Classes Of Starlike Mappings And Bloch Type Space In Several Complex Variables

In the thesis,we mainly give some properties for kinds of subclasses of biholomor-phic starlike mappings in several complex variables,and the operator theory between Bloch type space and weighted spaces was studied also.Focusing on these issues,we will divide this thesis into five chapters.In chapter 1,the background,some definitions and the main results of this thesis were introduced.In chapter 2,we consider a subclass of normalized biholomorphic mappings defined on the unit ball(or on the unit polydisc)in C~n,denoted by (?)(or (?)).Firstly,we obtain the growth theorem for (?).Secondly,we apply the growth theorem and a new type of the boundary Schwarz lemma to establish the distortion theorems of the Fréchet-derivative type and the Jacobi-determinant type for this sub-class.Furthermore,we obtain the two kinds of distortion theorems for subclasses of (?).In fact,if a mapping F∈(?),here,we allow the corresponding com-ponents in F to have different dimensions.Due to our works,the distortion theorems with g-starlike mapping(resp.starlike mapping)are partly established also.In chapter 3,on the basis of the previous studies in chapter 2,we get the second coefficient bound for a modified Carathéodory mapping,which is defined in the unit ball in C~2.Furthermore,we give an example of bounded support points for compact subclasses of g-parametric representation mappings.We also proved that a modified Roper-Suffridge extension operator can be embedded in more general g-Loewner chains.At last,we study the Kikuchi and Pell type results for the compact set of mappings which have g-parametric representation associated with a modified Roper-Suffridge ex-tension operator,which extend some earlier related results.In chapter 4,we unify the definitions for kinds of subclasses of starlike mappings on the bounded starlike circular domain?or polydisk D~nor the unit ball B_Xin a complex Banach space X.Moreover,we give an unified answer of Fekete-Szeg?problem for the kinds of subclasses of starlike mappings defined on?(or D~nor B_X)when the parameters are real.Furthermore,the same problems are also considered when the parameters are complex.Compare with some recent works,the critical processes of proofs are different:our arguments in this chapter are heavily based on the subordination techniques.In chapter 5,all the weighted composition operator,integral-type operator and weighted space are extended to the unit ball B_Xin a complex Banach space X.We successfully construct two test functions,which play a key role in the proof of our main results.Furthermore,we give the relations between the Bloch-type spaces and the little Bloch-type spaces,and the boundedness and compactness of the weighted composition operator from Bloch space(resp.little Bloch space)to the weighted space(resp.little weighted space)were studied.In fact,we also consider the boundedness and compact-ness of integral-type operator between the different Bloch-type spaces.Our main results extend the corresponding works on B~nto the case of B_Xof an infinite dimensional com-plex Banach space.This assumption that a set is relatively compact in B_Xis needed when consider the compactness of the operators.However,it is automatically satisfied when dim(X)<+∞.