n\) given in Theorem 5 is close to optimal.
scientific article; zbMATH DE number 5572508