Hua Deng, Saminathan Ponnusamy, Jinjing Qiao
In this paper, we present several necessary and sufficient conditions for a harmonic mapping to be normal. Also, we discuss maximum principle and five-point theorem for normal harmonic mappings. Furthermore, we investigate the convergence of sequences for sense-preserving normal harmonic mappings and show that the asymptotic values and angular limits are identical for normal harmonic mappings.
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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.
On the elliptic harmonic mappings and sense-preserving harmonic mappings
该文研究椭圆调和映射与保定向调和映射，建立多类Landau-Bloch型定理，给出系数估计与猜想，所得结果在部分情形下精确且改进了已有结论。
Results on normal harmonic and $φ$-normal harmonic mappings
本文研究平面调和映射中的正常性与$\phi$-正常性，建立Zalcman-Pang引理的调和映射版本，并引入扩展球面导数，给出$\phi$-正常的充分条件。
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…
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.
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.
On Harmonic Univalent Spirallike Mappings
本文研究了调和映射的遗传螺形性，给出了其系数条件、增长估计，并建立了与解析螺形映射的联系，提供了实例验证。
The aim of this paper is twofold. One is to introduce the class of harmonic $ν$-Bloch-type mappings as a generalization of harmonic $ν$-Bloch mappings and thereby we generalize some recent results of harmonic $1$-Bloch-type mappings investigated recently by Efraimidis et al. \cite{EGHV}. The other is to investigate some subordination principles for harmonic Bloch mappings and then establish Bohr's theorem for these mappings and in a general setting, in some cases.
In this note it is formulated the J. C. C. Nitsche type conjecture for bi-harmonic mappings. The conjecture has been motivated by the radial bi-harmonic mappings between annuli.
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 prove a Schwarz type lemma for harmonic mappings between the unit and a geodesic line in a Riemenn surface.
In this paper, we investigate the relationships between linear measure and harmonic mappings.
In this paper, we investigate the properties of locally univalent and multivalent planar harmonic mappings. First, we discuss the coefficient estimates and Landau's Theorem for some classes of locally univalent harmonic mappings, and then we study some Lipschitz-type spaces for locally univalent and multivalent harmonic mappings.
Landau type theorem for $α$-harmonic mappings
本文通过建立α-调和映射的系数估计，证明了单位圆盘上α-调和映射的Landau型定理。
In this paper, the main aim is to discuss the existence of the extreme points and support points of families of harmonic Bloch mappings and little harmonic Bloch mappings. First, in terms of the Bloch unit-valued set, we prove a necessary condition for a harmonic Bloch mapping (resp. a little harmonic Bloch mapping) to be an extreme point of the unit ball of the normalized harmonic Bloch spaces (resp. the normalized little harmonic Bloch spaces) in the unit disk $\mathbb{D}$. Then we show that a harmonic Bloch mapping $f$ is a support point of the unit ball of the normalized harmonic Bloch spa…
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 paper, we consider the convolutions of slanted half-plane mappings and strip mappings and generalize related results in general settings. We also consider a class of harmonic mappings containing slanted half-plane mappings and strip mappings and, as a consequence, we prove that the convex combination of such mappings is convex.
Certain geometric properties of planar harmonic mapping involving four parameter Wright functions
本文构造了与四参数Wright函数相关的调和映射，建立了其满足调和星形性和凸性的充分条件，并研究了调和近凸映射子类间的关系，所得结果新颖且具有意义。
In this paper, we first study the bounded mean oscillation of planar harmonic mappings, then a relationship between Lipschitz-type spaces and equivalent modulus of real harmonic mappings is established. At last, we obtain sharp estimates on Lipschitz number of planar harmonic mappings in terms of bounded mean oscillation norm, which shows that the harmonic Bloch space is isomorphic to $BMO_{2}$ as a Banach space..
Suppose $p\geq1$, $w=P[F]$ is a harmonic mapping of the unit disk $\mathbb{D}$ satisfying $F$ is absolutely continuous and $\dot{F}\in L^p(0, 2π)$, where $\dot{F}(e^{it})=\frac{\mathrm{d}}{\mathrm{d}t}F(e^{it})$. In this paper, we obtain Bergman norm estimates of the partial derivatives for $w$, i.e., $\|w_z\|_{L^p}$ and $\|\overline{w_{\bar{z}}}\|_{L^p}$, where $1\leq p<2$. Furthermore, if $w$ is a harmonic quasiregular mapping of $\mathbb{D}$, then we show that $w_z$ and $\overline{w_{\bar{z}}}$ are in the Hardy space $H^p$, where $1\leq p\leq\infty$. The corresponding Hardy norm estimates, …
In the present paper, we will study geometric properties of harmonic mappings whose analytic and co-analytic parts are (shifted) generated functions of completely monotone sequences.
