Zhi-Gang Wang, Xin-Zhong Huang, Zhi-Hong Liu, Rahim Kargar
In the present paper, we discuss several basic properties of a class of quasiconformal close-to-convex harmonic mappings with starlike analytic part, such results as coefficient inequalities, an integral representation, a growth theorem, an area theorem, and radii of close-to-convexity of partial sums of the class, are derived.
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Quasiconformal Extensions of Harmonic Univalent Mappings of the Unit Disk
本文研究单位圆盘内调和单叶映射的拟共形延拓条件，证明调和强星形映射可延拓至全平面并给出显式延拓函数，同时探讨圆盘外调和映射的可延拓性。
In this article, we establish the Bohr inequalities for the sense-preserving $K$-quasiconformal harmonic mappings defined in the unit disk $\mathbb{D}$ involving classes of Ma-Minda starlike and convex univalent functions, usually denoted by $\mathcal{S}^*(ψ)$ and $\mathcal{C}(ψ)$ respectively, and for $\log (f(z)/z)$ where $f$ belongs to the Ma-Minda classes or satisfies certain differential subordination. We also estimate Logarithmic coefficient's bounds for the functions in $\mathcal{C}(ψ)$ for the case $ψ(\mathbb{D})$ be convex.
Harmonic functions are natural generalizations of conformal mappings. In recent years, a lot of work have been done by some researchers who focus on harmonic starlike functions. In this paper, we aim to introduce two classes of harmonic univalent functions of the unit disk, called hereditarily $λ$-spirallike functions and hereditarily strongly starlike functions, which are the generalizations of $λ$-spirallike functions and strongly starlike functions, respectively. We note that a relation can be obtained between this two classes. We also investigate analytic characterization of hereditarily s…
Bohr inequality and Bohr-Rogosinski inequality for $K$-Quasiconformal harmonic mappings
本文研究单位圆盘上解析部分从属于凹单叶函数类、Ma-Minda凸与星形函数类的$K$-拟共形保向调和映射，证明了若干精确的Bohr型及Bohr-Rogosinski型不等式，推广了已有结果。
The hereditary property of convexity and starlikeness for conformal mappings does not generalize to univalent harmonic mappings. This failure leads us to the notion of fully starlike and convex mappings of order α, (0\leq α<1). A bound for the radius of fully starlikeness and fully convexity of order αis determined for certain families of univalent harmonic mappings. Convexity is not preserved under the convolution of univalent harmonic convex mappings, unlike in the analytic case. Given two univalent harmonic convex mappings f and g, the problem of finding the radius r_{0} such that f*g is a …
We investigate the univalency and the directional convexity of the convolution $ϕ\tilde{*}f=ϕ*h+\overline{ϕ*g}$ of the harmonic mapping $f=h+\bar{g}$ with a mapping $ϕ$ whose convolution with the mapping $z+\sum_{k=2}^{\infty}k^nz^k$ is starlike (and such a mapping $ϕ$ is called $n$-starlike). In addition, we investigate the directional convexity of (i) the convolution of an analytic convex mapping with the slanted half-plane mapping, and (ii) the partial sums of the convolution of a $6$-starlike mapping with the harmonic Koebe mapping and the harmonic half-plane mapping.
In this paper, we prove necessary and sufficient conditions for a sense-preserving harmonic function to be absolutely convex in the open unit disk. We also estimate the coefficient bound and obtain growth, covering and area theorems for absolutely convex harmonic mappings. A natural generalization of the classical Bernardi type operator for harmonic functions is considered and its connection between certain classes of uniformly starlike harmonic functions and uniformly convex harmonic functions is also investigated. At the end, as applications, we present a number of results connected with hyp…
Let $h$ and $g$ be two analytic functions in the unit disc $Δ$ that $g(0)=1$. Also let $β$ be a complex number with ${\rm Re}\{β\}>-1/2$. A function $f$ is said to be log--harmonic mapping if it has the following representation \begin{equation*} f(z)=z |z|^{2β} h(z)\overline{g(z)}\quad (z\in Δ). \end{equation*} A log--harmonic mapping $f$ is said to be starlike log--harmonic mapping of order $α$, where $0\leq α<1$, if \begin{equation*} {\rm Re}\left\{\frac{zf_z -\overline{z}f_{\overline{z}}}{f}\right\}>α\quad(z\in Δ). \end{equation*} In this paper, by use of the subordination principle, we stu…
In this paper, we extend Ahlfors's univalent criteria and Ahlfors's quasiconformal extension for analytic functions to harmonic mappings defined in the unit disk. Moreover, we give a general quasiconformal extension of harmonic Teichmüller mappings, whose maximal dilatation estimate is asymptotically sharp.
