The Properties Of Bounded Holomorphic Functions And Holomorphic Mappings

This thesis is concerned with the boundary Schwarz lemma on fixed point for bounded holomorphic functions and the Schwarz-Pick for the modulus of holomor-phic mappings.this paper is composed with three chapters.In chapter one,we briefly introduce the developmental background of holomor-phic function,and some definitions,notations in this thesis.In chapter two,we set Hk(D,D)be the set of all holomorphic mappings f from D into D such that z/O is a zero of order k of f(z)—f(0).In this chapter,we extend the boundary Schwarz lemma to HK(D,D).As a corollary,we improved the conclusions in Osserman[18]and Frolova[1].In chapter three,We offered an alternative and elementary proof of the result in Xu and Liu[21],our method is quite different from that which Xu and Liu have used in[21].And,we extend this conclusion to a general family of holomorphic mappings Hk(D,Bn).The work of this paper is extend the general bounded holomorphic functions and holomorphic mappings properties to Hk(D,D)and Hk(D,Bn),which is repre-sentative family of holomorphic mappings.And further enrich the contents of the schwarz lemma.