Univalence Of Quotient Of Analytic Functions

Let A denote the family of all analytic functions f in the unit disk D with the normalization f(0)= f'(0)-1.In this paper,we firstly consider the univalent radius of F(z)in the unit disk D defined by F(z)=z2/(1-?z)f(z)(|?|<1).Furthermore,a new subclasses H?m(?)of m-fold symmetric bi-univalent ?m is defined.By using a different appoach from the predecessors,the upper bounds of |am+1| and |a2m+1| of the function f in H?m(?)are obtained,and the results are more accurate.This paper has three main components:In the first part,the research background and preliminary knowledge are intro-duced.In the second part,the function F(z)is defined as follows:where f belongs to some subclasses of A or S.We mainly discussed the radius of univalence of F(z)when f(z)belongs to s?u?p(1/2)?g,or f(z)is of other function form.The results we obtained generalized some results of other scholars.In the third part,a new subclass H?m(?)of the class of m-fold symmetric bi-univalent ?m is introduced.The estimate for the initial coefficients of the function f in H?m(?)is studied.The results are more accurate than that of others.And also,the results generalize the conclusion of Zhigang Peng.