Extremal problems for quasiconformal maps of punctured plane domains (Q2781376)

The author proves that the affine mapping \(f= Kx+ iy\) with \(K>1\) of the plane punctured at \(z_{kl}= k+ il\) with \(k,l\in\mathbb{Z}\) is uniquely extremal in the class \([f]\) of all quasiconformal mappings \(g\) for which the composition \(f\circ g^{-1}\) is homotopic to a conformal map. It was shown by \textit{E. Reich} and \textit{K. Strebel} [Complex analysis -- Proc. 5th Rom.-Finn. Semin., Bucharest 1981, Part 1, Lect. Notes Math. 1013, 182-212 (1983; Zbl 0528.30009)] that \(f\) is extremal in this class. For the proof of the unique extremality the author uses a result concerning the unique Hahn-Banach extension of a functional [See \textit{V. Božin}, \textit{N. Lakic}, \textit{V. Marković} and \textit{M. Mateljević}, J. Anal. Math. 75, 299-338 (1998; Zbl 0929.30017)] and a criterion on unique extremality due to \textit{E. Reich} [Ann. Acad. Sci. Fenn., Ser. A I 6, 289-301 (1981; Zbl 0485.30025)]. In the paper the long-standing question of unique extremality is answered and a corollary related to the geometry of the corresponding Teichmüller space is derived, too. In the investigations of the paper a lot of estimates for meromorphic functions are obtained.