An Investigation Into The Properties Of Some Subclasses Of The Univalent Functions

This thesis is concerned systematically with some interesting properties of some subclasses ofunivalent functions. This paper is composed of fve chapters.In chapter one, we briefy introduce the developmental background of univalent function, andsome defnitions, notations in this thesis.In chapter two, we defne a widely representative function class Bh,pΣ(λ), and obtain coefcientestimates on the second and three items of Taylor expansion. In corollarys, we unify and improveall related research about the subclasses of holomorphic and bi-univalent functions.In chapter three, on the basis of previous works, we defne three subclasses of meromorphicand bi-univalent functions Σ (λ, α), ΣB (β, α) andΣ (α, β), and obtain some interesting results.In corollarys, we generalize some related works.In chapter four, we defne an interesting subclass Sλ(A, B; C, D) of close-to-convex function,and give several inclusion relationship, and found sharp coefcient inequalities, sharp growth andcovering theorems, we also show that the class is closed under convolution. The results presentedin this chapter unify many previous related results about many important subclasses of univalentfunctions.In chapter fve, we defne three subclasses of analytic and univalent functions by makinguse of the principle of diferential subordination and the Dziok-Srivastava convolution operator,and obtain many interesting properties, such as their inclusion relationships, closed classes underconvolution and so on. The results presented in this chapter provide generalizations of severalrecent works.The main result of this thesis is the breadth and the deep into further study under someknown result, we systematically improve and generalize some former researches. Though work ofthis paper,we have further knowledge for some properties of some important subclasses of univalentfunction.