A study on the proper holomorphic mappings and univalent holomorphic mappings on the unit ball

1.给出了具有参数表示的单叶全纯映象的增长定理和系数估计，并给出了大量的例子。我们得到了C^N中单位球B^n上某些全纯映象的星形半径、凸性、参数表示和Bloch半径的某些结果。设B是C^n中关于C^n上任意范数的单位球，给出了使{e^&lt；-t&gt；f(z，t)}，t&gt；0是B上的正规族的Loewner链f(z，t)是k重对称的充要条件。我们得到了映射是凸从属链的一些充要条件，并给出了C^N.5中欧氏单位球上的凸从属链的各种例子。给出了复Banach空间开单位球上一类有理映射的星形准则。当这些映射定义在Hilbert空间中时，我们还给出了它们是凸的一个充分条件。

1. We give a growth theorem and coefficient estimates for univalent holomorphic mappings which have parametric representation and we also give many examples.2. We obtain certain results related to radius of starlikeness, convexity, parametric representation and Bloch radius for some classes of holomorphic mappings on the unit ball B^n in C^n.3. Let B be the unit ball in C^n with respect to an arbitrary norm on C^n. We give a necessary and sufficient condition that a Loewner chain f(z, t), such that {e^<-t>f(z, t)},t>0 is a normal family on B, is k-fold symmetrical.4. We obtain certain necessary and sufficient conditions for a mapping to be a convex subordination chain and we give various examples of convex subordination chains on the Euclidean unit ball in C^n.5. We give starlike criteria for a class of rational mappings on the open unit ball of a complex Banach space. We also give a sufficient condition for these mappings to be convex when they are defined in Hilbert spaces.