In 1984, Clunie and Sheil-Small proved that a sense-preserving harmonic function whose analytic part is convex, is univalent and close-to-convex. In this paper, certain cases are discussed under which the conclusion of this result can be strengthened and extended to fully starlike and fully convex harmonic mappings. In addition, we investgate the properties of functions in the class $\mathcal{M}(α)$ $(|α|\leq 1)$ consisting of harmonic functions $f=h+\overline{g}$ with $g'(z)=αzh'(z)$, $\RE (1+{zh''(z)}/{h'(z)})>-{1}/{2} $ $ \mbox{for} |z|<1 $. The coefficient estimates, growth results, area t…
The main aim of this article is to establish certain relationships between $K$-quasiconformal harmonic mappings and John disks. The results of this article are the generalizations of the corresponding results of Ch.~Pommerenke \cite{Po}.
We prove that a harmonic quasiconformal mapping defined on a finitely connected domain in the plane, all of whose boundary components are either points or quasicircles, admits a quasiconformal extension to the whole plane if its Schwarzian derivative is small. We also make the observation that a univalence criterion for harmonic mappings holds on uniform domains.
In this paper, we investigate the relationships between linear measure and harmonic mappings.
In this article, we first discuss the Lipschitz characteristic and the linear measure distortion of $K$-quasiconformal harmonic mappings. Then we give some characterizations of the radial John disks with the help of Pre-Schwarzian of harmonic mappings.
In this article, we continue our investigations of the boundary behavior of harmonic mappings. We first discuss the classical problem on the growth of radial length and obtain a sharp growth theorem of the radial length of $K$-quasiconformal harmonic mappings. Then we present an alternate characterization of radial John disks. In addition, we investigate the linear measure distortion and the Lipschitz continuity on $K$-quasiconformal harmonic mappings of the unit disk onto a radial John disk. Finally, using Pommerenke interior domains, we characterize certain differential properties of $K$-qua…
In this paper we extend Rado-Choquet-Kneser theorem for the mappings with Lipschitz boundary data and essentially positive Jacobian at the boundary, without restriction on the convexity of image domain. It is an extension of a recent result of Alessandrini and Nesi \cite{ale}. Some applications for the family of quasiconformal harmonic mappings between Jordan domains are given.
Harmonic K-quasiconformal Koebe functions: construction and application to Pavlovic's problem
本文构造了调和K-准共形柯西函数，填补了几何函数论中的基础空白，提出了统一的极值函数框架，并探讨了调和K-准共形映射的极值理论。
We investigate harmonic mappings $f=h+\bar{g}$ defined in the unit disk, where $g$ and $h$ satisfy certain prescribed conditions and the analytic part $h$ belongs to the Ma and Minda class of starlike functions. Certain sharp radius results for univalency, close-to-convexity, fully starlikeness, fully convexity, and radius of strongly starlikeness are established. We also calculate the radius of uniformly starlikeness and convexity for these functions. Several results enhance the well-known radius results.
We prove that if the Schwarzian norm of a given complex-valued locally univalent harmonic mapping $f$ in the unit disk is small enough, then $f$ is, indeed, globally univalent and can be extended to a quasiconformal mapping in the extended complex plane.
