A counterexample theorem in quasiconformal mapping theory
Let \(\Delta\) denote the open unit disc, \(h\) a quasisymmetric homeomorphism of \(\partial\Delta\) onto itself, \(Q(h)\) the family of quasiconformal mappings of \(\Delta\) onto itself with \(h\) as boundary values. For \(f\in Q(h)\) let \(K[f]\) denote the maximal dilation of \(f\). Let \[ K^*_h=\underset {b\in Q(h).}{\text{g.l,b.}} K[f]. \] Let \(H_h=\text{g.l.b} K[f|\Delta-E]\) for all compact \(E\subset\Delta\). Let \(Q\) be a quadrangle obtained from \(\Delta\) by assigning four points on \(\partial\Delta\) as verties, \(h(Q)\) the quadrangle image of \(Q\) under \(h\); \(M(Q)\), \(M(h(Q))\) corresponding modules. Let \(K_h=\text{l.u.b}{M(h(Q)) \over M(Q)}\) for all such \(Q\). Evidently \(K_h\leq K^b_h\). There was a (rather dubious) conjecture that \(K_h=K_h^k\). \textit{J. M. Anderson} and \textit{A. Hinkkanen} [Comment. Math. Helv. 70, No. 3, 455--474 (1995; Zbl 0839.30017)] gave a counter example. \textit{S. Wu} [Comment. Math. Helv. 72, 593-604 (1997; Zbl 0944.30016)] and \textit{S. Yang} (Result. Math. 31, No. 1-2, 180-188 (1997; Zbl 0872.30011)], gave as a necessary condition for \(K_h=K^*_h\) that either is induced by an affine mapping or \(H_h=K^*_h\). They raised the question whether \(H_h=K^*_h\) implies that \(K_h= K^*_h\). The author gives a counterexample. The author considers some related topics concerning a quantity \(K_h'\) connected with the Dirichlet integral.
scientific article; zbMATH DE number 1782028