Distortion theorems for Bloch mappings on the unit polydisc D

In this article, we establish distortion theorems for some various subfamilies of Bloch mappings defined in the unit polydisc Dⁿ 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<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

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 subfamilies β(K) and β _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 Bloch mappings defined in D ⁿ . In particular, our results reduce to the classical results of Ahlfors and Landau when n = 1. This work was supported by the National Natural Science Foundation of China (Grant No. 10571164) and Specialized Research Fund for the Doctoral Program of Higher Education (SRFDP) (Grant No. 20050358052)     

Dispersion for Schrödinger Equation with Periodic Potential in 1D

We extend a result on dispersion for solutions of the linear Schrödinger equation, proved by Firsova for operators with finitely many energy bands only, to the case of smooth potentials in 1D with infinitely many bands. The proof consists in an application of the method of stationary phase. Estimates for the phases, essentially the band functions, follow from work by Korotyaev. Most of the paper is devoted to bounds for the Bloch functions. For these bounds we need a detailed analysis of the quasimomentum function and the uniformization of the inverse of the quasimomentum function.     

Julia''''s lemma and Bloch constants

The classical Julia's lemma is improved, branch points of Bloch functions are investigated, and new lower bounds of Bloch constants for functions with branch points are obtained.     

Julia’s lemma and bloch constants

The classical Julia’s lemma is improved, branch points of Bloch functions are investigated, and new lower bounds of Bloch constants for functions with branch points are 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.     

多复变量的K-拟共形映射

本文研究Cn里的单位球B到Cn里的K-拟共形全纯映射.证明的最后结果是:若f是B到Cn里的K-拟共形全纯映射,满足det(f'(0))=1,则f(B)包含一个半径至少是(CnK)1-n∫01(1+t)N-1/(1-t)2exp{-(n+1)t/1-t}dt的单叶球,其中Cn>1是只依赖于n的常数,当n→∞时,Cn→(10)~(1/2).     

Estimates on Bloch constants for planar harmonic mappings

The Bloch constants for quasiregular harmonic mappings and open planar harmonic mappings are considered. Better estimates are obtained. The results, presented in this paper, improve the one made by Chen et al. and Grigoryan. This work was supported by the Research Foundation for Doctor Programme (Grant No. 20050574002) and the National Natural Science Foundation of China (Grant No. 10471048)     

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.     

对Bloch定理中覆盖圆的估计

本文研究了Block定理覆盖圆问题.利用微分与积分估值法,对覆盖圆的半径做了更精确地估计,得到了两个相关定理,扩大了覆盖圆的范围.     

调和函数类的一些界的估计结果北大核心CSCD

研究了单位开圆盘内保向复值的调和函数类,其中函数的解析部分满足一定的从属条件.完整的给出了该函数类Bloch常数的界、系数估计、增长和偏差不等式的界.进一步,通过指定一些关键参数,证明其pre-Schwarzian导数的范数也是有界的.     

On the Bloch Constant for Κ-Quasiconformal Mappings in Several Complex Variables

We study the Bloch constant for Κ-quasiconformal holomorphic mappings of the unit ball B of C ⁿ . The final result we prove in this paper is: If f is a Κ-quasiconformal holomorphic mappig of B into C ⁿ such that det(f′(0)) = 1, then f(B) contains a schlicht ball of radius at least where C _n > 1 is a constant depending on n only, and as n→∞. Received June 24, 1998, Accepted January 14, 1999     

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.     

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.     

第一类典型域上的Bloch常数

引入第一类典型域R_I(m,n)上的全纯映照子族H_k(R_I(m,n)),当k→+∞时,该映照族就是R_I(m,n)上的局部双全纯映照族.建立了H_k(R_I(m,n))上的Bonk偏差定理.当k=1和k→+∞时,其结果分别都回到了FitzGerad和龚升关于典型域R_I(m,n)上的Bonk偏差定理.当m=n=1时,其结果又回到了Liu和Minda在单位圆盘上的偏差定理.应用偏差定理,给出了映照族H_k(R_I(m,n))上的Bloch常数估计,其结果补全了从k=1和k→+∞之间的R_I(m,n)上Bloch常数估计的所有结果,而且把单位球上的Bloch常数估计推广到R_I(m,n)上.     

单位圆盘到自身的调和映照的傅立叶系数估计

利用函数论的一般方法给出单位圆盘到自身的调和映照的傅立叶系数的界限,改进并推广了P.Duren和王晓英的相关结果.     

Bounds for the fundamental solutions of elliptic and parabolic equations

It is shown that any elloptic or parabolic operator in nondivergence form with measurable coefficients has a global fundamental solution verifying certain pointwise bounds.     

拟共形映射和HAUSDORFF维数

讨论了标准Cantor集 F 的Hausdorff维数dim_H(F)在拟共形映射 f 下的变化, 给出了dim_H(f(F))的准确上下界.     

Bloch constant and Landau's theorem for planar p-harmonic mappings

In this paper, our main aim is to introduce the concept of planar p-harmonic mappings and investigate the properties of these mappings. First, we discuss the p-harmonic Bloch mappings. Two estimates on the Bloch constant are obtained, which are generalizations of the main results in Colonna (1989) [9]. As a consequence of these investigations, we establish a Bloch and Landau's theorem for p-harmonic mappings.     

Bloch space in the unit ball of

In this paper we obtain higher-dimensional versions of the Holland-Walsh characterization of the Bloch space and the Stroethoff characterization of the little Bloch space.