Cluster sets of pseudo-analytic functions

1. Let w=T(z) be a quasiconformal mapping, with bounded dilatation, of |z| <1 onto |w|<1. Then, w=T(z) can be extended to a topological mapping of the closed disc |z|•...1 onto |w|•...1 (cf. Ahlf ors [2], Mori [1]). It had been an open question whether the extended T is necessarily absolutely continuous on |z|=1. This question has been settled by Beurling-Ahlfors [1]: The answer is in the negative. This is a striking result, from the view-point of the theory of cluster sets. It is well-known that a large part of this theory can be discussed system atically by applying the theory of functions of class (U) in Seidel's sense [1] (see, for example, Tsuji [2]). However, by the result of Beurling-Ahlfors, it is easily seen that we cannot extend the theory of functions of class (U) to the case of