In this paper,we introduce the subfamilies Hm(RIV(n))of holomorphic mappings defined on the Lie ball RIV(n)which reduce to the family of holomorphic mappings and the family of locally biholomorphic mappings when m=1 and m→+∞,respectively.Various distortion theorems for holomophic mappings Hm(RIV(n))are established.The distortion theorems coincide with Liu and Minda’s as the special case of the unit disk.When m=1 and m→+∞,the distortion theorems reduce to the results obtained by Gong for RIV(n),respectively.Moreover,our method is different.As an application,the bounds for Bloch constants of Hm(RIV(n))are given.
WANG JianFei1,LIU TaiShun2 &XU HuiMing1 1College of Mathematics,Physics and Information Engineering,Zhejiang Normal University,Jinhua 321004, China 2D epartment of Mathematics,Huzhou Teachers College,Huzhou 313000,China
In this paper, a class of biholomorphic mappings named quasi-convex mapping of order a in the unit ball of a complex Banach space is introduced. When the Banach space is confined to Cn, we obtain the relation between this class of mappings and the convex mappings. Furthermore, the growth and covering theorems of this class of mappings are given on the unit ball of a complex Banach space X. Finally, we get the second order terms coefficient estimations of the homogeneous expansion of quasi-convex mapping of order a defined on the polydisc in Cn and on the unit ball in a complex Banach space, respectively.
LIU Taishun & XU Qinghua Department of Mathematics, Huzhou Teachers College, Huzhou, 313000, China
In this article, a normalized biholomorphic mapping f defined on bounded starlike circular domain in Cn is considered, where z = 0 is a zero of order k ＋ 1 of f（z） - z. The sharp growth, covering theorems for almost starlike mappings of order α and starlike mappings of order α are established. Meanwhile, the construction of the above mappings on bounded starlike circular domain in Cn is also discussed, it provides the extremal mappings for the growth, covering theorems of the above mappings.
In this paper,we give a definition of Bloch mappings defined in the unit polydisk D^n, which generalizes the concept of Bloch functions defined in the unit disk D.It is known that Bloch theorem fails unless we have some restrictive assumption on holomorphic mappings in several complex variables.We shall establish the corresponding distortion theorems for subfamiliesβ(K)andβ_(loc)(K) of Bloch mappings defined in the polydisk D^n,which extend the distortion theorems of Liu and Minda to higher dimensions.As an application,we obtain lower and upper bounds of Bloch constants for various subfamilies of Bloeh mappings defined in D^n.In particular,our results reduce to the classical results of Ahlfors and Landau when n=1.
WANG JianFei~(1+) LIU TaiShun~2 1 College of Mathematics and Physics
In this paper, the authors establish distortion theorems for various subfamilies Hk（B） of holomorphic mappings defined in the unit ball in C^n with critical points, where k is any positive integer. In particular, the distortion theorem for locally biholomorphic mappings is obtained when k tends to -∞. These distortion theorems give lower bounds on [det f′（z）[ and Re det f′（z）. As an application of these distortion theorems, the authors give lower and upper bounds of Bloch constants for the subfamilies βk（M） of holomorphic mappings. Moreover, these distortion theorems are sharp. When B is the unit disk in C, these theorems reduce to the results of Liu and Minda. A new distortion result of Re det f′（z） for locally biholomorphic mappings is also obtained.
