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.     

DISTORTION THEOREMS FOR BLOCH MAPPINGS ON THE UNIT POLYDISC Dn

In this article,we establish distortion theorems for some various subfamilies of Bloch mappings defined in the unit polydisc D n with critical points,which extend the results of Liu and Minda to higher dimensions.We obtain lower bounds of | det(f′(z))| and det(f′(z)) for Bloch mapping f.As an application,some lower and upper bounds of Bloch constants for the subfamilies of holomorphic mappings are given.     

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.     

Distortion theorem for Bloch mappings on the unit ball <Emphasis Type="Italic">ℬ</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

In this paper, we obtain a version of subordination lemma for hyperbolic disk relative to hyperbolic geometry on the unit disk D. This subordination lemma yields the distortion theorem for Bloch mappings f ∈ H（B^n） satisfying ｜｜f｜｜0 = 1 and det f＇（0） = α ∈ （0, 1], where｜｜f｜｜0 = sup{（1 - ｜z｜^2 ）n＋1/2n det（f＇（z））[1/n ： z ∈ B^n}. Here we establish the distortion theorem from a unified perspective and generalize some known results. This distortion theorem enables us to obtain a lower bound for the radius of the largest univalent ball in the image of f centered at f（0）. When a = 1, the lower bound reduces to that of Bloch constant found by Liu. When n = 1, our distortion theorem coincides with that of Bonk, Minda and Yanagihara.     

P-Bloch函数的偏差定理和单叶半径

In this paper, we establish distortion theorems for both normalized p-Bloch functions with branch points and normalized locally univalent p-Bloch functions defined on the unit disk, respectively. These distortion theorems give lower bounds on |f′(z)| and ■f′(z). As applications of these distortion theorems, the lower bounds of the radius of the largest schlicht disk on these Bloch functions are given, respectively. Notice that when p = 1, our results reduce to that of Liu and Minda.     

Sharp Distortion Theorems for a Subclass of Biholomorphic Mappings Which Have a Parametric Representation in Several Complex Variables

In this paper, the sharp distortion theorems of the Fr′echet-derivative type for a subclass of biholomorphic mappings which have a parametric representation on the unit ball of complex Banach spaces are established, and the corresponding results of the above generalized mappings on the unit polydisk in C~n are also given. Meanwhile, the sharp distortion theorems of the Jacobi determinant type for a subclass of biholomorphic mappings which have a parametric representation on the unit ball with an arbitrary norm in C~n are obtained, and the corresponding results of the above generalized mappings on the unit polydisk in C~n are got as well. Thus, some known results in prior literatures are generalized.     

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.     

Sharp distortion theorems for a subclass of close-to-convex mappings

We introduce the class of strongly close-to-convex mappings of order c~ in the unit ball of a complex Banach space, and then, we give the sharp distortion theorems for this class of mappings in the unit ball of a complex Hilbert space X or the unit polydisc in Cn. As an application, a sharp growth theorem for strongly close-to-convex mappings of order α is obtained.     

SHARP DISTORTION THEOREMS FOR A CLASS OF BIHOLOMORPHIC MAPPINGS IN SEVERAL COMPLEX VARIABLES

In this paper,we first establish the sharp growth theorem and the distortion theorem of the Frechet derivative for biholomorphic mappings defined on the unit ball of complex Banach spaces and the unit polydisk in Cⁿ with some restricted conditions.We next give the distortion theorem of the Jacobi determinant for biholomorphic mappings defined on the unit ball of Cⁿ with an arbitrary norm and the unit polydisk in Cⁿ under certain restricted assumptions.Finally we ...     

ON THE SHARP GROWTH, COVERING THEOREMS FOR NORMALIZED BIHOLOMORPHIC MAPPINGS IN C^n

In this article.a normalized biholomorphic mapping f defined on bounded starlike circular domain in C″is considered,where z=0 is a zero of order k 1 of f(z)(?) The sharp growth.covering theorems for almost starlike mappings of order a and starlile mappings of order wave established.Meanwhile,the construction of the above mappings on bounded starlike circular domain in C~n is also discussed,it provides the extremal mappings for the growth,covering theorems of the above mappings.     

