The Estimates of All Homogeneous Expansions for a Subclass of ε Quasi-convex Mappings in Several Complex Variables

The authors obtain the estimates of all homogeneous expansions for a subclass of ε quasi-convex mappings on the unit ball in complex Banach spaces. Moreover, the estimates of all homogeneous expansions for the above generalized mappings on the unit polydisk in Cnare also obtained. Especially, the above estimates are only sharp for a subclass of starlike mappings, quasi-convex mappings and quasi-convex mappings of type A. The results are the generalization of many known results.     

The representation theorem of onto isometric mappings between two unit spheres of l∞-type spaces and the application on isometric extension problem

In this paper, we first derive the representation theorem of onto isometric mappings in the unit spheres of real l∞-type spaces, then we conclude that such mappings can be extended to the whole space as (real) linear isometries.     

RESEARCH ANNOUNCEMENTS The New Necessary and Sufficient Conditions for the Uniformly Continuous and Continuous of Duality Mappings

In[1],Ma shaoqin gave the necessary and sufficient condition for the uniformlycontinuous and continuous of duality mappings.In the note,We give very simplecriteria for the uniformly continuous and continuous of duality mappings.And by[1],the necessary and sufficient conditions for the norm of a real Banach Space to beuniformly Frechet differentiable and Frechet differentiable are described.     

The isometric extension problem in the unit spheres of l~p(Г)(p >1) type spaces

In this paper, we first derive the representation theorem of onto isometric mappings in the unit spheres of lp(Г)(p > 1,p ≠ 2) type spaces, and then we conclude that such mappings can be extended to the whole space as real linear isometries by using the previous result of the author.     

On quasi-convex mappings of orderαin the unit ball of a complex Banach space Dedicated to Professor Sheng GONG on the occasion of his 75th birthday

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.     

The Classification of Proper Holomorphic Mappings Between Special Hartogs Triangles of Different Dimensions

The authors discuss the proper holomorphic mappings between special Hartogs triangles of different dimensions and obtain a corresponding classification theorem.     

The Comparison Theorem on Cartan-Hartogs Domain of the Fourth Type

As is known to all, theory of invariant metric is very important in several complex analysis. The Bergman, Caratheodory and Kobayashi metrics are important biholomorphic invariants. They play very important role in studying the boundary geometry of the domain and biholomorphic mappings extending smoothly to the boundaries of the relevant domains.     

Bloch constant of holomorphic mappings on the unit polydisk of C~n

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.     

The Study for Some Subclasses of Biholomorphic Mappings by an Unified Method

LetΩ∈Cn be a bounded starlike circular domain with 0∈Ω. In this paper, we introduce a class of holomorphic mappings Mg onΩ. Let f(z) be a normalized locally biholomorphic mapping onΩsuch that Jf-1(z)f(z)∈Mg and z=0 is the zero of order k + 1 of f(z)-z. We obtain the growth and covering theorems for f(z). Especially, as corollaries, we unify and generalize many known results. Moreover, in view of proofs of corollaries, the essential relations among the subclasses of starlike mappings are shown.     

Uniqueness Theorem of Meromorphic Mappings with Moving Targets

Some uniqueness theorems of meromorphic mappings with moving targets are given under the inclusion relations between the zeros sets of meromorphic mappings.     

Bloch Constant of Holomorphic Mappings on the Unit Ball of C~n

In this paper,the authors establish distortion theorems for various subfamilies H_k(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 subfamiliesβ_k(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 biholomorphie mappings is also obtained.     

The Roper-Suffridge extension operator and classes of biholomorphic mappings Dedicated to Professor Sheng GONG on the occasion of his 75th birthday

In this paper we give a survey about the Roper-Suffridge extension operator and the developments in the theory of univalent mappings in several variables to which it has led. We begin with the basic geometric properties (most of which now have a number of different proofs) and discuss relations with the theory of Loewner chains and generalizations and modifications of the operator, some of which are very recent.     

Fixed Points of Nonexpansive Type Mappings in Topological Spaces

Recently. Penot[1] and Kirk[2]obtained the abstract versions of Kirk's fixed point theorem of [3] for nonexpansive mappings, respectively. Gille spie and Williams [4, 5] replaced the reflexive and normal structure used in Kirk [3] and Kannan [6], by uniformly normal structure to obtain the fixed point theorems for nonexpansive and Kannan mappings. Kaukich [7] also extended the result of [3] to the generalized nonexpansive mappings.     

