Estimating the error in the Koebe construction
\textit{P. Koebe} [``Über eine neue Methode der konformen Abbildung und Uniformisierung'', Gött. Nachr. 1912, 861--878 (1912; JFM 43.0520.01)] proposed an iterative method, the so-called Koebe construction, for approximating the unique conformal map \(f\) of a nondegenerate \(n\)-connected domain \(D\) with \(\infty\in D\) and \(0\not\in D\) onto a circular domain \(C\) such that \(f(z)= z+ O(z^{-1})\). \textit{D. Gaier} [Arch. Ration. Mech. Anal. 3, 149--178 (1959; Zbl 0088.28702)] and \textit{P. Henrici} [Applied and computational complex analysis. Volume III: Discrete Fourier analysis, Cauchy integrals, construction of conformal maps, univalent functions. Reprint. New York, NY: Wiley (1993; Zbl 1107.30300)] calculated upper bounds on the error in the construction, but these depend on prior knowledge of \(C\). Here the authors calculate a bound for the error in the Koebe construction that requires only knowledge of \(D\), using a result of \textit{R. E. Thurman} [Trans. Am. Math. Soc. 346, No. 2, 605--616 (1994; Zbl 0820.30013)] on the distortion of capacity by conformal maps and a generalization of the Schwarz-Pick lemma by \textit{Z.-X. He} and \textit{O. Schramm} [Ann. Math. (2) 137, No. 2, 369--406 (1993; Zbl 0777.30002)].
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