On The Distortiod And Explodable Properties For Quasiconformal Mappings

In this paper we mainly study the following three problems:(1)In part II, we study some boundary properties for domains under quasiconformal mappings, and give a necessary and sufficient condition to judge whether a set is explodable or not. One hyperbolic area distortion theorem of radial quasiconformal mapping is proved, it improves the results obtained by Porter and Resdis recently.(2)In part III, we consider a comparative problem of Reich Extension method and Douady-Earle Extension method. It is proved that under the same boundary value q5 on 3A, ifK is sufficiently small, the estimate of the maximal dilatation of Reich Extension is still smaller than the one of Douady-Earle Extension.(3)In part IV, we explore some relative problems on domains with Poincar?metric. Similar to Chuaqui and Osgood work, on a family of quasidisks, we obtain a rather sharp estimate for the upper bound of the logarithmic gradient of Poincar?metric. And we also obtain a sharper estimate for the logarithmic gradient of Poincar?metric on an arbitrary hyperbolic domain.