A counterexample theorem in quasiconformal mapping theory

Given a quasisymmetric homeomorphism h of the unit circle onto itself, deno te by K-h* H-h and K-h the extremal maximal dilatation, boundary dilatation and maximal dilatation of h, respectively. It is proved that there exists a family of quasisymmetric homeomorphisms h such that K-h < H-h = K-h*. Thi s gives a negative answer to a problem asked independently by Wu and Yang. Furthermore, some related topics are also discussed.