Boundary Schwarz lemma for solutions to non-homogeneous biharmonic equations

Schwarz lemma for harmonic mappings between Riemann surfaces

We prove a Schwarz type lemma for harmonic mappings between the unit and a geodesic line in a Riemenn surface.

The Schwarz-Pick lemma of high order in several variables

We prove a high order Schwarz-Pick lemma for mappings between unit balls in complex spaces in terms of the Bergman metric. From this lemma, Schwarz-Pick estimates for partial derivatives of arbitrary order of mappings are deduced.

A Schwarz-Pick lemma for the modulus of holomorphic mappings between the unit balls in complex spaces

In this paper we prove a Schwarz-Pick lemma for the modulus of holomorphic mappings between the unit balls in complex spaces. This extends the classical Schwarz-Pick lemma and the related result proved by Pavlovic.

α-调和方程解的Schwarz引理与Schwarz-Pick引理

Schwarz Lemma and Schwarz-Pick Lemma for solutions of the $α$-harmonic equation

本文研究α>-1时α-调和方程解的Schwarz型与Schwarz-Pick型不等式，通过三角函数积分得到超几何函数形式的不等式，改进了已有结果。

A boundary Schwarz lemma for holomorphic mappings from the polydisc to the unit ball

In this paper, we prove a general Schwarz lemma at the boundary for holomorphic mappings from the polydisc to the unit ball in any dimensions. For the special case of one complex variable, the obtained results give the classic boundary Schwarz lemma.

A note on Schwarz-Pick lemma for bounded complex-valued harmonic functions in the unit ball of R^n

In this paper we prove a Schwarz-Pick lemma for bounded complex-valued harmonic functions in the unit ball of R^n.

A note on the Schwarz lemma for harmonic functions

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.

Schwarz lemma for holomorphic mappings in the unit ball

In this note we establish a Schwarz type inequality for holomorphic mappings between unit balls $B_n$ and $B_m$ in corresponding complex spaces.

A Schwarz-Pick lemma for the modulus of holomorphic mappings from the polydisk into the unit ball

In this paper we prove a Schwarz-Pick lemma for the modulus of holomorphic mappings from the polydisk into the unit ball. This result extends some related results.

On harmonic functions and the Schwarz lemma

We study the Schwarz lemma for harmonic functions and prove sharp versions for the cases of real harmonic functions and the norm of harmonic mappings.

Schwarz lemmas via the pluricomplex Green's function

We prove a version of the Schwarz lemma for holomorphic mappings from the unit disk into the symmetric product of a Riemann surface. Our proof is function-theoretic and self-contained. The main novelty in our proof is the use of the pluricomplex Green's function. We also prove several other Schwarz lemmas using this function.

Schwarz lemma and Kobayashi metrics for holomorphic and pluriharmonic functions

The Schwarz lemma as one of the most influential results in complex analysis and it has a great impact to the development of several research fields, such as geometric function theory, hyperbolic geometry, complex dynamical systems, and theory of quasi-conformal mappings. In this note we mainly consider various version of Schwarz lemma and its relatives related to holomorphic functions including several variables.

On a multi-point Schwarz-Pick lemma

We consider the multi-point Schwarz-Pick lemma and its associate functions due to Beardon-Minda and Baribeau-Rivard-Wegert. Basic properties of the associate functions are summarized. Then we observe that special cases of the multi-point Schwarz-Pick lemma give Schur's continued fraction algorithm and several inequalities for bounded analytic functions on the unit disk.

全纯盘和极小盘在边界处的Schwarz引理

The Schwarz lemma for holomorphic and minimal disks at the boundary

本文证明了$\mathbb{C}^n$中单位球上全纯盘的边界Schwarz引理，并利用Forstnerič和Kalaj关于共形调和映射的Schwarz-Pick引理，进一步证明了极小共形盘的边界Schwarz引理。

The Schwarz lemma at the boundary

The most classical version of the Schwarz lemma involves the behavior at the origin of a bounded, holomorphic function on the disc. Pick's version of the Schwarz lemma allows one to move the origin to other points of the disc. In the present paper we explore versions of the Schwarz lemma at a boundary point of a domain (not just the disc). Estimates on derivatives of the function, and other types of estimates as well, are considered. We review recent results of several authors, and present some new theorems as well.

