A counterexample theorem in quasiconformal mapping theory

<正> Given a quasisymmetric homeomorphism h of the unit circle onto itself, denote by Kh* , Hh and Kh the extremal maximal dilatation, boundary dilatation and maximal dilatation of h, respectively. It is proved that there exists a family of quasisymmetric homeomoiphisms h such that Kh < Hh = Kh* . This gives a negative answer to a problem asked independently by Wu and Yang. Furthermore, some related topics are also discussed.