On Coefficient Estimation Problem Of Several Analytic Function Classes

Univalent analytic function theory is an important branch of analytic function theory.In univalent function theory,a typical problem is to estimate the coefficient functional of the function.The coefficient functional of univalent function can also be extended to some analytic functions.In this article,we mainly study the coefficient estimation and related problems of the analyticalα order prestarlike function and a certain univalent starlike function S(α,β).There are three chapters in this article,and the details is as follows:In the first chapter,we introduce the research background of univalent analytic function theory and coefficient functionals.Then we introduce the related concepts and symbols of univalent functions and some subclasses.Finally we introduce the main research content of this article.In the second chapter,We first established the coefficient relationship between the class of α order pretarlike function Ra and the class of function Qwhose real part is greater than1/2.Then by using the coefficient relationship,we estimate the sharp upper bound of coefficient for the α order pretarlike function.Finally we estimates the upper bound of second order Hankel determinants H2(2)and H2(3),Fekete-Szeg(?) functional,Zalcman functional and third order Hermitian Toeplitz determinant for the α order pretarlike function.Parts of our results are sharp.In the third chapter,On the basis of the definition of the class of starlike function S(α,β)given by Kuroki and Owa,first of all,we give the integral representation for the class of starlike function.Next,by using convolution,we derive a necessary and sufficient condition for functions to be in the class S(α,β).Then we derive a sufficient condition for the coefficients when the function belongs to class S(α,β).Finally we estimate the upper bound of second order Hankel determinant H2(2)and Fekete-Szeg(?) functional for the class S(α,β).The upper bound of Fekete-Szeg(?) functional is sharp.