We prove that the Logarithm of the Jacobian of a sense preserving harmonic mappings between surfaces is superharmonic, provided that the Gaussian curvature of the image domain is non-negative.
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 ${\mathcal B}_{H}(λ)$ of uniformly locally univalent harmonic mappings. Finally, we show that the subclass of $k$-quasiconformal harmonic mappings in ${\mathcal B}_{H}(λ)$ and the class ${\mathcal B}_{H}(λ)$ are contained in the Hardy space of a specific exponent depending on the $λ$, respectively, and we also discuss the growth of coefficients for harmonic mappings in ${\mathcal B}_{H}(λ)$.
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。
Complex-valued harmonic functions that are univalent and sense-preserving in the open unit disk are widely studied. A new methodology is employed to construct subclasses of univalent harmonic mappings from a given subfamily of univalent analytic functions. The notion of harmonic Alexander integral operator is introduced. Also, the radius of convexity for certain families of harmonic functions is determined.
In this paper we introduce a definition of the pre-Schwarzian and the Schwarzian derivatives of any locally univalent harmonic mapping $f$ in the complex plane without assuming any additional condition on the (second complex) dilatation $ω_f$ of $f$. Using the new definition for the Schwarzian derivative of harmonic mappings, we prove analogous theorems to those by Chuaqui, Duren, and Osgood. Also, we obtain a Becker-type criterion for the univalence of harmonic mappings.
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.
We introduce a class of Weierstrass-Enneper lifts of harmonic mappings which satisfy a criterion for univalence introduced by Duren, Osgood and the first author in [J. Geom. Anal. (2007)].
We prove some sharp inequalities for complex harmonic functions on the unit disk. The results extend a M. Riesz conjugate function theorem and some well-known estimates for holomorphic functions. We apply some of results to the isoperimetric inequality for harmonic mappings.
In this paper, we investigate some properties of planar harmonic mappings. First, we generalize the main results in \cite{CPW3} and \cite{HT}, and then discuss the relationship between area integral means and harmonic Hardy spaces or harmonic weighted Bergman spaces. At the end, coefficient estimates of mappings in weighted Bergman spaces are obtained.
In this paper, we first establish the Schwarz-Pick lemma of higher-order and apply it to obtain a univalency criteria for planar harmonic mappings. Then we discuss distortion theorems, Lipschitz continuity and univalency of planar harmonic mappings defined in the unit disk with linearly connected images.
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 investigate the properties of hyperbolic harmonic mappings in the unit ball $\mathbb{B}^{n}$ in $\IR^n$ $(n\geq 2)$. Firstly, we establish necessary and sufficient conditions for a hyperbolic harmonic mapping to be in the Bloch space $\mathcal{B}(\mathbb{B}^{n})$ and the generalized Bloch space $\mathcal{L}_{\infty,ω}\mathcal{B}_{α,\mathrm{a}}^{0}(\mathbb{B}^{n})$, respectively. Secondly, we discuss the relationship between the integral means of hyperbolic harmonic mappings and that of their gradients. The obtained results are the generalizations of Hardy and Littlewood's rel…
In this paper, we study non sense-preserving harmonic mappings $f=h+\overline{g}$ in $\mathbb{D}$ when its analytic part $h$ is convex and injective in $\mathbb{D}$ and obtain radius of injectivety.
We study the order of affine and linear invariant families of planar harmonic mappings in the unit disk and determine the order of the family of mappings with bounded Schwarzian norm. The result shows that finding the order of the class $\mathcal{S}_H$ of univalent harmonic mappings can be formulated as a question about Schwarzian norm and, in particular, our result shows consistency between the conjectured order of $\mathcal{S}_H$ and the Schwarzian norm of the harmonic Koebe function.
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 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.
Several properties of a class of generalized harmonic mappings
研究定义一类二阶齐次偏微分方程解的实核 α-调和映射，并分析其表示定理、Lipschitz 连续性、单叶性等性质。
In this paper, we discuss some properties on hyperbolic-harmonic mappings in the unit ball of $\mathbb{C}^{n}$. First, we investigate the relationship between the weighted Lipschitz functions and the hyperbolic-harmonic Bloch spaces. Then we establish the Schwarz-Pick type theorem for hyperbolic-harmonic mappings and apply it to prove the existence of Landau-Bloch constant for mappings in $α$-Bloch spaces.
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.
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…
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.
The main purpose of this paper is to study the concept of normal function in the context of harmonic mappings from the unit disk $\mathbb{D}$ to the complex plane. In particular, we obtain necessary conditions for that a function $f$ to be normal.
In this paper, we determine the sharp estimates for Toeplitz determinants of a subclass of close-to-convex harmonic mappings. Moreover, we obtain an improved version of Bohr's inequalities for a subclass of close-to-convex harmonic mappings, whose analytic parts are Ma-Minda convex functions.