Sharp bounds for the growth and distortion of the analytic part of convex K-quasiconformal harmonic mappings
本文给出了保向凸K-拟共形调和映射解析部分增长与畸变的精确上下界。
Topological Characteristics of Harmonic Quasiconformal Unit Disk Automorphisms in the Uniform Topology
本文研究赋予一致收敛拓扑的单位圆盘调和拟共形自同胚空间 $HQ(\mathbb{D})$，探讨并揭示了其可分性、紧性、道路连通性与完备性等重要拓扑性质。
On Koebe-type functions for harmonic quasiconformal mappings
本文研究了谐波准共形映射中Koebe型函数的性质，建立了相关的一致性条件、范数、系数不等式及生长和面积定理，改进了已有结果。
Let $D$ and $Ω$ be Jordan domains with Dini's smooth boundaries and and let $f:D\mapsto Ω$ be a harmonic homeomorphism. The object of the paper is to prove the following result: If $f$ is quasiconformal, then $f$ is Lipschitz. This extends some recent results, where stronger assumptions on the boundary are imposed, and somehow is optimal, since it coincides with the best condition for Lipschitz behavior of conformal mappings in the plane and conformal parametrization of minimal surfaces.
We prove that for harmonic quasiconformal mappings $α$-Hölder continuity on the boundary implies $α$-Hölder continuity of the map itself. Our result holds for the class of uniformly perfect bounded domains, in fact we can allow that a portion of the boundary is thin in the sense of capacity. The problem for general bounded domains remains open.
The Bohr's phenomenon for certain K-quasiconformal harmonic mappings
本文研究单位圆盘中K-拟共形保向调和映射的玻尔现象，建立了改进玻尔不等式、精细玻尔不等式及玻尔-罗戈辛斯基不等式的多个精确版本。
It is known for some time that there exists an energy-minimal diffeomorphism between two doubly-connected domains $Ω$ and $D$ provided that $\mathrm{Mod}(Ω)\le \mathrm{Mod}{D}$ and that diffeomorphism is harmonic \cite{tedi}. In this note we give a short proof of the fact that for given annuli $Ω$ and $D$ satisfying the condition $\mathrm{Mod}(Ω)\le \mathrm{Mod}(D)$ there exist a $K-$quasiconformal harmonic diffeomorphism $f:Ω\onto D$, where $K=K(τ)$, $τ=\mathrm{Mod}(Ω)/ \mathrm{Mod}(D)$ and $\lim_{τ\to 1}K(τ)=1$.
Let $L_H$ denote the set of all normalized locally one-to-one and sense-preserving harmonic functions in the unit disc $Δ$. It is well-known that every complex-valued harmonic function in the unit disc $Δ$ can be uniquely represented as $f = h + \overline{g}$, where $h$ and $g$ are analytic in $Δ$. In particular the decomposition formula holds true for functions of the class $L_H$. For a fixed analytic function $h$, an interesting problem arises - to describe all functions $g$, such that $f$ belongs to $L_H$. The case when $f\in L_H$ and $h$ is the identity mapping was considered [2]. More gen…
Bohr Phenomenon for $K$-quasiconformal Harmonic Mappings Involving One Parameter
本文研究$K$-拟共形调和映射的Bohr型不等式，涉及参数或凸组合，并推广至凹单叶函数类，推广了已有结果。
If $Ω$ is a simply connected domain in $\overline{\mathbb C}$ then, according to the Ahlfors-Gehring theorem, $Ω$ is a quasidisk if and only if there exists a sufficient condition for the univalence of holomorphic functions in $Ω$ in relation to the growth of their Schwarzian derivative. We extend this theorem to harmonic mappings by proving a univalence criterion on quasidisks. We also show that the mappings satisfying this criterion admit a homeomorphic extension to $\overline{\mathbb C}$ and, under the additional assumption of quasiconformality in $Ω$, they admit a quasiconformal extension …
In this paper, we obtain coefficient criteria for a normalized harmonic function defined in the unit disk to be close-to-convex and fully starlike, respectively. Using these coefficient conditions, we present different classes of harmonic close-to-convex (resp. fully starlike) functions involving Gaussian hypergeometric functions. In addition, we present a convolution characterization for a class of univalent harmonic functions discussed recently by Mocanu, and later by Bshouty and Lyzzaik in 2010. Our approach provide examples of harmonic polynomials that are close-to-convex and starlike, res…
Hardy spaces of harmonic quasiconformal mappings and Baernstein's theorem
本文研究单位圆盘内规范化 $K$-拟共形调和映射，证明其几何子类解析与余解析部分的 Baernstein 型极值定理，并据此给出积分均值估计及属于 Hardy 空间 $h^p$ 的 $p$ 范围。
Harmonic quasiconformal mappings between $\mathscr{C}^1$ smooth Jordan domains
本文证明具有 $\mathscr{C}^1$ 边界的 Jordan 域间的调和拟共形映射对任意 $\alpha<1$ 全局 Hölder 连续但非 Lipschitz，推广了 Warschawski 的经典定理。
Bohr-type inequalities for classes of analytic maps and K-quasiconformal harmonic mappings
本文通过结合经典Bohr不等式的改进版本与面积测度方法，推广并精化了解析自映射类及K-拟共形调和映射类的Bohr型不等式，并证明结果最优。
The purpose of this paper is to explore conditions which guarantee Lipschitz-continuity of harmonic maps w.r.t. quasihyperbolic metrics. For instance, we prove that harmonic quasiconformal maps are Lipschitz w.r.t. quasihyperbolic metrics.
