Distortion Theorems Of Bloch Mappings In Several Complex Variables And Bohr's Theorem On Classical Domains

This thesis is divided into two parts. The first part, we study distortion theorems of subclasses of Bloch mappings defined in the unit polydisk, the unit ball and classical domains systematically. By making use of distortion theorems, we can obtain the estimates of Bloch constants in several complex variables. The rest of the thesis we give a new form of Bohr theorem by a new idea, and establish a new multidimensional analogue of Bohr theorem on the classical domains by using a new method. In particular, when classical domains reduce to the unit disc in C, our Bohr theorem reduces to that of Bohr. The thesis consists of five chapters.In the first chapter, we introduce some definitions, notations, and the main results of this thesis briefly.In Chapter 2, We obtain distortion theorems for various subfamilies of Bloch mappings defined in the R_I(m,n). As an application of these distortion theorems, we give estimates of Bloch constants for these subfamilies of holomorphic mappings. These results enable us to contain some known results and also lead to new results. In particular, we extend the work of Bloch constant, which established by C. H. FitzGerald, S. Gong, D. Minda, X. Liu and others.In Chapter 3, we obtain a version of subordination lemma for hyperbolic disk relative to hyperbolic geometry on the unit disk. This subordination lemma yields distortion theorems for Bloch mappings on the unit ball and the unit polydisk. Here we establish the distortion theorem from a unified perspective and generalize some known results. This distortion theorem enables us to obtain a lower bound estimate for the radius of the largest univalent ball in the image. We extend the results of M. Bonk, D. Mind and H. Yanagihara to several complex variables.In chapter 4, we give a new generalization of Bonk distortion theorem to the unit polydisk D~n. By making use of this distortion theorem, we establish the estimates of Bloch constants of some subfamilies of Bloch mappings defined in D~n. In a similar method, we obtain the lower estimate of the Bloch constant of locally biholomorphic mappings defined in D~n. When n=1, our result reduces to that of Ahlfors.In the last chapter, we give a new generalization of Bohr's theorem in higher dimen- sions. We establish the Bohr theorem in the classical domains. Moreover, we obtain the constant 1/3 is also the best possible. In particular, we propose an open problem. We answer the problem in two special cases. However, the general problem has not been solved.The significance of distortion theorems of subclasses of Bloch mappings lies in extending some known results. In particular, we reveal the important relations of the subclasses of Bloch mappings and unify all the related results, which make the geometric function theory much richer. The significance of Bohr's theorem in higher dimensions establishes the classical Bohr theorem from one variable case to holomorphic mappings between classical domains. The result is very interesting and beautiful. In particular our result coincides with Bohr's theorem in one complex variable.