Growth of the coefficient of quasiconformality and the boundary correspondence of automorphisms of a ball (Q1078711)

By a well-known theorem due to J. Väisälä and F. W. Gehring, a quasiconformal homeomorphism of the unit ball \(B^ n\) onto \(B^ n\) has a homeomorphic extension to the closure \(\bar B^ n\). Various generalizations and refinements of this result have been found by V. A. Zorich, R. Näkki, P. Caraman and by others. The reviewer extended the above result to the case of proper quasiregular mappings in 1976. In the paper under review the above result is proved for homeomorphisms which need not be quasiconformal but which have the property that their coefficient of quasiconformality in \(B^ n(r)\) grows with a certain rate, like O(log(1/(1-r))). The result is closely connected with recent work of \textit{V. A. Zorich} (to appear in Dokl. Akad. Nauk SSSR) ''Asymptotic of the coefficient of quasiconformality and the boundary behavior of automorphisms of a disc''.