On the degree of convergence of lemniscates in finite connected domains (Q707201)

Let \(\Omega\) be an unbounded region in the complex plane of connectivity \(q\) bounded by mutually disjoint Jordan curves \(\Gamma_j, j=1,\dots,q\). The boundary \(\Gamma=\partial \Omega\) can be approximated by lemniscates which lie in \(\Omega\). The author gives a quantitative version of this theorem. Let \(G(z)\) be the Green's function of \(\Omega\) with pole at infinity and let \(\Gamma_{1+s}\) be the level line \(\{z\in \Omega: G(z)=\ln(1+s)\}\). Define \(S_n\) as the infimum of all \(s\) for which there exists a polynomial \(P_n\) of degree \(\leq n\) such that the lemniscate \(J(P_n):=\{z: | P_n(z)| =1\}\) is outside \(\Gamma\) but inside \(\Gamma_{1+s}\). The author shows that the estimate \(S_n=O(\ln n/n)\) holds if the boundary curve \(\Gamma\) is Dini-smooth. This generalizes a result of \textit{V. Andrievskij} [J. Approximation Theory 105, No. 2, 292--304 (2000; Zbl 0963.41012)] who proved such an estimate for the simply connected case \(q=1\).