Differential Subordination Relations And Extremal Value Properties Of Some Classes Of Analytic Functions

The content of this paper is composed of five parts.In the first part of this paper, we briefly introduce the background of the problem-researching and their significance in theory and practice. We also introduce some unsolved problems. In addition, we briefly introduce the researches of this paper.In the second part of this paper, by using subordination relation and inequality, we introduce two sbuclasses H(λ,a,b) and f(β,A,b) of univa-lent functions with negative coefficients. We obtain their coefficient estimates, distortion properties, covering theorems and extreme points.In the third part of this paper, we introduce a new generalized class N(λ,α,β,σ) of non-Bazilevic functions. We discuss the Fekete-Szego inequality of N(λ,α,β,σ). The sharp results are obtained, it generalizes the releated results of some authors.In the fourth part of this paper, we introduce another new generalized class N(λ,α,β,σ) of non-Bazilevic functions. The subordination relations, inclusion relations, distortion theorems and inequality properties are discussed by using differential subordination method.In the fifth part of this paper, the α+iμ type A-Bazilevic functions B_n(λ,α,μ,A,B,g(z)) is introduced. The subordination relations and inequality properties of the subclass B_n{λ,α,μ,A,B,z) are discussed by using differential subordination method. The results obtained generalize and improve the related results of some authors, and we obtain some other new results too.