Some Problems Of The Successive Coefficients Of Univalent Functions

Content: Suppose the growth of is a difficult and interesting problem (Goluzinproblem). Many authors have studied it, but it yet hasn't been completely solved. In this thesis, firstly, we investigate Goluzin problem of the successive coefficientsof univalent functions with maximal growth on k directions. Let k=2 , , we obtain and the exponent can not be improved. Letk, the similar result is also obtained.Secondly, we study Goluzin problem of circularly symmetric functions. Let f is a circularly symmetric function, , with constructing a positive realfunction and using integral method, the difference of the modular of successive coefficients of k symmetric functions of circularly symmetric functions is also obtained, which is a sharp estimate. In addition, the integral representation of circularly symmetric functions is also obtained.Finally, we indicate similarity between circularly symmetric functions and starlike functions, putting forward two problems to further study.