A generalized Schwarz lemma for two domains related to $μ$-synthesis

We present a set of necessary and sufficient conditions that provides a Schwarz lemma for the tetrablock $\mathbb E$. As an application of this result, we obtain a Schwarz lemma for the symmetrized bidisc $\mathbb G_2$. In either case, our results generalize all previous results in this direction for $\mathbb E$ and $\mathbb G_2$.

Extremal functions of boundary Schwarz lemma

In this paper, we present an alternative and elementary proof of a sharp version of the classical boundary Schwarz lemma by Frolova et al. with initial proof via analytic semigroup approach and Julia-Carathéodory theorem for univalent holomorphic self-mappings of the open unit disk $\mathbb D\subset \mathbb C$. Our approach has its extra advantage to get the extremal functions of the inequality in the boundary Schwarz lemma.

Schwarz lemma and distortion for harmonic functions via length and area

We give sharp estimates for distortion of harmonic by means of area and length of the corresponding surface.

Schwarz-Pick-Ahlfors引理的一个新变形

A new variant of the Schwarz-Pick-Ahlfors lemma

我们证明了一个“一般收缩引理”，类似于施瓦茨-挑选-Ahlfors引理和它的许多推广，但不同之处在于适用于地图的一个有限的磁盘到一个磁盘，而不是要求域的地图是完整的。结论是到原点的距离都收缩了，通过一个极限过程，我们可以恢复原来的Ahlfors引理，即{ \em all}距离都收缩了。证明的方法也是不同的，因为它涉及到收缩的施瓦茨-皮克-阿尔福斯型引理的比较定理黎曼几何。

Schwarz type lemmas and their applications in Banach spaces

The main purpose of this paper is to develop some methods to investigate the Schwarz type lemmas of holomorphic mappings and pluriharmonic mappings in Banach spaces. Initially, we extend the classical Schwarz lemmas of holomorphic mappings to Banach spaces, and then we apply these extensions to establish a sharp Bloch type theorem for pluriharmonic mappings on homogeneous unit balls of $\C^n$ and to obtain some sharp boundary Schwarz type lemmas for holomorphic mappings in Banach spaces. Furthermore, we improve and generalize the classical Schwarz lemmas of planar harmonic mappings into the sh…

Geometric Versions of Schwarz's Lemma for Spherically Convex Functions

We prove several sharp distortion and monotonicity theorems for spherically convex functions defined on the unit disk involving geometric quantities such as spherical length, spherical area and total spherical curvature. These results can be viewed as geometric variants of the classical Schwarz lemma for spherically convex functions.

平面域上挤压函数的Schwarz型引理

A Schwarz-type lemma for squeezing function on planar domains

本文利用最大值原理，为有限连通平面域上的挤压函数建立了Schwarz型引理，并由此直接推导出双连通域上挤压函数的显式公式。

Linear connectivity, Schwarz-Pick lemma and univalency criteria for planar harmonic mappings

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.

Boundary Schwarz lemma for harmonic mappings having zero of order $p$

Suppose $w$ is a sense-preserving harmonic mapping of the unit disk $\mathbb{D}$ such that $w(\mathbb{D})\subseteq\mathbb{D}$ and $w$ has a zero of order $p\geq1$ at $z=0$. In this paper, we first improve the Schwarz lemma for $w$, and then, we establish its boundary Schwarz lemma. Moreover, by using the automorphism of $\mathbb{D}$, we further generalize this result.

Schwarz's Lemmas for mappings satisfying Poisson's equation

For $n\geq3$, $m\geq1$ and a given continuous function $g:~Ω\rightarrow\mathbb{R}^{m}$, we establish some Schwarz type lemmas for mappings $f$ of $Ω$ into $\mathbb{R}^{m}$ satisfying the PDE: $Δf=g$, where $Ω$ is a subset of $\mathbb{R}^{n}$. Then we apply these results to obtain a Landau type theorem.

The Nehari-Schwarz lemma and infinitesimal boundary rigidity of bounded holomorphic functions

We survey a number of recent generalizations and sharpenings of Nehari's extension of Schwarz' lemma for holomorphic self-maps of the unit disk. In particular, we discuss the case of infinitely many critical points and its relation to the zero sets and invariant subspaces for Bergman spaces, as well as the case of equality at the boundary.

