A remark on ``an approximation condition and extremal quasiconformal extension''
Let \(D\) be the unit disc in the complex plane and \(b\) a sense-preserving quasisymmetric homeomorphism of \(\partial D\) onto itself. Let \(Q\) be the quadrangle \(D(z_1, z_2,z_3,z_4)\) with domain \(D\) and vertices \(z_1,z_2,z_3,z_4\) on \(\partial D\). Let \(M(Q)\) be the module of \(Q\). Let \[ K_0(f)= {\underset {Q}{\text{l.u.b.}}} {M\bigl(f(Q) \bigr) \over M(Q)} \quad \text{and} \quad K_1(b)= \text{g.l.b.}K \] such that \(f\) has a \(K\)-quasiconformal extension to \(D\cup \partial D\). \textit{J. M. Anderson} and \textit{A. Hinkanen} [Comment. Math. Helv. 70, 455-474 (1995; Zbl 0839.30017)] gave an example where \(K_1(f) >K_0(f)\). The authors present the following necessary and sufficient condition for \(K_1(f)= K_0(f)\). Suppose \(f(z)\) is an extremal quasiconformal mapping of \(D\) onto itself with complex dilation \(\mu(z)\) \((\| \mu\|_\infty =k_1<1)\). Then for its boundary function \(f\) a necessary and sufficient condition for \(K_1(f)= K_0(f)\) is \[ {\underset {Q}{\text{l.u.b}}} {\mathcal R}\iint_D \mu(z) \Phi^{'2}_Q (z)dA= k_1 \] where \(\Phi_Q(z)\) maps \(Q=D(z_1,z_2, z_3,z_4)\) conformally onto the rectangle \[ R= \{\zeta= \xi+i \eta,\;0<\xi <a,\;0<\eta <b,\;ab=1\}. \]
scientific article; zbMATH DE number 1159089