Proper Holomorphic Mappings And Boundary Behaviour Of Loewner Chain

Proper mapping is an important object in several complex variables. The proper mapping of one complex variable is mainly considered in the Riemann sur-face. However, in recent years, many mathematicians pay special attention to the intersection of the proper mapping, fractal geometry and boundary property, which is the one of the hotspots of domestic and foreign scholars, and they have gained some achievements. The Loewner differential equation is an important element of univalent functions and it is one of the most useful tools in solving the extremal problem of univalent functions. In1999, Oded Schramm think a family the random conformal map with a parameter can be obtained by solving a differential equation describing the Brownian motion. In fact, the differential equation is obtained by taking the driving function for a standard Brownian motion in the Loewner differ-ential equation (i.e. λ(t)=(?)Bt). People call the stochastic version of differ-ential equation Stochastic Loewner Evolution, denoted (SLE). It is a new research direction and it has great significance in mathematics and statistical physics. This direction, which has the tool of one complex variable and the basis of the Loewner differential equation, is a frontier of scientific research. It connects with modern probability theory, physics, conformal field theory and penetrats each other. Pro-fessor G. F Lawler, at Cornell University is one of the most active people in this area and in2005, he published the book which first introduced the theory.(Confor-mally Invariant Processes In The Plane (Mathematical Surveys And Monographs, vol.114)). Two mathematicians have got the Fields Medal for outstanding work in this area in2006and2010.In this paper,we first introduce some knowledge and results of the proper mapping and the Loewner chain,and we give our main results. The center of this paper is section2and section3. In section2, we study the topological degree of bounded analytic functions and give the expression of a proper mapping the upper half-plane to itself. In section3,we consider the boundary properties of Loewner chain in the following aspects:(1)we prove Lt is increasing;(2)we estimate the length of Lt and obtain the relations of driving functions, upper half-plane capacity and logarithm capacity;(3)we discuss the derivative of Loewner chain gt(z) on the boundary, and estimates for the scope of this derivative and its growth speed;(4)we study the properties of the Loewner chain gt(z) for some special driving functions.