The Estimate Of Coefficient And The Distortion Theorem Of Roper-Suffridge Extension Operator

The estimate of the coe?cient and the distortion theorem of biholomorphic mappings playan important role in function theory of several complex variables; in this thesis,we study a specialclass of biholomorphic starlike mappings F(z) which keep the property of starlikeness in Bn.From the starlikeness function, we discuss the estimation of homogeneous expansion and thedistortion theorem of the operator F(z) during the two dimensions. The whole thesis containsthree chapters.In the first chapter, we introduce brie?y the background of the development of the geometricfunction theory in several complex variables, some notations, basic concepts, definitions and themain results of the thesis.In the second chapter, we prove the estimate of the coe?cient of the operator F(z) whichwe use the method which comes from the one complex variable and obtain some new results.In the third chapter, we prove the distortion of the operator F(z) during the two dimensions,we also use the method which comes from the one complex variable. Then we prove that theresult can come back to the situation of the one dimension.The main results of this thesis are based on the known results, but extend and improvethem. So we have a deep realization about the estimation of the coe?cient and the distortiontheorem of the starlike mapping F(z).