Hyperbolic metric on the strip and the Schwarz lemma for HQR mappings

We give simple proofs of various versions of the Schwarz lemma for real valued harmonic functions and for holomorphic (more generally harmonic quasi\-re\-gu\-lar, shortly HQR) mappings with the strip codomain. Along the way using the principle of subordination and the corresponding conformal mapping, depicted on the Figure 1, we get a simple proof of a new version of the Schwarz lemma for real valued harmonic functions (see Theorems 4 and 5) and Theorem 6 related to holomorphic mappings. Using the Schwarz-Pick lemma related to distortion for harmonic mappings and the elementary properties of t…

The Boundary Schwarz lemma for harmonic and pluriharmonic mappings and some generalisations

The main purpose of this paper is to develop some methods to investigate the Schwarz type lemmas of holomorphic mappings and pluriharmonic mappings in Hilbert and Banach spaces. Initially, we extend the classical Schwarz lemmas of holomorphic mappings to Hilbert and Banach spaces. Furthermore, we improve and generalize the classical Schwarz lemmas of planar harmonic mappings into the sharp forms of Banach spaces, and present some applications to sharp boundary Schwarz type lemmas for pluriharmonic mappings in Hilbert and Banach spaces and the obtained results provide the improvements and gener…

The Schwarz-Pick lemma for slice regular functions

The celebrated Schwarz-Pick lemma for the complex unit disk is the basis for the study of hyperbolic geometry in one and in several complex variables. In the present paper, we turn our attention to the quaternionic unit ball B. We prove a version of the Schwarz-Pick lemma for self-maps of B that are slice regular, according to the definition of Gentili and Struppa. The lemma has interesting applications in the fixed-point case, and it generalizes to the case of vanishing higher order derivatives.

A Schwarz lemma for two families of domains and complex geometry

We make sharp estimates to obtain a Schwarz type lemma for the symmetrized polydisc $\gn$ and for the extended symmetrized polydisc $\Gn$. We explicitly construct an interpolating function under certain condition. To do so, we followed the methods described in \cite{Young-LMS}. Also we find a few geometric interplay between the members of the family $\Gn$ and its closure $\widetildeΓ_n$.

四块体的Schwarz引理

Schwarz Lemma for the tetrablock

我们描述了所有复测地线在四块通过原点，从而获得的形式的所有极值的Schwarz引理的四块。文中还给出了Lempert函数和测地线的其它一些极值。该文件可以被视为一个延续的结果Abouhajar，白色和年轻。证明依赖于复测地线在一般域的必要形式，这也证明了在文件中。

The Schwarz type Lemmas and the Landau type theorem of mappings satisfying Poisson's equations

For a given continuous function $g:~Ω\rightarrow\mathbb{C}$, we establish some Schwarz type Lemmas for mappings $f$ in $Ω$ satisfying the {\rm PDE}: $Δf=g$, where $Ω$ is a subset of the complex plane $\mathbb{C}$. Then we apply these results to obtain a Landau type theorem, which is a partial answer to the open problem in \cite{CP}. %The obtained results are

A Schwarz lemma for harmonic mappings between the unit balls in real Euclidean spaces

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.

关于“Schwarz型引理、Landau型定理及非齐次双调和方程解的Lipschitz型空间”的评注

Remarks on "Schwarz-type lemma, Landau-type theorem, and Lipschitz-type space of solutions to inhomogeneous biharmonic equations"

本文改进非齐次双调和方程Dirichlet问题解属于Lipschitz型空间的既有定理，去除控制函数的增长限制，并放宽边界条件。

谱单位球的Sharp Schwarz型引理

Sharp Schwarz-type lemmas for the spectral unit ball

我们提供了两个施瓦茨型引理的概括-第一个结果Globevnik和其他由于Ransford和白色-全纯映射到谱单位球。第一个问题涉及C中的单位圆盘到谱单位球的映射，而第二个问题涉及自映射。上述结果适用于从原点映射到原点的全纯映射。我们将这些结果推广到谱单位球上的一般全纯映射。我们还表明，我们的结果是尖锐的。

