David Kalaj, Marijan Markovic
Concrete sharp constants in a pointwise estimate of the gradient of a harmonic function in the unit disk are obtained under the assumption that function belong to Hardy space $h^p$, $p\ge 1$. This generalizes some recent result of Maz'ya & Kresin and a result of Colonna related to the Bloch constant of harmonic mappings of the unit disk into itself.
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A harmonic self-homeomorphism of a disk does not increase the area of any concentric disk.
We extend the classical Schwarz-Pick inequality to the class of harmonic mappings between the unit disk and a Jordan domain with given perimeter. It is intriguing that the extremals in this case are certain harmonic diffeomorphisms between the unit disk and a convex domain that solve the Beltrami equation of second order.
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.
The aim of this paper is to obtain the Schwarz-Pick type inequality for $α$-harmonic functions $f$ in the unit disk and get estimates on the coefficients of $f$. As an application, a Landau type theorem of $α$-harmonic functions is established.
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}.
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 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…
We prove some isoperimetric type inequalities for real harmonic functions in the unit disk belonging to the Hardy space $h^p$, $p>1$ and for complex harmonic functions in $h^4$. The results extend some recent results on the area. Further we discus some Riesz type results for holomorphic functions.
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。
Bohr phenomenon for analytic and harmonic mappings on shifted disks
本文研究了移位圆盘上分析函数和调和映射的Bohr不等式及其改进形式，推导了多个精确结果。
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…
In this paper we extend Rado-Choquet-Kneser theorem for the mappings with weak homeomorphic Lipschitz boundary data and Dini's smooth boundary but without restriction on the convexity of image domain, provided that the Jacobian satisfies a certain boundary condition. The proof is based on a recent extension of Rado-Choquet-Kneser theorem by Alessandrini and Nesi \cite{ale} and it is used the approximation principle.
Complex harmonic mean
本文研究复数调和均值的性质，证明其在特定圆盘内的限制，并揭示其圆环结构。
In the article the authors consider the class ${\mathcal H}_0$ of sense-preserving harmonic functions $f=h+\overline{g}$ defined in the unit disk $|z|<1$ and normalized so that $h(0)=0=h'(0)-1$ and $g(0)=0=g'(0)$, where $h$ and $g$ are analytic in the unit disk. In the first part of the article we present two classes $\mathcal{P}_H^0(α)$ and $\mathcal{G}_H^0(β)$ of functions from ${\mathcal H}_0$ and show that if $f\in \mathcal{P}_H^0(α)$ and $F\in\mathcal{G}_H^0(β)$, then the harmonic convolution is a univalent and close-to-convex harmonic function in the unit disk provided certain conditions…
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\|_…
Closures of Harmonic Bergman Besov Spaces in the Weighted Harmonic Bloch Spaces on the Unit Ball
本文通过可逆径向微分算子，利用自然水平集刻画了Rn单位球上加权调和Bloch空间中调和Bergman Besov空间的闭包，并首先证明了该空间包含于加权调和小Bloch空间。
In this paper we prove a Schwarz-Pick lemma for bounded complex-valued harmonic functions in the unit ball of R^n.
We study two problems concerning harmonic measure on "champagne subdomains" of the unit disk. These domains are obtained by removing from the unit disk little disks around sequences of points with a uniform distribution with respect to the pseudohyperbolic metric of the unit disk. We find (I) a necessary and sufficient condition on the decay of the radii of the little disks for the exterior boundary to have positive harmonic measure, and (II) describe sampling and interpolating sequences for Bergman spaces in terms of the harmonic measure on such ``champagne subdomains''.
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…
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, 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…
We prove that a positive function on the unit disk admits a harmonic majorant if and only if a certain logarithmic Lipschitz upper envelope of it (relevant because of the Harnack inequality) admits a superharmonic majorant. We discuss the logarithmic Lipschitz regularity of this superharmonic majorant, and show that in general it cannot be better than that of the Poisson kernel. We provide examples to show that mere superharmonicity of the data does not help with the problem of existence of a harmonic majorant.
We study proper rational maps from the unit disk to balls in higher dimensions. After gathering some known results, we study the moduli space of unitary equivalence classes of polynomial proper maps from the disk to a ball, and we establish a normal form for these equivalence classes. We also prove that all rational proper maps from the disk to a ball are homotopic in target dimension at least $2$.
