In this article, first, a sufficient condition for a starlike mapping of order a f（x） defined on the unit ball in a complex Banach space is given. Second, the sharp estimate of the third homogeneous expansion for f ...In this article, first, a sufficient condition for a starlike mapping of order a f（x） defined on the unit ball in a complex Banach space is given. Second, the sharp estimate of the third homogeneous expansion for f is established as well, where f（z） = （f1（z）, f2（z）,..., fn（z））＇ is a starlike mapping of order a or a normalized biholomorphic starlike mapping defined on the unit polydisk in Cn, and D2fk（0）（z2） /2i= zk（∑l=1^b akzzl）, k = 2t l=1 k = 1, 2,..., n. Our result states that the Bieberbaeh conjecture in several complex variables （the case of the third homogeneous expansion for starlike mappings of order α and biholomorphic starlike mappings） is partly proved.展开更多
In this article, the generalized Roper-Suffridge extension operator in Banach spaces for locally biholomorphic mappings is introduced. It is proved that this operator preserves the starlikeness on some domains in Bana...In this article, the generalized Roper-Suffridge extension operator in Banach spaces for locally biholomorphic mappings is introduced. It is proved that this operator preserves the starlikeness on some domains in Banach spaces but does not preserves convexity for some cases. Moreover, the growth theorem, covering theorem, and the radius of starlikeness are discussed. Some results of Roper and Suffridge, Gong and Liu, Graham et al in C^n are extended to Hilbert spaces or Banach spaces.展开更多
基金Project supported by National Natural Science Foundation of China(10971063,11061015)Major Program of Zhejiang Provincial Natural Science Foundation of China(D7080080)Guangdong Natural Science Foundation(06301315)
文摘In this article, first, a sufficient condition for a starlike mapping of order a f（x） defined on the unit ball in a complex Banach space is given. Second, the sharp estimate of the third homogeneous expansion for f is established as well, where f（z） = （f1（z）, f2（z）,..., fn（z））＇ is a starlike mapping of order a or a normalized biholomorphic starlike mapping defined on the unit polydisk in Cn, and D2fk（0）（z2） /2i= zk（∑l=1^b akzzl）, k = 2t l=1 k = 1, 2,..., n. Our result states that the Bieberbaeh conjecture in several complex variables （the case of the third homogeneous expansion for starlike mappings of order α and biholomorphic starlike mappings） is partly proved.
基金This research is partly supported by the National Natural Science Foundation of China (10471048) the Doctoral Foundation of the Education Committee of China(20050574002)+1 种基金 the Natural Science Foundation of Fujian Province, China (Z0511013)the Education Commission Foundation of Fujian Province, China (JB04038)
文摘In this article, the generalized Roper-Suffridge extension operator in Banach spaces for locally biholomorphic mappings is introduced. It is proved that this operator preserves the starlikeness on some domains in Banach spaces but does not preserves convexity for some cases. Moreover, the growth theorem, covering theorem, and the radius of starlikeness are discussed. Some results of Roper and Suffridge, Gong and Liu, Graham et al in C^n are extended to Hilbert spaces or Banach spaces.
