The non-explosion of quasi-conformal mapping

R.M.Protor defines the hyperbolic metric under K-quasi-conformal mapping.The set with finite hyperbolic area is said to be k-explodable and the quasi-conformal mapping is explodable if there exists a quasi-conformal mapping and its hyperbolic area is infinite,Based on the [1] study on the radial mapping defined in the unit disk,and the estimation of the hyperbolic area distortion,we study the more general function class and find the condition which makes them nonexplodable.Then the quasi-conformal harmonic mapping in the unit disk is studied and its non-explosion has been proved.