The Effect of Planar Harmonic Mappings on the Lebesgue Measure of Sets
本文研究了平面调和映射对复平面上可测集勒贝格测度的影响，通过建立精确的面积变形不等式并结合Hardy空间方法，得出了关于调和映射的刚性现象和极值变形的结论。
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…
We give a sharp bound for the Lebesgue constant associated to Leja sequences in the complex unit disk, confirming a conjecture made by Calvi and Phung in 2011
A Unified Study of Bohr's Inequality for analytic and harmonic mappings on the Unit Disk
该研究统一探讨单位圆盘上解析与调和映射的Bohr不等式改进形式，引入可微递增函数序列及Hurwitz-Lerch Zeta函数，所得结果均为最优。
Some Bohr-type inequalities for several subclasses of harmonic functions
本文针对单位圆盘上的若干复值调和函数子类建立了Bohr型不等式，并证明了所得结果均为精确的。
Schwarz-Pick Lemma for Invariant Harmonic Functions on the Complex Unit Ball
本文研究复单位球上不变调和函数的Schwarz-Pick型不等式，证明了梯度和径向导数的sharp常数一致，并推导出两个推论。
Some differential inequalities and criteria for univalency in the unit disk
本文借助Jack引理建立解析函数的微分不等式，并据此提出单位圆盘内新的单叶性判定准则。
In this note we consider some generalizations of the Schwarz lemma for harmonic functions on the unit disk, whereby values of such functions and the norms of their differentials at the point $z=0$ are given.
We provide some new sharp embeddings for p-Carleson and related measures in the unit disk of the complex plane
In this note, we discuss the coefficient regions of analytic self-maps of the unit disk with a prescribed fixed point. As an application, we solve the Fekete-Szegő problem for normalized concave functions with a prescribed pole in the unit disk.
Hilbert Space of Complex-Valued Harmonic Functions in the Unit Disc
该研究提出复值调和函数希尔伯特空间这一解析函数希尔伯特空间的扩展形式，证明其具备与原空间类似的多种性质，拓展了相关理论的适用范围。
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…
We prove that, for any finite Blaschke product $w=B(z)$ in the unit disk, the corresponding Riemann surface over the $w$--plane contains a one-sheeted disk of the radius $0.5$. Moreover, it contains a unit one-sheeted disk with a radial slit. We apply this result to obtain a universal sharp lower estimate of the Bloch seminorm for finite Blaschke products.
We consider a class $\THO$ of typically real harmonic functions on the unit disk that contains the class of normalized analytic and typically real functions. We also obtain some partial results about the region of univalence for this class.
Carleson-Type Measures and Kernel Estimates for Potential-Harmonic Weighted Bergman Spaces on the Unit Ball
本文研究了C^n单位球上的加权Bergman空间，建立了Carleson嵌入的刻画，并证明了再生核函数的逐点和范数估计。
Revisiting Bohr Inequalities with Analytic and Harmonic Mappings on unit disk
本文针对单位圆盘上的自解析映射类与调和映射类，通过引入子圆盘面积测度、高阶导数量及连续函数序列，建立了经典 Bohr 不等式的多种改进与精细化版本，并给出应用。
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.
We first give an exposition on holomorphic isometries from the Poincaré disk to polydisks and from the Poincaré disk to the product of the Poincaré disk with a complex unit ball. As an application, we provide an example of proper holomorphic map from the unit disk to the complex unit ball that is irrational, algebraic and holomorphic on a neighborhood of the closed unit disk. We also include some new results on holomorphic isometries.
Sharp pointwise estimate of $α-$harmonic functions
本文研究单位圆盘上属于 Hardy 类 h^p 的 α-调和映射，建立了函数值及其导数的锐点态估计，并证明了固定原点的 α-调和自映射的 Schwarz 型引理。
On planar harmonic functions with identical Jacobians
本文在已知一个代表函数的前提下，完整刻画了开单位圆盘中具有相同雅可比行列式的保向调和函数族。
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.
Suppose that finitely many disjoint open arcs have been selected on the unit circle, each of length less than $π$. Let $L_0$ be a longest among them. One can treat the unit disk as a hyperbolic plane in the Poincare disk model. From this perspective each arc $L$ of the selected set determines a hyperbolic half-plane bounded by the geodesic curve $I$ joining endpoints of the arc $L$. Remove from the unit disk all these hyperbolic half-planes. The remaining domain is simply connected and contains the origin. Now map this domain conformally onto the unit disk so that the origin stays fixed. After…
We continue the study of analytic functions in the unit disk of finite order with arbitrary set of singular points on the unit circle, introduced in \cite{FG}. The main focus here is made upon the inverse problem: the existence of a function from this class with a given singular set and zero set subject to certain Blaschke-type condition. We also discuss the local analog of the main result from \cite{FG} similar to the standard local Blaschke condition for analytic and bounded functions in the unit disk.
We study radial behavior of harmonic functions in the unit disk belonging to the Korenblum class. We prove that functions which admit two-sided Korenblum estimate either oscillate or have slow growth along almost all radii.
