单叶调和映射的线性极值问题

利用线性极值问题有解的必要条件，研究了从单位圆盘到自身的单叶调和映射的傅立叶系数，得到其确界估计。     

Linear Connectivity,Schwarz–Pick Lemma and Univalency Criteria for Planar Harmonic Mapping

In this paper, we first establish a Schwarz–Pick lemma for higher-order derivatives of planar harmonic mappings, and apply it to obtain univalency criteria. Then we discuss distortion theorems,Lipschitz continuity and univalency of planar harmonic mappings defined in the unit disk with linearly connected images.     

平面上具有有界Fréchet导数的调和映照单叶半径的精确估计

给定单位圆盘D={z||z|1}上调和映照f(z)=h(z)+g(z),其中h(z)和g(z)为D上的解析函数,满足f(0)=0,λf(0)=1,ΛfΛ.通过引入复参数λ,|λ|=1,本文研究调和映照Fλ(z)=h(z)+λg(z)和解析函数Gλ(z)=h(z)+λg(z)的性质,得到Fλ(z)和Gλ(z)单叶半径的精确估计.作为应用,本文得到单位圆盘D上某些K-拟正则调和映照Bloch常数的更好估计,改进和推广由Chen等人所得的相应结果.     

The Distortion Theorems for Harmonic Mappings with Negative Coefficient Analytic Parts

Some sharp estimates for coefficients, distortion and the growth order are obtained for harmonic mappings $f ∈ TL^α_H$which are locally univalent harmonic mappings in the unit disk $\mathbb{D}:=\{z:|z| < 1\}$. Moreover, denoting the subclass $TS^α_H$ of the normalized univalent harmonic mappings, we also estimate the growth of $|f|,$ $f ∈ TS^α_H,$ and their covering theorems.     

The Distortion Theorems for Harmonic Mappings with Negative Coefficient Analytic Parts

Some sharp estimates for coefficients, distortion and the growth order are obtained for harmonic mappings $f \in TL^{\alpha}_H,$ which are locally univalent harmonic mappings in the unit disk $\mathbb{D}:=\{z:|z|<1\}.$ Moreover, denoting the subclass $TS^{\alpha}_H$ of the normalized univalent harmonic mappings, we also estimate the growth of $|f|,$ $f \in TS^α_H,$ and their covering theorems.     

Uniformly locally univalent harmonic mappings associated with the pre-Schwarzian norm

In this paper, we consider the class of uniformly locally univalent harmonic mappings in the unit disk and build a relationship between its pre-Schwarzian norm and uniformly hyperbolic radius. Also, we establish eight ways of characterizing uniformly locally univalent sense-preserving harmonic mappings. We also present some sharp distortions and growth estimates and investigate their connections with Hardy spaces. Finally, we study subordination principles of norm estimates.     

Harmonic mappings onto stars

A general version of the Radó-Kneser-Choquet theorem implies that a piecewise constant sense-preserving mapping of the unit circle onto the vertices of a convex polygon extends to a univalent harmonic mapping of the unit disk onto the polygonal domain. This paper discusses similarly generated harmonic mappings of the disk onto nonconvex polygonal regions in the shape of regular stars. Calculation of the Blaschke product dilatation allows a determination of the exact range of parameters that produce univalent mappings.     

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.     

Quasiconformal harmonic mappings between \fancyscriptC1,m{{\fancyscript{C}}^{1,\mu}} Euclidean surfaces

The conformal deformations are contained in two classes of mappings quasiconformal and harmonic mappings. In this paper we consider the intersection of these classes. We show that, every K quasiconformal harmonic mapping between surfaces with boundary is a Lipschitz mapping. This extends some recent results of several authors where the same problem has been considered for plane domains. As an application it is given an explicit Lipschitz constant of normalized isothermal coordinates of a disk-type minimal surface in terms of boundary curve only. It seems that this kind of estimates are new for conformal mappings of the unit disk onto a Jordan domain as well.     

Properties of Some Classes of Planar Harmonic and Planar Biharmonic Mappings

The aim of this paper is to investigate some properties of planar harmonic and biharmonic mappings. First, we use the Schwarz lemma and the improved estimates for the coefficients of planar harmonic mappings to generalize earlier results related to Landau’s constants for harmonic and biharmonic mappings. Second, we obtain a new Landau’s Theorem for a certain class of biharmonic mappings. At the end, we derive a relationship between the images of the linear connectivity of the unit disk \mathbbD{\mathbb{D}} under the planar harmonic mappings f=h+[`(g)]{f=h+\overline{g}} and under their corresponding analytic counterparts F = h − g.     

