Abstract: To study the relationship between the maximal dilatation of quasi-symmetric self-homoemorphism of the unit circle and the maximal dilatation of its extremal extension, it was proved in [1] that if the maximal dilatation K_q(h) of a qusi-symmetric self-homoemorphism h cannot be arrive at some quadrilateral with the unit disk △ as its domain and vertices on the unit circle $\pd$, then K_q(h)≤ H(h), where H(h) is the boundary dilatation of h. The main resluts in [1] quite depend on this result. But its proof there is very complicated. In this paper, the authors give another elementary and simple proof. Furthermore, they use this result to study the problem when the maximal dilatation of a qusi-symmetric self-homoemorphism of the unit circle could be arrive at some quadrilateral.

Key words: Quasiconformal mapping, Quasi-symmetric
self-homoemorphism, Teichmüller space