On Slice Regular Functions And Holomorphic Self-mappings Of Strongly Pseudoconvex Domains

This PhD dissertation is mainly devoted to the study of quaternionic and octonionic slice regular functions,and to that of geometric properties at the regular boundary fixed points of holomorphic self-mappings of bounded strongly pseudoconvex domains in C~n.It consists of four chapters,and its main content is as follows:The first chapter is meant as an introduction to the historical background and re-search status of slice regular functions,as well as the main results and research methods of this dissertation.The second chapter deals with geometric properties of quaternionic slice regular functions.First of all,we prove a new convex combination identity for slice regular functions that preserve one slice.Using this identity as a main tool,the sharp growth and distortion theorems together with Koebe one-quarter theorem are established for slice regular extension to the quaternionic unit ball of univalent functions on the unit disc of the complex plane.It turns out that this new convex combination identity is a very important tool and plays a key role in the study of slice regular functions.Secondly,we study in depth by using the Schwarz-Pick lemma the boundary behavior of slice regular self-mappings of the unit ball and of the right half-space in the quaternionic space.In particular,we obtain the precise asymptotic behavior at infinity of slice regular self-mappings of the quaternionic right half-space,and in turn a Burns-Krantz rigidity theorem.Moreover,we find accidentally that the boundary Schwarz lemma proved by Gentili and Vlacci in 2008 is generally incorrect.Finally,we adopt a completely new approach to prove a correct version of boundary Schwarz lemma,and improve considerably a well-known Osserman type estimate.The main aim of the third chapter is the extensive study of octonionic slice reg-ular functions,mainly focusing on analytic and geometric properties.First,we prove a new splitting lemma by means of the well-known Cayley-Dickson process,and then use it to define regular product,regular conjugate and symmetrization for slice regular functions.Our definition can connect effectively octonionic slice regular functions to holomorphic functions and mappings of one complex variable.Secondly,by combin-ing the method introduced in the proof of boundary Schwarz lemma on the quaternionic unit ball and the classical interior Schwarz lermma in several complex variables as well as some new techniques,we prove a general boundary Schwarz lemma for slice regular functions mapping a symmetric slice domain with C2 boundary to a convex domain with C1 boundary.As applications,we obtain two Landau-Toeplitz type theorems for slice regular functions on the unit ball with respect to regular diameter and slice diameter respectively,together with an interesting Cauchy type estimate.Finally,we use a new tool to show that octonionic slice regular functions satisfy a certain openness property and a special case of the minimal principle.In the fourth chapter(the last chapter),we prove a boundary Schwarz lemma for holomorphic self-mappings of strongly pseudoconvex domains in C~n,which general-izes the corresponding result on the unit ball B~n(?)C~n that is previously proved by Liu,Wang and Tang.This result was also independently proved by Liu and Tang.