The Successive Coefficients Of Some Univalent Functions

Content: In this thesis, we investigate the Goluzin problem on the successive coefficients of univalent functions, respectively solving the problem of Goluzin on two subcate-gories of univalent functions and discussing Fekele - Szego problem of a new subcategory of univalent functions.The Goluzin problem is that the successive coefficients of univalent functions restrict each other,which is die peculiar property, saying specifically, we let , and try to compute regularity of growth of Fekete - Szego proved the following result in 1933. Let the function f(z) S be given by , and this equality is held for every u. Later, it is called the Fekele - Szeg6 problem that we estimate the maximum value of .This thesis consists of four chapters. In chapter1 ,we introduce the related background and preparatory. In chapler2.a new subcalegory S* of univalent functions is established . We prove that the calegory of starlike functions are a subcategory of S* . Also the difference of the modular of successive coefficients of fk (z), with f( z) S , is estimated. In chap-ter3, we establish a new subcategroy S ( a) of univalenl functions and its exlensive category 5 (a , B). The difference of die modular of successive coefficients of S (a)and S (a,B), widi f( z} S, is eslimaled. In chapler4, a new class Sc (B) of univalent functions is introduced. We discuss die Fekele - Szegti inquality for Sc(B) and die sharp result is obtained.