The Order of Hypersubstitutions of Type （2, 2）

Hypersubstitutions are mappings which map operation symbols to terms of the corresponding arities. They were introduced as a way of making precise the concept of a hyperidentity and generalizations to M-hyperidentities. A variety in which every identity is satisfied as a hyperidentity is called solid. If every identity is an M-hyperidentity for a subset M of the set of all hypersubstitutions, the variety is called M-solid. There is a Galois connection between monoids of hypersubstitutions and sublattices of the lattice of all varieties of algebras of a given type. Therefore, it is interesting and useful to know how semigroup or monoid properties of monoids of hypersubstitutions transfer under this Galois connection to properties of the corresponding lattices of M-solid varieties. In this paper, we study the order of each hypersubstitution of type （2, 2）, i.e., the order of the cyclic subsemigroup generated by that hypersubstitution of the monoid of all hypersubstitutions of type （2, 2）. The main result is that the order is 1, 2, 3, 4 or infinite.     

Bloch Constant of Holomorphic Mappings on the Unit Ball of ℂ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>*

In this paper, the authors establish distortion theorems for various subfamilies Hk(B) of holomorphic mappings defined in the unit ball in Cn with critical points, where k is any positive integer. In particular, the distortion theorem for locally biholomorphic mappings is obtained when k tends to oo. These distortion theorems give lower bound son det f(z) and Redet 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.     

Ball-covering property of Banach spaces that is not preserved under linear isomorphisms

By a ball-covering B of a Banach space X, we mean that it is a collection of open balls off the origin whose union contains the sphere of the unit ball of X. The space X is said to have a ball-covering property, if it admits a ball-covering consisting of countably many balls. This paper, by constructing the equivalent norms on l~∞, shows that ball-covering property is not invariant under isomorphic mappings, though it is preserved under such mappings if X is a Gateaux differentiability space; presents that this property of X is not heritable by its closed subspaces; and the property is also not preserved under quotient mappings.     

SHARP ESTIMATES OF ALL HOMOGENEOUS EXPANSIONS FOR A SUBCLASS OF QUASI-CONVEX MAPPINGS OF TYPE B AND ORDER α IN SEVERAL COMPLEX VARIABLES

In this article,first,the sharp estimates of all homogeneous expansions for a subclass of quasi-convex mappings of type B and order α on the unit ball in complex Banach spaces are given.Second,the sharp estimates of all homogeneous expansions for the above generalized mappings on the unit polydisk in C~n are also established.In particular,the sharp estimates of all homogeneous expansions for a subclass of quasi-convex mappings(include quasi-convex mappings of type A and quasi-convex mappings of type B) in several complex variables are get accordingly.Our results state that a weak version of the Bieberbach conjecture for quasi-convex mappings of type B and order α in several complex variables is proved,and the derived conclusions are the generalization of the classical results in one complex variable.     

Approximation of Nearest Common Fixed Point of Nonexpansive Mappings in Hilbert Spaces

The purpose of this paper is to study the convergence problem of the iteration scheme xn＋l = λn＋1y ＋ （1 - λn＋1）Tn＋1xn for a family of infinitely many nonexpansive mappings T1, T2,... in a Hilbert space. It is proved that under suitable conditions this iteration scheme converges strongly to the nearest common fixed point of this family of nonexpansive mappings. The results presented in this paper extend and improve some recent results.     

Points of Coincidence and Common Fixed Points for Ⅱ-Expansive Mappings on Complex Valued Metric Spaces

We use the two mappings satisfying Ⅱ-expansive conditions on complex valued metric spaces to construct the convergent sequences and prove that the unique limit of the sequences is the point of coincidence or common fixed point of the two mappings.Also,we discuss the uniqueness of points of coincidence or common fixed points and give the existence theorems of unique fixed points.The obtained results generalize and improve the corresponding conclusions in references.     

Invariant Curves of Reversible Mappings with Small Twist

In this paper,we are concerned with the existence of invariant curves of reversible mappings.A variant of the classical small twist theorem is given.