On weakly non-decreasable quasiconformal mappings

The notion of non-decreasable dilatation for quasiconformal mappings, which was introduced by Edgar Reich, plays an important role in the theory of extremal quasiconformal mappings. It is an interesting open problem so far whether an extremal quasiconformal mapping with non-decreasable dilatation exists in every Teichmüller equivalence class. In this paper, we have partially solved this problem. It is proved that for every Teichmüller equivalence class [ f ] , there exists an extremal quasiconformal mapping g in [ f ] with weakly non-decreasable dilatation.