Schwarz Lemma At The Boundary For Several Complex S And Its Applications

Firstly, Schwarz lemmas at the boundary on some domains in Cn are introduced. Secondly, by Schwarz lemma at the boundary of several complex variables distortion theorems at the extreme points for normalized biholomorphic starlike mappings and subclass of starlike mappings and normalized biholomorphic spirallike mappings of type β and subclass of spirallike mappings of type β on the unit ball in Cn are obtained. Finally, a subclass of normalized biholomorphic convex mappings on the unit ball in complex Banach space is discussed.In chapter1, the research background of geometric function theory of several complex variables, notations and definitions are briefly introduced, and the main results of this thesis are listed.In chapter2, Schwarz lemmas at the boundary of several complex variables on the unit ball Bn, unit polydisc Dn and a peculiar Reinhardt domain BP1,P2in Cn are studied, respectively. These results are the generalization of the Schwarz lemma at the boundary of one complex variable in the space of higher dimensions.In chapter3, the distortion theorems of determinant at the extreme points and distortion theorems of matrix on the complex tangent space at the extreme points for normalized biholomorphic starlike mappings, almost starlike mappings of order a and starlike mappings of order a on the unit ball Bn are established, respectively.In chapter4, the distortion theorems of determinant at the extreme points and distortion theorems of matrix on the complex tangent space at the extreme points for normalized biholomorphic spirallike mappings of type β, almost spirallike mappings of type β and order a and spirallike mappings of type β and order a on the unit ball Bn are obtained, respectively. Although starlike mappings is a proper subclass of spirallike mappings of type β, they have the same distortion theorems of determinant and distortion theorems of matrix on the complex tangent space at the extreme points on the unit ball in Euclid space.In chapter5, a new subclass of normalized biholomorphic convex mappings on the unit ball in complex Banach space is defined, and the growth theorem, covering theorem and distortion theorem of matrix are charactered. These results are just the well konwn properties of convex mappings in special case.The main results of this thesis are not only the extension and deepening for the theory of one complex variable in higher dimensions but also the perfection for the geometric function theory of several complex variables.