Area Theorems and Quasiconformal Extensions of Harmonic Mappings with a Pole
本文针对单位圆盘中具有单极点且允许对数奇异性的单叶调和映射，建立了广义面积定理，并给出了显式k-拟共形延拓的充分条件。
It is proved the following theorem, if $w$ is a quasiconformal harmonic mappings between two Riemann surfaces with smooth boundary and aproximate analytic metric, then $w$ is a quasi-isometry with respect to Euclidean metric.
Let $ \mathcal{H}(\mathbb{D}) $ be the linear space of analytic functions on the unit disk $ \mathbb{D}=\{z\in\mathbb{C}: |z|<1\} $ and let $ \mathcal{B}=\{w\in \mathcal{H}(\mathbb{D}: |w(z)|<1)\} $. The classical Bohr's inequality states that if a power series $ f(z)=\sum_{n=0}^{\infty}a_nz^n $ converges in $ \mathbb{D} $ and $ |f(z)|<1 $ for $ z\in\mathbb{D} $, then \begin{equation*} \sum_{n=0}^{\infty}|a_n|r^n\leq 1\;\;\mbox{for}\;\; r\leq \frac{1}{3} \end{equation*} and the constant $ 1/3 $ is the best possible. The constant $ 1/3 $ is known as Bohr radius. A function $ f : \mathbb{D}\righ…
Coefficients and Integral Mean Estimates for $K$-Quasiconformal Harmonic Mappings
本文证明了K-拟共形调和映射相关的系数猜想成立，确定了其所属Hardy空间与加权Bergman空间的$p$范围，改进了相关估计并讨论了奇性映射的平行结果。
The Bohr's Phenomenon for the class of K-quasiconformal harmonic mappings
本文研究K-准共形保形调和映射类中的改进玻尔不等式及其精炼版本，通过引入非负量S_ρ(h)和替代主项系数的方法，获得单位圆盘内调和映射的精确玻尔-罗戈辛斯基半径。
We derive a quasiconformal extension to 3-space of the Weierstrass-Enneper lifts of a class of harmonic mappings defined in the unit disk. The extension is based on fibrations of space by circles in domain and image that correspond to each other in a natural way. Convexity plays an essential role in the analysis. As a corollary we derive a sufficient condition for the underlying harmonic mapping to be univalent in the disk, with an explicit quasiconformal extension to the extended plane that generalizes the well known formula by Ahlfors-Weill.
