A PROBLEM IN EXTREMAL QUASI-CONFORMAL EXTENSIONS

A constant K-0((m))(h) is introduced for every quasisymmetric mapping h of the unit circle and every integer m greater than or equal to 4 wh ich contains the constant K-0(h) (indicated by the change in module of the quadrilaterals with vertices on the circle) as a special case. A necessary and sufficient condition is established for K-0((m))(h) = K- 1(h). It is shown that there are infinitely many quasisymmetric mappin gs of the unit circle having the property that K-0((m))(h) < K-1(h), w here K-1(h) is the maximal dilatation of h.