ALMOST QUASICONFORMAL MAPPINGS WITH GIVEN BOUNDARY VALUES AND A COMPLEX DILATATION BOUND

In the extremal problems of quasiconformal mappings with given boundary valuesand a complex dilatation bound which are discussed by Reich,the extremal mapping isrequired to have no conformal point set of positive measure on the defining set T of thecomplex dilatation bound b(w).Under the additional assumptions that T\T has measurezero and b(w)is continuous a.e.Chen Jixiu proved that the extremal mapping may berelaxed to have a conformal positive measure set and a finite number of singularity pointson T.In this paper,the author proves that when the additional assumptions are given up,the same relaxations still hold and the extremal mapping is also allowed to have a countablenumber of singularity points on T.