DISTORTION THEOREMS FOR SUBCLASSES OF STARLIKE MAPPINGS ALONG A UNIT DIRECTION IN C^n

In this paper, we obtain a distortion theorem of Jacobian matrix for biholomorphic subclasses of starlike mappings along a unit direction on the unit polydisc. These results extend the classical distortion theorem of starlike mappings to higher dimensions. We then give an upper bound estimate for biholomorphic subclasses of starlike mappings along a unit direction in a complex Banach space.     

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.     

A refinement about estimation of expansion coefficients for normalized biholomorphic mappings

A refining estimation of homogeneous expansion for / is discussed, where f belongs to a subclass of all normalized biholomorphic mappings defined on the unit polydisk in Cn or the unit ball in complex Banach spaces, and x = 0 is a zero of order k 1 of f(x)-x. Moreover, an estimation of homogeneous expansion for subordinate mappings defined on the unit ball in complex Banach spaces is also given.     

Bloch Constant on α-Bloch Mappings of the Unit Ball

This note is denoted to establishing sharp distortion theorems for subclasses of α-Bloch mappings defined in the unit ball of Cn with critical points.Furthermore,the estimates of Bloch constant with respect to these subclasses are given.     

Sharp Growth Theorems and Coefficient Bounds for Starlike Mappings in Several Complex Variables

Let B be the unit ball in a complex Banach space. Let S^＊k＋1（B） be the family of normalized starlike mappings f on B such that z = 0 is a zero of order k ＋ 1 of f（z） - z. The authors obtain sharp growth and covering theorems, as well as sharp coefficient bounds for various subsets of S^＊k＋1（B）.     

Sharp distortion theorems for some subclasses of star like mappings on Bnp in Cn

We mainly establish the distortion theorems of Jacobi determinant for three subclasses of starlike mappings on Bpn,where Bp={z=(z1,……,zn)T∈Cn:∑nl=1|zl|p<1}p>1.In particular,the above distortion theorems are sharp if Bpn is the unit polydisk in Cn.Our results reduce to the corresponding classical results in one dimension of complex function theory.     

Properties of New Holomorphic Mappings with Respect to Conic Domains

This paper is mainly about holomorphic mappings associated with conic regions which are closely connected with k-ST(α). We introduce new subclasses of starlike(spirallike) functions,namely, S_c~p(k, α)(S_c~p(k, α, β)), and discuss their coefficient estimates and the Fekete–Szeg?–Goluzin's problem. Then we generalize S_c~p(k, α, β) on the unit ball B~n in C~n, that is, k-conic spirallike mappings of type β and order α. We obtain the growth, covering and distortion theorems of the generalized mappings. Besides that, we construct k-conic spirallike mappings of type β and order α on B~n through S_c(k, α, β) by the generalized Roper-Suffridge extension operators.     

New Subclasses of Biholomorphic Mappings and the Modified Roper-Suffridge Operator

The authors propose a new approach to construct subclasses of biholomorphic mappings with special geometric properties in several complex variables. The RoperSuffridge operator on the unit ball B~n in C~n is modified. By the analytical characteristics and the growth theorems of subclasses of spirallike mappings, it is proved that the modified Roper-Suffridge operator [Φ_(G,γ)(f)](z) preserves the properties of S_Ω~*(A, B), as well as strong and almost spirallikeness of type β and order α on B~n. Thus, the mappings in S_Ω~*(A, B), as well as strong and almost spirallike mappings, can be constructed through the corresponding functions in one complex variable. The conclusions follow some special cases and contain the elementary results.     

单位球上α-Bloch映照的Bloch常数（英文）

This note is denoted to establishing sharp distortion theorems for subclasses of α-Bloch mappings defined in the unit ball of C~n with critical points. Furthermore, the estimates of Bloch constant with respect to these subclasses are given.     

Coefficient Inequalities for p-Valent Functions

In the present paper, the authors introduce a new subclass of p-valent analytic functions with complex order defined on the open unit disk U={z:z∈C and |z|1} and obtain coefficient inequalities for the functions in these class. Application of these results for the functions defined by the convolution are also obtained.