We study quasiconformal mappings of the unit disk that have planar extension with controlled distortion. For these mappings we prove a bound for the modulus of continuity of the inverse map, which somewhat surprisingly is almost as good as for global quasiconformal maps. Furthermore, we give examples which improve the known bounds for the three point property of generalized quasidisks. Finally, we establish optimal regularity of such maps when the image of the unit disk has cusp type singularities.
Schwarz-Pick Lemma for $(α, β)$-Harmonic Functions in the Unit Disc
本文研究单位圆盘中满足参数条件的 $(α,β)$-调和函数，建立基于边界 $L^p$ 范数的原点导数精确估计、任意点导数范数渐近精确估计，并推广至高阶导数及既有函数类。
In this paper we define Gevrey polyanalytic classes of order N on the unit disk D and we obtain for these classes a characteristic expansion into N-analytic polynomials on suitable neighborhoods of D. As an application of our main theorem, we perform for the Gevrey polyanalytic classes of order N on the unit disk D, an analogue to E. M. Dyn'kin's theorem. We also derive, for these classes, their characteristic degree of the best uniform approximation on D by N-analytic polynomials.
We present a family of sense-preserving harmonic mappings in the unit disk related to the classical generalized (analytic) Koebe functions. We prove that these are precisely the mappings that maximize simultaneously the real part of every Taylor coefficient as well as the growth and distortion of functions in affine and linear invariant families of complex-valued harmonic functions.
We obtain an explicit uniform upper bound for the derivative of a conformal mapping of the unit disk onto a convex domain. This estimate depends only on the outer and inner radii of the domain, and on a curvature radius of its boundary. Its proof is based on a Möbius invariant metric of hyperbolic type, introduced by Kulkarni and Pinkall in 1994.
Quaternionic metamonogenic functions in the unit disk
本研究构造了单位圆盘内一组除低阶小子空间外正交的四元数亚单演函数，可在四元数Hilbert模L²中逼近所有亚单演函数，并用于表示极坐标下虚时间波动方程的含时解。
We consider two parametric families of special functions: One is defined by a power series generalizing the classical Mathieu series, and the other one is a generalized Mathieu type power series involving factorials in its coefficients. Using criteria due to Fejer and Ozaki, we provide sufficient conditions for these functions to be close-to-convex or starlike inside the unit disk, and thus univalent.
The purpose of this paper is to study the Schwarz-Pick type inequalities for harmonic or pluriharmonic functions. By analogy with the generalized Khavinson conjecture, we first give some sharp estimates of the norm of harmonic functions from the Euclidean unit ball in $\mathbb{R}^n$ into the unit ball of the real Minkowski space. Next, we give several sharp Schwarz-Pick type inequalities for pluriharmonic functions from the Euclidean unit ball in $\mathbb{C}^n$ or from the unit polydisc in $\mathbb{C}^n$ into the unit ball of the Minkowski space. Furthermore, we establish some sharp coefficien…
For normalized harmonic functions $f(z)=h(z)+\bar{g(z)}$ in the open unit disk $\mathbb{U}$, a sufficient condition on $h(z)$ for $f(z)$ to be $p$-valent in $\mathbb{U}$ is discussed. Moreover, some interesting examples and images of $f(z)$ satisfying the obtained condition are enumerated.
We give improvements of estimates of invariant metrics in the normal direction on strictly pseudoconvex domains. Specifically we will give the second term in the expansion of the metrics. This depends on an improved localisation result and estimates in the one variable case. Finally we will give a new characterisation of the unit disk in $\mathbb C$ in terms of the asymptotic behaviour of quotients of invariant metrics.
In this paper, we prove the Khavinson conjecture for hyperbolic harmonic functions on the unit ball. This conjecture was partially solved in \cite{JKM2020}.
For $α> -1$ and $β>0, $ let $\mathcal{B}_{\mathcal{H}}^0(α, β)$ denote the class of sense preserving harmonic mappings $f=h+\overline{g}$ in the open unit disk $\mathbb{D}$ satisfying $|zh''(z)+α(h'(z)-1)|\leq β-|zg''(z)+αg'(z)|.$ First, we establish that each function belonging to this class is close-to-convex in the open unit disk if $β\in (0, 1+α]$. Next, we obtain coefficient bounds, growth estimates and convolution properties. We end the paper with applications and will construct harmonic univalent polynomials belonging to this class.
A quantitative version of an inequality obtained in \cite[Theorem~2.1]{mathz} is given. More precisely, for normalized $K$ quasiconformal harmonic mappings of the unit disk onto a Jordan domain $Ω\in C^{1,μ} $ ($0<μ\le 1$) we give an explicit Lipschitz constant depending on the structure of $Ω$ and on $K$. In addition we give a characterization of q.c. harmonic mappings of the unit disk onto an arbitrary Jordan domain with $C^{2,α}$ boundary in terms of boundary function using the Hilbert transformations. Moreover it is given a sharp explicit quasiconformal constant in terms of the boundary fu…
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