Estimate For The Initial Coefficients Of Bi-univalent Functions

In this paper,we firstly investigate the coefficients of bi-univalent functions.We obtain the estimation of |a2| and |a3|,where|a2| and |a3| are the coefficients of the bi-univalent function f which belong to two subclasses.Furthermore,we study the radius of starlikeness of the quotient of two univalent functions which belong to s*(?)and s*(?)respectively.This paper consists of four parts:The first part of this paper is the introduction of the background of investigation and pre-knowledge.We review the history of estimation of the coefficients of univalent functions and bi-uivalent functions.We introduce the basic knowledge,terminology and symbols involved in this paper.In the second part,by using the method different from other authors,we investi-gate the bounds of the initial coefficients of the bi-uivalent functions belonging to two subclasses and obtain more accurate estimate.In the third part of this paper,we investigate the radius of starlikeness of the quo-tient of two univalent functions.Finally,we summarize this paper and make a prospect for the future work.