On The Estimate Of Boundary Distortion For Quasiconformal Mappings And Teh Comparison Of Their Dilatations

In this paper we study the estimate of boundary distortion for quasiconforrnal mappings and the comparison of their dilatations. At first, we study some distortion theories for quasihomographies of the unit circle, and give a better estimate for p when quasihomographies take as p -quasisymmetric functions. Our results improve the corresponding results obtained byZajac recently. Secondly, for the quasisymmetric function of the form , wherea has a period a> 0, we derive the estimates of belonging to F1 (M,1). As an application we consider the distortion estimate problem for theboundary of quasiconformal mapping with fixed point zero and a pair of symmetric points on the boundary. Finally, we consider the problem for the comparison of Reich Extension and Douady-Earle Extension .As far as the maximal dilatation is concerned, we proved that even for sinai] K Reich Extension is better than Douady-Earle Extension.