【作者】 宁菊红；

【导师】 叶中秋；

【作者基本信息】 江西师范大学 ， 应用数学， 2003， 硕士

【摘要】 若f（z）∈S，（f（z）/z）^λ=1+sum from n=1 to ∞（c_nzⁿ），λ∈(0，1]，相邻系数‖C_n∣-∣C_n-1‖的增长是一个难而有趣的问题（Goluzin问题）。许多作者对此进行了研究，但该问题至今没有完全解决。 论文首先研究了具有k个增长方向单叶函数的Goluzin问题。当k=2，λ≥1/2时，得到，‖C_n∣-∣C_n-1‖≤A_λn^λ-1+ε，并且指出指数λ-1是不可改进的。当k=m（m≥3），1＞λ＞1/2时，也得到了类似的结果。 论文接着研究了圆对称函数的Goluzin问题。当f为园对称函数，λ=1/k（k=2，3，…）时，通过构造一个正实部函数，利用积分方法，得到了k次圆对称函数相邻系数模之差的精确估计。另外，还得到了圆对称函数的积分表示。 在论文的最后，指出圆对称函数与星形函数具有某些相似处，提出了两个值得进一步探讨的问题。更多还原

【Abstract】 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.更多还原