复Banach空间中一类螺形映射子族的性质

Properties of a subclass of spirallike mappings in complex Banach space

首先定义了一个新的螺形映射子族S_(B)[β,A,B],其中β∈(−π/2,π/2),−1≤B<A≤1.然后利用Loewner理论给出了映射族S_(B)[β,A,B]在复Banach空间单位球上的增长定理和沿某单位方向的偏差定理.最后给出了欧氏空间单位球B^(n)上正规化双全纯映射族成为S_(B)[β,A,B]的充分条件.特别地,作为主要结果的应用,当β,A,B取某些特殊值时,可以很容易地得到一些熟知的结果.

First,a new subclass of spirallike mappings S_(B)[β,A,B]is defined,whereβ∈(−π/2,π/2),−1≤B<A≤1.Then the growth theorem and distortion theorem along a unit direction for S_(B)[β,A,B]were obtained using the Loewner theory on the unit ball in complex Banach space.Finally,some sufficient conditions for normalized biholomorohic mappings to be S_(B)[β,A,B]were also characterized on the unit ball B^(n)in Euclidean space.In particular,as an application of the main results,whenβ,A and B takes some special values we can obtain some well-known results.