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中文摘要：
首先建立了Cn中单位多圆柱上一类近于凸映照子族精确的偏差定理，同时在复Banach空间单位球上也建立了该类映照精确的偏差定理的下界估计．其次在复Banach空间单位球上建立了准星形映照精确的偏差定理．所得结果将单复变中近于凸函数和星形函数的偏差定理推广至高维情形，并且对龚升提出的一个公开问题给出肯定的回答．
英文摘要：
In this paper, firstly, the sharp distortion theorem for a subclass of close-to- convex mappings defined in the unit polydisk of Cn is established. Furthermore, the sharp lower bound estimate of the distortion theorem for a subclass of close-to-convex mappings in the unit ball of complex Banach spaces is also established. Secondly, the sharp distortion theorem for quasi-starlike mappings in the unit ball of complex Banach spaces is given. The authors extend the distortion theorem for close-to-convex functions and starlike functions in the theory of one complex variable to higher dimensions, and give an affirmative answer to an open problem concerning distortion theorem for quasi-starlike mappings proposed by Gong Sheng.
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