Suppose that $f$ is a $K$-quasiconformal self-mapping of the unit disk $\mathbb{D}$, which satisfies the following: $(1)$ the biharmonic equation $Δ(Δf)=g$ $(g\in \mathcal{C}(\overline{\mathbb{D}}))$, (2) the boundary condition $Δf=φ$ ($φ\in\mathcal{C}(\mathbb{T})$ and $\mathbb{T}$ denotes the unit circle), and $(3)$ $f(0)=0$. The purpose of this paper is to prove that $f$ is Lipschitz continuos, and, further, it is bi-Lipschitz continuous when $\|g\|_{\infty}$ and $\|φ\|_{\infty}$ are small enough. Moreover, the estimates are asymptotically sharp as $K\to 1$, $\|g\|_{\infty}\to 0$ and $\|φ\|_…
On the coefficients estimate of K-quasiconformal harmonic mappings
本文针对Wang等人提出的K-拟共形调和映射系数猜想，证明其对星形、近凸及典型实K-拟共形调和映射成立，还建立了凸K-拟共形调和映射的精确系数估计，为该猜想提供了肯定性支撑。
The present article concerns the Bohr radius for $K$-quasiconformal sense-preserving harmonic mappings $f=h+\overline{g}$ in the unit disk $\mathbb{D}$ for which the analytic part $h$ is subordinated to some analytic function $φ$, and the purpose is to look into two cases: when $φ$ is convex, or a general univalent function in $\ID$. The results state that if $h(z) =\sum_{n=0}^{\infty}a_n z^n$ and $g(z)=\sum_{n=1}^{\infty}b_n z^n$, then $$\sum_{n=1}^{\infty}(|a_n|+|b_n|)r^n\leq \dist (φ(0),\partialφ(\ID)) ~\mbox{ for $r\leq r^*$} $$ and give estimates for the largest possible $r^*$ depending o…
We consider extensions of differential fields of mappings and obtain a lower energy bound for quasiconformal extension fields in terms of the topological degree. We also consider the related minimization problem for the $q$-harmonic energy, and show that the energy minimizers admit higher integrability.
Generalized Bohr inequalities for K-quasiconformal harmonic mappings and their applications
本文研究K-准共形调和映射在单位圆盘上的广义Bohr不等式，通过引入新的主函数序列改进了传统方法，并推导了新的不等式及Gaussian超几何函数下的Bohr定理扩展。
We consider the class univalent log-harmonic mappings on the unit disk. Firstly, we obtain necessary and sufficient conditions for a complex-valued continuous function to be starlike or convex in the unit disk. Then we present a general idea, for example, to construct log-harmonic Koebe mapping, log-harmonic right half-plane mapping and log-harmonic two-slits mapping and then we show precise ranges of these mappings. Moreover, coefficient estimates for univalent log-harmonic starlike mappings are obtained. Growth and distortion theorems for certain special subclass of log-harmonic mappings are…
The classical Bohr inequality states that if $ f $ is an analytic function with the power series representation $ f(z)=\sum_{n=0}^{\infty}a_nz^n $ in the unit disk $ \mathbb{D}:=\{z\in\mathbb{C} : |z|<1\} $ such that $ |f(z)|\leq 1 $ for all $ z\in\mathbb{D} $, then \begin{equation*} \sum_{n=0}^{\infty}|a_n|r^n\leq 1\;\; \text{for}\;\; |z|=r\leq\frac{1}{3} \end{equation*} and the constant $ 1/3 $ cannot be improved. The constant $ r_0=1/3 $ is known as Bohr radius and the inequality $ \sum_{n=0}^{\infty}|a_n|r^n\leq 1 $ is known as Bohr inequality. Let $ \mathcal{H} $ be the class of complex-v…
The theory of $q$-analogs frequently occurs in a number of areas, including the fractals and dynamical systems. The $q$-derivatives and $q$-integrals play a prominent role in the study of $q$-deformed quantum mechanical simple harmonic oscillator. In this paper, we define a symmetric $q$-derivative operator and study new family of univalent functions defined by use of that operator. We establish some new relations between functions satisfying analytic conditions related to conical sections.
Let $h$ be a quasiconformal (qc) mapping of the unit disk $\mathbb{U}$ onto a Lyapunov domain. We show that $h$ maps subdomains of Lyapunov type of $\mathbb{U}$, which touch the boundary of $\mathbb{U}$, onto domains of similar type. In particular if $h$ is a harmonic qc (hqc) mapping of $\mathbb{U}$ onto a Lyapunov domain, using it, we prove that $h$ is co-Lipschitz (co-Lip) on $\mathbb{U}$. This settles an open intriguing problem.