某类双调和映射的Landau型定理

令f(z)=h(z)+g(z)为开单位圆盘U内的调和映射.本文首先建立了有界调和映射的精确系数估计,其次得到了满足有界性条件|h(z)|+|g(z)|≤M的正规化调和映射的精确系数估计.作为其应用,得到了双调和映射的一些Landau型定理.这些结果改进和推广了早期作者的相关结果.     

-energy integral and harmonic mappings

In this paper, we discuss harmonic mappings on the unit disk with respect to any metric by using the -energy integral that was first introduced by Li in 1997 to treat quasiconformal harmonic mappings on the Poincaré disk, instead of the total energy integral. Some basic properties of harmonic mappings are given. Moreover, we give a new proof of the uniqueness theorem of Markovic and Mateljevic, which is more explicit and natural.

     

Differential inequalities in linear- and affine-invariant families of harmonic mappings

In linear- and affine-invariant families of harmonic mappings of the unit disk we prove some differential inequalities such as the sharp two-sided estimate of the Jacobian and an estimate of the curvature of the image of the circle.     

Hadamard Products of Convex Harmonic Mappings

Functions f in the class $ K_H $ are convex, univalent, harmonic, and sense preserving in the unit disk. Such functions can be expressed as $ f = h + \overline {g} $ where h and g are analytic functions. If $ f \in K_H $ has $ h(0) = 0, g(0) = 0, h'(0) = 1$ , and $ g'(0) = 0 $ , then $ f \in K_H^0 $ . For $ f \in K_H^0 $ and } analytic in the unit disk, an integral representation for $ f\tilde {*}\varphi = h*\varphi + \overline {g*\varphi } $ is found. With } a strip mapping, $ f\tilde {*}\varphi $ is shown to be in $ K_H^0 $ . In a 1958 paper, Pólya and Schoenberg conjectured that if f and g are conformal mappings of the unit disk onto convex domains, then the Hadamard product f 2 g of f and g has the same property. It is known that the analogue of that result for harmonic mappings is false. In this paper, some examples are given in which the property of convexity is preserved for Hadamard products of certain convex harmonic mappings. In addition, an integral formula is used to determine the geometry of the Hadamard product from the geometry of the factors. This is true in particular for the convolution of strip mappings with certain functions $ f_n \in K_H^0 $ which take the unit disk to regular n -gons.     

Uniformly locally univalent harmonic mappings

The primary aim of this paper is to characterize the uniformly locally univalent harmonic mappings in the unit disk. Then, we obtain sharp distortion, growth and covering theorems for one parameter family \({{\mathcal {B}}}_{H}(\lambda )\) of uniformly locally univalent harmonic mappings. Finally, we show that the subclass of k-quasiconformal harmonic mappings in \({{\mathcal {B}}}_{H}(\lambda )\) and the class \({{\mathcal {B}}}_{H}(\lambda )\) are contained in the Hardy space of a specific exponent depending on \(\lambda \), respectively, and we also discuss the growth of coefficients for harmonic mappings in \({{\mathcal {B}}}_{H}(\lambda )\).     

Harmonic mappings with hereditary starlikeness

We study a hereditary starlikeness property for planar harmonic mappings on a disk and on an annulus. While such a property is a common trait of conformal mappings, it may be absent in harmonic mappings. It turns out that a sufficient condition for a harmonic mapping f to possess this hereditary property is to have a harmonic argument — a striking feature of conformal mappings that does not extend to all harmonic mappings.     

On harmonic K-quasiconformal mappings associated with asymmetric vertical strips

In this paper,we discuss the sense-preserving univalent harmonic mappings from the unit disk D onto asymmetrical vertical strips Ω_α={ω:α-π/2sinαR(ω)α/2sinα},π/2≤απSuch results as analytic representation formula,coefficient estimates,distortion theorem and area theorem are derived.     

A^p（φ）上Toeplitz算子的换位子

本文使用不变加权面积平均值性质刻画了单位圆盘内的调和函数．由此我们探讨了加权Bergman空间A^p(φ)上的Toeplitz算子，给出了两个具有界调和符号的Toeplitz算子交换或本质交换的一些充要条件．     

On the conformal mappings in the unit disk fixing arbitrarily many boundary points

In this paper, we give a method of constructing conformal mappings defined in the unit disk which can fix arbitrarily many points on the unit circle.