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

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.