On the dilatation of extremal quasiconformal mappings of polygons
Let \(h\) be a quasisymmetric mapping of the boundary of the unit disk \(\mathbb{D}_z\) onto the boundary of \(\mathbb{D}_w\) and let \(f\) be a quasiconformal extension of \(h\) into the disk. It is called extremal and denoted by \(f_0\) if its maximal dilatation \(K_0\) is smallest possible. We always assume \(K_0 > 1\). The disk \(\mathbb{D}_z\) becomes a quadrilateral \(Q\) if we mark four different points \(z_j\), \(j = 1,\dots,4\), in the positive direction on its boundary \(\partial \mathbb{D}_z\). The mapping \(f_0\) takes the vertices \(z_j\) into points \(w_j = f_0(z_j)\) on \(\partial\mathbb{D}_w\) and thus the quadrilateral \(Q\) into a quadrilateral \(Q' = f_0(Q)\) inscribed in \(\mathbb{D}_w\). It follows from the definition of quasiconformality that the conformal moduli \(M\) and \(M'\) of \(Q\) and \(Q'\) respectively satisfy (for general properties of quasiconformal mappings, see [\textit{O. Lehto}, and \textit{K. I. Virtanen}, Quasiconformal mappings in the plane (1973; Zbl 0267.30026)]) \[ \frac{1}{K_0} M\leq M'\leq K_0M.\tag{1} \] It has been a question for some time, if the bound \(K_0\) is best possible in the inequality (1), in other words, if the maximal dilatation \(K_0\) of the extremal quasiconformal extension \(f_0\) of \(h\) can be determined by the ratio of the moduli of inscribed quadrilaterals, \[ \sup\;\frac{M'}{M} = K_0.\tag{2} \] The question has recently been answered in the negative by \textit{J. M. Anderson} and \textit{A. Hinkkanen} [Comment. Math. Helv. 70, No. 3, 455-474 (1995; Zbl 0839.30017) by laborious computations of a counterexample (horizontal stretching of a parallelogram) and by \textit{E. Reich} [Proc. Am. Math. Soc. 125, No. 5, 1479-1481 (1997; Zbl 0866.30016)] who reduced it to an approximation problem for holomorphic functions. More counterexamples are given by \textit{S. Wu} [ibid. 72, No. 4, 593-604 (1997)].
scientific article; zbMATH DE number 1278815