Bohr's type inequalities are studied in this paper: if f is a holomorphic mapping from the unit ball B^n to B^n, f(0)=p, then we have sum from k=0 to∞|Dφ_P(P)[D^kf(0)(z^k)]|／k!||Dφ_P(P)||＜1 for|z|＜max{1／2+|P|,(1-|p|)／2^(1／2)andφ_P∈Aut(B^n) such thatφ_(p)=0. As corollaries of the above estimate, we obtain some sharp Bohr's type modulus inequalities. In particular, when n=1 and |P|→1, then our theorem reduces to a classical result of Bohr.
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
Let φ be a holomorphic self-map of Bn and ψ ∈ H（Hn）. A composition type operator is defined by Tψ,φ（f） = ψf o φ for f ∈ H（Bn）, which is a generalization of the multiplication operator and the composition operator. In this article, the necessary and sufficient conditions are given for the composition type operator Tψ,φ to be bounded or compact from Hardy space HP（Bn） to μ-Bloch space Bμ（Bn）. The conditions are some supremums concerned with ψ,φ, their derivatives and Bergman metric of Bn. At the same time, two corollaries are obtained.
This note induces some generalized Roper-Suffridge extension operators such that they are used to construct some almost starlike mappings of order α and starlike mappings of order a on different domains.
On bounded symmetric domain Ω of C^n, we investigate the properties of functions in weighted Bergman spaces A^P（Ω,dvs） for 0 〈 p ≤ ＋∞ and -1 〈 s 〈 4-∞. Based on the estimate of Bergman kernel, we obtain some characterizations of functions in A^P（Ω, dvs） in terms of a class of linear operators D^αB. Making use of these characterizations, we extend A^P（Ω,dvs） to the weighted Bergman spaces Aα^p,B（Ω,dvs） in a very natural way for 1 〈 p 〈 4-∞ and any real number s, that is, -∞ 〈 s 〈 ＋∞. This unified treatment covers some classical Bergman spaces, Besov spaces and Bloch spaces. Meanwhile, the boundedness of Bergman projection operators on Aα^P,β（Ω, dvs） and the dual of Aα^P,B（Ω, dvs） are given.
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.
