<正> §1.若 k 次对称函数(?)在单位圆(?)中正则单叶,此种函数之全体组成函数族 S_k.若 k 次对称函数(?)在区域1<|ζ|<∞中正则单叶,则此种函数之全体组成函数族Σ_k.关于 S_2 中函数之系数之模数,作者曾有估计,至于S_3中函数之系数之模数,陈建功教授在1933年证明(?)是有界的,且不大于 e~3.1934年列文亦
Abstract
Let the k-symmetric funtions(?)be regular andschlicht in the unit circle |z|<1,and the k-symmetric functions F_k(?)(?)be regular and schlicht in the region 1<|ξ|<∞.In this note,we obtain the following theorems:(?)In particular if a_3~((2))=0,then(?)In particular if a_4~((3))=0,then(?)(?)which is equivalent to(?)(?) (?)(?)whereξ_1,ξ-2 are any two points in the region 1<|ξ|<∞,and w_1,w_2,……,w_(2k)are the roots of x~(2k)-1=0.The theorems 1,3 and 2,improve Takahashi's results~[4](?)andLevin's results~[3] |α_n~((2))|<3.39 respectively.The theorem 4 is an extension of Basilewitsch's~[8] inequality(?)