Abstract: The Beurling-Ahlfors' extension is studied under relatively general conditions and its dilatation fonction is estimated. Particularly, the classic Deurling -Ahlfors' theorem can be obtained under the M-condition. 

Key words: quasiconfornal mapping, quasiconformal honeomorphism, Beurling-Ahlfors'exten- sion, dilatation function, quasisymmetric function