Coefficient Estimates For Some Classes Of Bi-univalent Functions

In recent years,the problem of estimating the coefficients of various forms of bi-univalent functions has been a hot topic in the research of geometric function theory.Some scholars have defined new classes of bi-univalent functions by using various operators,and others have extended bi-univalent functions to m-fold symmetric bi-univalent functions,so as to study the coefficient estimates of the classes of bi-univalent functions.It is of great research value and theoretical significance in geometric function theory.In 2013,T.Panigrahi and G.Murugusundaramoorthy defined some classes of bi-univalent functions associated with Hohlov operator,and obtained the first three coefficient estimates of functions in the class TΣa,b;c(α,t)and class MΣa,b;c(β,t).In 2015,G.Murugusundaramoorthy and K.Vijaya further discussed the second Hankel determinant of functions in the class MΣa,b;c(β,t).In the same year,H.M.Srivastava and S.Gaboury et al.generalized bi-univalent functions to obtain some classes of m-fold symmetric bi-univalent functions,and found the first two coefficient estimates of functions in the classes.In 2019,S.Altinkaya and S.Yalcin further-discussed the second Hankel determinant of functions in the class HΣM(β)of m-fold symmetric bi-univalent functions.The purpose of this paper is to define some new classes of bi-univalent functions by using operators and some conditions,and to study the coefficient estimates of functions in the classes.This paper consists of the following three chapters.The first chapter is the introduction and preliminary knowledge,which introduces the research background and current situation of this paper.On the basis of previous research,we study new classes MΣa,b;c(λ,φ)and HΣm(τ,μ,δ,β).The second chapter,we obtain the estimates of coefficients |a2| and |a3},and discuss the upper bound of the Fekete-Szfg(?) inequality |a3-μa22| and the second Hankel determinant |a2a4-a32| for functions in the class MΣa,b;c(λ,φ)associated with Hohlov operator of bi-univalent functions by applying the Schwarz function.The third chapter,we found the estimates of coefficients |am+1| and |a2m+1|,and discuss the upper bound of the Fekete-Szeg(?) inequality |a2m1+1-μam+12| and the second Hankel determinant|am+1a3m+1-a2m+12| for functions in the class HΣm(τ,μ,δ,β)of mfold symmetric bi-univalent functions by applying the positive real part function.