Coefficient Estimates Of Some Analytic Function Classes And Some Applications Of Differential Subordination

The content of this paper will be mainly divided into five parts.In the part of Introduction in the first chapter, we briefly introduced the developmental course of some important problems and researches in the theory of Univalent Function, we also introduced developmental status of recent researches and some unsolved problems. In addition, we briefly introduced researches and innovations of this paper.In the second part of this paper, we introduced a new class of analytic functions F(α,β). As a generalized class of starlike functions, we studied the Fekete-Szego problem of it and obtained the accurate Fekete-Szego inequalities of F(α,β), we also got the extreme functions of it.The third part of this paper deepened and generalized the second part's results. In the third part, we studied the Fekete-Szego problem of a new class of analytic functions F_λ~*(α,β) defined by Ruscheweyh Derivative D~λ f{z) and we obtained the accurate estimation of supremum and the expressions of its extreme functions.In fourth part of this paper, a subclass A(k,p,λ,α) of p-valent function class A(k,p) is introduced, sufficient conditions for p-valent starlike functions subclass of order β and p-valent convex functions subclass of order β in class A(k,p) were obtained, and we also got the coefficient estimates of A(k,p, λ,α). In addition, we got some properties of Hadamard product of the subclass H(k,p,λ,α) of A{k,p,λ,α) by using of the Cauchy-Schwarz Inequality.In the fifth part of this paper, three new classes of analytic functions are introduced by using Hardamard product. Those three classes of analytic functions are T_g~r(p;A,B),R_g~r (p;A,B) and Q_g~r(p;A,B). By study in these classes, we respectively obtained some new differential subordination relations of them.All the Theoremes and Corollaries obtained in this paper are sharp, and they improved and generalized some former researches.