Coefficient Estimates And Convolution Properties Of Some Analytic Function Classes

The problems of coefficient estimates and extreme values of univalent functions are highly emphasized all the time by mathematicians from all over the world. This paper studies these problems on some subclasses of univalent functions and gets some meaningful results. The results presented here would provide extensions of those given in earlier works, which then deeply consummates the study of these problems.This paper is divided into five parts:The first chapter is the exordium which gives a simple introduction of the phylogeny of univalent functions and some classes and their notations that will appear and be used in the present paper. In the second chapter, a subclass G (λ,α)of analytic functions with negative coefficients is introduced. The necessary conditions, coefficient estimates and distortion theorems for functions belonging to this class are provided. The inclusion relationship and the convexity for this class are also discussed. Two extended subclasses CΩ(λ,α) and C ?Ω(λ,α) of convex functions, which are defined by Ruscheweyh derivative are discussed in the third chapter. It studies coefficient estimates, distortion theorems and expressions for functions belonging to the two classes. At the same time, it gives the conclusion relationship of them with respect to real numberλ. The forth chapter studies two classes M (λ,α) and N (λ,α) which are generalized from M (α) and N (α) introduced by S.Owa and J.Nishivaki. The coefficient estimates, results of convolution and conclusion relationships are provided in this chapter. The sixth chapter introduces the notions of functions starlike and convex with respect to k-symmetric points of complex order. Two subclasses Ss( k )(b ) and C s( k)(b )are discussed. The coefficient inequalities, integral representations and convolution conditions for these classes are provided.