Affine balancing of complex dilatations: affine invariants and optimal distortion of harmonic quasiconformal mappings
本文针对调和拟共形映射，引入仿射外接半径与仿射直径作为仿射不变量，刻画最优仿射平衡下的最小畸变，并建立锐利双边界及二阶估计，统一极值畸变问题。
Holomorphic motions, Assouad dimension and quasiconformal mappings
本文研究全纯运动下集合的拟Assouad维数变化，证明其倒数在Fuhrer--Ransford--Younsi意义下下调和，由此得到拟共形失真界并改进Smirnov定理，方法无需最优Sobolev正则性。
A Lewy-type theorem for pluriharmonic mappings in $\C^2$ and bi-Lipschitz quasiconformal harmonic maps
本研究证明了$\mathbb{C}^2$中多重调和一一映射的实雅可比行列式非零的四维Lewy型定理，还给出了调和拟共形同胚为双Lipschitz映射的判定准则。
We extend the result of Lavrentiev which asserts that the harmonic measure and the arc-length measure are $A_\infty$ equivalent in a chord-arc Jordan domain. By using this result we extend the classical result of Lindelöf to the class of quasiconformal (q.c.) harmonic mappings by proving the following assertion. Assume that $f$ is a quasiconformal harmonic mapping of the unit disk $\mathbf{U}$ onto a Jordan domain. Then the function $A(z)=\arg(\partial_φ(f(z))/z)$ where $z=re^{iφ}$, is well-defined and smooth in $\mathbf{U}^*=\{z: 0<|z|<1\}$ and has a continuous extension to the boundary of th…
We extend a well-known theorem by Jones and Makarov [JM] on the singularity of boundary distortion of planar conformal mappings. We use a different technique to recover the previous result and, moreover, generalize the result for quasiconformal mappings of the unit ball $\B^n\subset \mathbb{R}^n$, $n\ge 2$. We also establish an estimate on the Hausdorff (gauge) dimension of the boundary of the image domain outside an exceptional set of given size on the sphere $\partial \B^n$. Furthermore, we show that this estimate is essentially sharp. [JM] P. W. Jones and N. Makarov: Density properties of h…
We provide two new formulas for quasiconformal extension to $\overline{\mathbb{C}}$ for harmonic mappings defined in the unit disk and having sufficiently small Schwarzian derivative. Both are generalizations of the Ahlfors-Weill extension for holomorphic functions.
In \cite{kamz} the author proved that every quasiconformal harmonic mapping between two Jordan domains with $C^{1,α}$, $0<α\le 1$, boundary is bi-Lipschitz, providing that the domain is convex. In this paper we avoid the restriction of convexity. More precisely we prove: any quasiconformal harmonic mapping between two Jordan domains $Ω_j$, $j=1,2$, with $C^{j,α}$, $j=1,2$ boundary is bi-Lipschitz.
A counterexample to the Clunie--Sheil-Small coefficient-difference conjecture via quasiconformal maps
通过构造调和K-拟共形映射，否证了Clunie--Sheil-Small系数差猜想，并揭示了其Hardy空间性质与共形情形的差异，同时给出了Beltrami系数在Sobolev空间时的精确范围。
The concept of a conformal deformation has two natural extensions: quasiconformal and harmonic mappings. Both classes do not preserve the conformal type of the domain, however they cannot change it in an arbitrary way. Doubly connected domains are where one first observes nontrivial conformal invariants. Herbert Groetzsch and Johannes C. C. Nitsche addressed this issue for quasiconformal and harmonic mappings, respectively. Combining these concepts we obtain sharp estimates for quasiconformal harmonic mappings between doubly connected domains. We then apply our results to the Cauchy problem fo…
We determine the representation theorem, distortion theorem, coefficients estimate and Bohr's radius for log-harmonic starlike mappings of order $α$, which are generalization of some earlier results. In addition, the inner mapping radius of log-harmonic mappings is also established by constructing a family of $1$-slit log-harmonic mappings. Finally, we introduce pre-Schwarzian, Schwarzian derivatives and Bloch's norm for non-vanishing log-harmonic mappings, several properties related to these are also obtained.
For a univalent smooth mapping $f$ of the unit disk $\ID$ of complex plane onto the manifold $f(\ID)$, let $d_f(z_0)$ be the radius of the largest univalent disk on the manifold $f(\ID)$ centered at $f(z_0)$ ($|z_0|<1$). The main aim of the present article is to investigate how the radius $d_h(z_0)$ varies when the analytic function $h$ is replaced by a sense-preserving harmonic function $f=h+\overline{g}$. The main result includes sharp upper and lower bounds for the quotient $d_f(z_0)/d_h(z_0)$, especially, for a family of locally univalent $Q$-quasiconformal harmonic mappings $f=h+\overline…
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