The Schwarz Lemma at the Boundary of the Symmetrized Bidisc

The symmetrized bidisc ${\textbf{G}}_{2}$ is defined by $${\textbf{G}}_{2}:=\{(z_1+z_2,z_1z_2)\in\mathbb{C}^2: |z_1|<1,|z_2|<1,\; z_1,z_2\in\mathbb{C}\}.$$ It is a bounded inhomogeneous pseudoconvex domain without $\mathcal{C}^1$ boundary, and especially the symmetrized bidisc hasn't any strongly pseudoconvex boundary point and the boundary behavior of both Carathéodory and Kobayashi metrics over the symmetrized bidisc is hard to describe precisely. In this paper, we study the boundary Schwarz lemma for holomorphic self-mappings of the symmetrized bidisc ${\textbf{G}}_2$, and our boundary Schw…

近埃尔米特流形的受控穷竭函数与 Schwarz 型引理

Tamed exhaustion functions and Schwarz type lemmas for almost Hermitian manifolds

针对近埃尔米特流形，论文借助 Hessian 比较定理构造特殊穷竭函数，并据此建立近全纯映射的 Schwarz 型引理及 Liouville 型定理。

多圆盘上的Schwarz引理

A Schwarz lemma on the polydisk

有一个已知的推广经典施瓦茨引理的全纯函数从多圆盘到圆盘。本文刻画了在这个广义不等式中处处满足等式的函数：它们是n +1 × n+1对称酉的传递函数，特别是有理的，内的，属于多圆盘的Schur-Agler类.我们还提出了一些充分条件的函数是这种类型。

The Real Schwarz Lemma: The Sequel

A decade ago, when teaching complex analysis, the third named author posed the question on whether or not there is an analogue to the Schwarz lemma for real analytic functions. This led to the note [MT], indicating that it is possible to have a real analytic automorphism $f$ of $(-1,1)$ with $f'(0)$ arbitrarily large. In this note we provide other families with this property, and moreover show that we can always find such a function so that $f'(0)$ equals any desired real number. We end with some questions on related problems.

Minimal surfaces and Schwarz lemma

We prove a sharp Schwarz type inequality for the Weierstrass- Enneper representation of the minimal surfaces. It states the following. If $F:\mathbf{D}\to Σ$ is a conformal harmonic parameterization of a minimal disk $Σ$, where $\mathbf{D}$ is the unit disk and $|Σ|=πR^2$, then $|F_x(z)|(1-|z|^2)\le R$. If for some $z$ the previous inequality is equality, then the surface is an affine disk, and $F$ is linear up to a Möbius transformation of the unit disk.

边界上的新Schwarz-Pick引理与全纯映射的刚性

A new Schwarz-Pick Lemma at the boundary and rigidity of holomorphic maps

本文秉承Burns-Krantz边界Schwarz引理精神，利用Harnack型不等式和边界Hopf引理，为单位圆盘和$\mathbb C^N$强凸域上的共形伪度量与全纯自映射建立了新的边界Schwarz-Pick引理及刚性定理。

A form of Schwarz's lemma and a bound for the Kobayashi metric on convex domains

We present a form of Schwarz's lemma for holomorphic maps between convex domains $D_1$ and $D_2$. This result provides a lower bound on the distance between the images of relatively compact subsets of $D_1$ and the boundary of $D_2$. This is a natural improvement of an old estimate by Bernal-González that takes into account the geometry of $\partial{D_1}$. Using similar techniques, we also provide a new estimate for the Kobayashi metric on bounded convex domains.

Weighted Lipschitz continuity, Schwarz-Pick's Lemma and Landau-Bloch's theorem for hyperbolic-harmonic mappings in $\mathbb{C}^{n}$

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.

复单位球上不变调和函数的Schwarz-Pick引理

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。

Schwarz lemmas for mappings with bounded Laplacian

We establish some Schwarz type Lemmas for mappings defined on the unit disk with bounded Laplacian. Then we apply these results to obtain boundary versions of the Schwarz lemma.

