The Distortion Theorems For The Subfamilies Of Holomorphic Functions

The function theory of a complex variable is an important branch of the analysis theory, in which the distortion estimates for the subfamilies of holomorphic functions are very interesting research subjects. Many mathematicians have studied it and obtained many valuable research results.The present thesis deals with the distortion estimates for some classes of the subfam-ilies of holomorphic functions, such as p-Bloch functions and subfamilies of holomorphic functions S(A, B) and so on. We obtain the distortion theorems for these functions, and by using distortion theorems, we can get the estimates of schlicht radius for these functions, respectively, namely the estimates of radius of the largest schlicht disk on the Riemann surface. These results are all meaningful. The structure of this thesis is listed as follows.In the first chapter, we introduce the background of the development of function theory of a complex variable, some definitions and notations and the main results of the thesis.In the second chapter, we study the distortion estimates for the subfamilies of p-Bloch functions and establish distortion theorems for βp(n) and β∞p by using Liu and Minda lemma and Julia’s lemma of half plane. As applications of these distortion theorems, we obtain their estimates of schlicht radius, respectively.In the third chapter, we study holomorphic functions So and establish growth the-orem, distortion theorem and the estimates of high-order derivative for the functions by using the expression theorem of positive real part functions. In addition, we extend the functions to holomorphic functions S(A.B). and we can get growth theorem.The significance of this thesis lies in generalizing many known results. In particular, we give the important relations of the subclasses of holomorphic functions and unify some known results, which make the geometric function theory of a complex variable much richer.