近于凸映照子族全部项齐次展开式的精确估计

The sharp estimates for each item in the homogeneous polynomial expansions of a subclass of close-to-convex mappings

本文建立了Cn中单位多圆柱上近于凸映照子族和一类近于准凸映照全部项齐次展开式的精确估计.与此同时,作为推论给出了Cn中单位多圆柱上近于凸映照子族和一类近于准凸映照精确的增长定理和精确的偏差定理上界估计.所得主要结论表明Cn中单位多圆柱上关于近于凸映照子族和一类近于准凸映照的Bieberbach猜想成立,而且与单复变数的经典结论相一致.

In this paper,the sharp estimates for each item in the homogeneous polynomial expansions of a subclass of close-to-convex mappings and a class of close-to-quasi-convex mappings in the unit polydisc of C^n are established.Meanwhile,as corollaries,the sharp growth theorem and the sharp upper bound for the distortion theorem of a subclass of close-to-convex mappings and a class of close-to-quasi-convex mappings in the unit polydisc of C^n are given as well.Our main result shows that the Bieberbach conjecture holds for a subclass of close-to-convex mappings in the unit polydisc of C^n.It reduces to the corresponding classical result in the theory of one complex variable.