关于边界Schwarz引理和某些映射的刚性定理

On the Boundary Schwarz lemma and the rigidity theorem for certain mappings

本文研究了从$B_{\ell_p^n} imes\mathbb{D}^{m}$到$\mathbb{D}^{m}$的全纯映射，给出了向量值全纯函数的边界Schwarz引理的简明证明，并扩展了现有结果，同时获得了单位球$B_{\ell_p^n}$上多调和自映射的边界Schwarz引理，以及对于$B_{\ell_p^n}$的全纯自映射的边界刚性定理。

A converse to the Schwarz lemma for planar harmonic maps

A sharp version of a recent inequality of Kovalev and Yang on the ratio of the $(H^1)^\ast$ and $H^4$ norms for certain polynomials is obtained. The inequality is applied to establish a sharp and tractable sufficient condition for the Wirtinger derivatives at the origin for harmonic self-maps of the unit disc which fix the origin.

带边界且具有离散高斯曲率的广义双曲圆填充的存在性、刚性与离散Schwarz–Pick引理

Existence, Rigidity, and Discrete Schwarz--Pick Lemma for Generalized Hyperbolic Circle Packings with Boundary and Discrete Gaussian Curvatures

本文针对带有限多边形胞腔分解的紧带边曲面，研究具有指定边界顶点测地曲率、内部顶点总测地曲率及对偶圆圆心离散高斯曲率的广义双曲圆填充，给出其存在的充要条件与唯一性，建立含比较结果及刚性的离散Schwarz–Pick引理。

A boundary Schwarz Lemma for holomorphic mappings between unit balls of different dimensions

In this paper, we give a general boundary Schwarz lemma for holomorphic mappings between unit balls in any dimensions. It is proved that if the mapping $f\in C^{1+α}$ at $z_0\in \partial \mathbb B^n$ with $f(z_0)=w_0\in \partial \mathbb B^N$ for any $n,N\geq 1$, then the Jacobian matrix $J_f(z_0)$ maps the tangent space $T_{z_0}(\partial \mathbb B^n)$ to $T_{w_0}(\partial \mathbb B^N)$, and the holomorphic tangent space $T^{(1,0)}_{z_0}(\partial \mathbb B^n)$ to $T^{(1,0)}_{w_0}(\partial \mathbb B^N)$ as well.

单位圆盘内 $(α, β)$-调和函数的 Schwarz-Pick 引理

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。

关于有界域Bergman度量的Schwarz引理

On the Schwarz Lemma for Bergman metrics of bounded domains

本文提出基于概率论Cauchy-Schwarz不等式的新Schwarz引理，用于有界域Bergman度量的研究。

Schwarz-Pick Lemma for Harmonic and Hyperbolic Harmonic Functions

We establish some inequalities of Schwarz-Pick type for harmonic and hyperbolic harmonic functions on the unit ball of and we disprove a recent conjecture of Liu [Schwarz-Pick Lemma for Harmonic Functions, International Mathematics Research Notices, 2021].

附录"缺陷充分因子的阿贝尔簇，施瓦茨引理，和双曲超曲面的低程度，“美国数学杂志119（1997年），1139-1172

Addendum to ``Defects for ample divisors of abelian varieties, Schwarz lemma, and hyperbolic hypersurfaces of low degrees,'' Amerian Journal of Mathematics 119 (1997), 1139-1172

这是一个补充，我们以前的文件缺陷的充分因子的一个阿贝尔簇。它修改了原始论文的一个论点，以处理那里的一个困难。同时，通过将计数函数替换为在重数处截断的计数函数，改进了原文献中的结果，该重数由阿贝尔簇的维数和除数的Chern数的显函数给出.

阿贝尔簇的充分因子的缺陷、Schwarz引理和低次双曲超曲面

Defects for Ample Divisors of Abelian Varieties, Schwarz Lemma, and Hyperbolic Hypersurfaces of Low Degrees

本文主要证明了关于阿贝尔簇的充分因子的亏损关系的一个定理。主要定理设$A$是复维数$n$的交换阿贝尔簇，$D$是$A$中的充分因子.设$f:{\bf C}\rightarrow A$是一个全纯映射.则映射$f$和除数$D$的缺陷为零。主要定理的推论。在阿贝尔簇$A$中的充分因子$D$的补是双曲的，在这个意义上，从$\bf C$到$A-D$不存在非常数全纯映射。