多复变数一类α次星形映射齐次展开式各项的精细估计  

The Refined Estimates of All Homogeneous Expansions for a Subclass of Starlike Mappings of Order α in Several Complex Variables

作　　者：刘小松 刘太顺[2] Xiao Song LIU;Tai Shun LIU(School of Mathematics and Computation Science,Lingnan Normal University,Zhanjiang 524048,P.R.Chin;Department of Mathematics,Huzhou Teachers College,Huzhou 313000,P.R.China)

机构地区：[1]岭南师范学院数学与统计学院,湛江524048 [2]湖州师范学院数学系,湖州313000

出　　处：《数学学报（中文版）》2018年第6期1029-1036,共8页Acta Mathematica Sinica：Chinese Series

基　　金：国家自然科学基金资助项目(11471111);广东省自然科学基金资助项目(2014A030307016)

摘　　要：本文首先给出复Banach空间单位球上一类α次星形映射齐次展开式各项的精细估计,特别当这些映射又是k折对称映射时,估计还是精确的.其次建立C^n中单位多圆柱上上述推广映射齐次展开式各项的精细估计,同样当这些映射又是k折对称映射时,估计仍是精确的.由此证明了多复变数中关于α次星形映射的弱Bieberbach猜想,且所得到的估计都能回到单复变数的情形.We first obtain the refined estimates of all homogeneous expansions for a subclass of starlike mappings of order α on the unit ball in complex Banach spaces, especially the estimates are all sharp if these mappings are k-fold symmetric mappings. We next establish the refined estimates of all homogeneous expansions for the above generalized mappings on the unit polydisk in Cn. In particular, the estimates are also all sharp if these mappings are k-fold symmetric mappings. It is shown that we have proved a weak version of the Bieberbach conjecture for starlike mappings of order α in the case of several complex variables, and the derived results reduce to the classical results in the case of one complex variable.

关 键 词：齐次展开式 k+1阶零点 k折对称映射 α次星形映射 Bieberbach猜想 

分 类 号：O174.56[理学—数学]