We construct sense-preserving univalent harmonic mappings which map the unit disk onto a domain which is convex in the horizontal direction, but with varying dilatation. Also, we obtain minimal surfaces associated with such harmonic mappings. This solves also a recent problem of Dorff and Muir (Abstr. Appl. Anal. (2014)). In several of the cases, we illustrate mappings together with their minimal surfaces pictorially with the help of \texttt{Mathematica} software.
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…
Landau-type Theorem and Bloch constant for elliptic harmonic mappings
研究 $(K,K')$-椭圆调和映射，通过估计其 Taylor 系数证明 Landau 型定理，并给出该映射类的 Bloch 常数。
In this paper, we give a new generalization of the Bohr inequality in refined form both for bounded analytic functions, and for sense-preserving harmonic functions with analytic part being bounded.
In this paper, we define both the upper and lower order of a sense-preserving harmonic mapping in $\mathbb{D}$. We generalize to the harmonic case some known results about holomorphic functions with positive lower order and we show some consequences of a function having finite upper order. In addition, we improve a related result in the case when there is equality in a known distortion theorem for harmonic mappings with finite upper order. Some examples are provided to illustrate the developed theory.
We establish two-point distortion theorems for sense-preserving planar harmonic mappings $f=h+\overline{g}$ which satisfies the univalence criteria in the unit disc such that, Becker's and Nehari`s harmonic version. In addition, we find the sharp two-point distortion theorem when $h$ is a convex function, and normalized mappings such that $h(\D)$ is a $c$-linearly connected domain. To do this, we use the order of this family.
In this paper, we show that harmonic Bloch mappings are Lipschitz continuous with respect to the pseudo-hyperbolic metric. This result improves the corresponding result of Theorem 1 of [P. Ghatage, J. Yan, and D. Zheng, Composition operators with closed range on the Bloch space, Proc. Amer. Math. Soc. 129 (2000), 2039-2044]. Furthermore, we prove the similar property for harmonic quasiregular Bloch-type mappings.
In this paper we prove a Schwarz lemma for harmonic mappings between the unit balls in real Euclidean spaces. Roughly speaking, our result says that under a harmonic mapping between the unit balls in real Euclidean spaces, the image of a smaller ball centered at origin can be controlled. This extends the related result proved by Chen in complex plane.
In this paper, we study harmonic mappings by using the shear construction, introduced by Clunie and Sheil-Small in 1984. We consider two classes of conformal mappings, each of which maps the unit disk D univalently onto a domain which is convex in the horizontal direction, and shear these mappings with suitable dilatations ω. Mappings of the first class map the unit disk D onto four-slit domains and mappings of the second class take D onto regular n-gons. In addition, we discuss the minimal surfaces associated with such harmonic mappings. Furthermore, illustrations of mappings and associated m…
The paper is devoted to the study of compositions of polyharmonic mappings in simply connected domains. More precisely, we determine necessary and sufficient conditions of polyharmonic mapping $f$ such that $f\circ F$ (resp. $F\circ f$) is $l$-harmonic for any analytic function (or harmonic mapping but not analytic, or $q$-harmonic mapping but not $(q-1)$-harmonic) $F$.
A subclass of complex-valued close-to-convex harmonic functions that are univalent and sense-preserving in the open unit disc is investigated. The coefficient estimates, growth results, area theorem, boundary behavior, convolution and convex combination properties for the above family of harmonic functions are obtained.
On the pre-Schwarzian norm estimate of special close-to-convex harmonic mappings
研究单位圆盘内具有特殊解析部分的近凸调和映射类，得到了精确的预Schwarzian范数估计，并讨论了调和Bloch映射理论及解析与共解析部分的增长与偏差定理。
A Zalcman-Pang type rescaling result for $ϕ$-normal harmonic mappings and applications
本文研究单位圆盘上调和映射的正规性与$ϕ$-正规性，建立涉及扩展球面导数的充分条件，证明Zalcman-Pang型重标度引理，并导出新的$ϕ$-正规性准则及Lappan型刻画。
The first author proved that the harmonic convolution of a normalized right half-plane mapping with either another normalized right half-plane mapping or a normalized vertical strip mapping is convex in the direction of the real axis. provided that it is locally univalent. In this paper, we prove that in general the assumption of local univalency cannot be omitted. However, we are able to show that in some cases these harmonic convolutions are locally univalent. Using this we obtain interesting examples of univalent harmonic maps one of which is a map onto the plane with two parallel slits.
Landau-Bloch type theorem for elliptic and $K$-quasiregular harmonic mappings
本文改进了准正则和椭圆调和映射的系数界，并建立了平面中(K,K')-椭圆和K-准正则调和映射的Landau-Bloch型定理，还证明了单位盘上K-准共形自映射的系数估计。
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。
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