scholarly journals Schwarz lemma at the boundary and rigidity property for holomorphic mappings on the unit ball of $\mathbb {C}^n$

2016 ◽  
Vol 145 (4) ◽  
pp. 1709-1716 ◽  
Author(s):  
Xiaomin Tang ◽  
Taishun Liu ◽  
Wenjun Zhang
Keyword(s):  
Unit Ball ◽  
Holomorphic Mappings ◽  
Schwarz Lemma ◽  
Rigidity Property

scholarly journals SCHWARZ LEMMA FOR HOLOMORPHIC MAPPINGS IN THE UNIT BALL

2017 ◽  
Vol 60 (1) ◽  
pp. 219-224 ◽  
Author(s):  
DAVID KALAJ
Keyword(s):  
Unit Ball ◽  
Harmonic Functions ◽  
Type Inequality ◽  
Holomorphic Mappings ◽  
Schwarz Lemma ◽  
Complex Spaces

AbstractIn this note, we establish a Schwarz–Pick type inequality for holomorphic mappings between unit balls Bn and Bm in corresponding complex spaces. We also prove a Schwarz-Pick type inequality for pluri-harmonic functions.

scholarly journals Boundary Schwarz lemma and rigidity property for holomorphic mappings of the unit polydisc in Cn

Filomat ◽  
2020 ◽  
Vol 34 (9) ◽  
pp. 2813-2818
Author(s):  
Ziyan Huang ◽  
Di Zhao ◽  
Hongyi Li
Keyword(s):  
Fixed Point ◽  
Unit Disk ◽  
Complex Plane ◽  
Complex Space ◽  
Holomorphic Mappings ◽  
Schwarz Lemma ◽  
Higher Dimensional ◽  
Rigidity Property ◽  
Boundary Schwarz Lemma

In this paper, we generalize the classical Schwarz lemma at the boundary from the unit disk D in the complex plane to the unit polydisc Dn in higher-dimensional complex space. Two boundary Schwarz lemmas for holomorphic mappings of Dn and corresponding rigidity properties are established without the restriction of the interior fixed point.

scholarly journals Schwarz Lemma for Harmonic Mappings in the Unit Ball

2017 ◽  
Vol 12 (2) ◽  
pp. 545-554
Author(s):  
David Kalaj
Keyword(s):  
Unit Ball ◽  
Schwarz Lemma ◽  
Harmonic Mappings

scholarly journals A Schwarz lemma for hyperbolic harmonic mappings in the unit ball

2021 ◽  
Vol 127 (3) ◽  
Author(s):  
Jiaolong Chen ◽  
David Kalaj
Keyword(s):  
Unit Ball ◽  
Explicit Form ◽  
Smooth Function ◽  
Harmonic Mapping ◽  
Sharp Inequality ◽  
Schwarz Lemma ◽  
Harmonic Mappings ◽  
Manuscripta Math ◽  
Sharp Constant ◽  
Mapping Theory

Assume that $p\in [1,\infty ]$ and $u=P_{h}[\phi ]$, where $\phi \in L^{p}(\mathbb{S}^{n-1},\mathbb{R}^n)$ and $u(0) = 0$. Then we obtain the sharp inequality $\lvert u(x) \rvert \le G_p(\lvert x \rvert )\lVert \phi \rVert_{L^{p}}$ for some smooth function $G_p$ vanishing at $0$. Moreover, we obtain an explicit form of the sharp constant $C_p$ in the inequality $\lVert Du(0)\rVert \le C_p\lVert \phi \rVert \le C_p\lVert \phi \rVert_{L^{p}}$. These two results generalize and extend some known results from the harmonic mapping theory (D. Kalaj, Complex Anal. Oper. Theory 12 (2018), 545–554, Theorem 2.1) and the hyperbolic harmonic theory (B. Burgeth, Manuscripta Math. 77 (1992), 283–291, Theorem 1).

Bloch Constant of Holomorphic Mappings on the Unit Ball of ℂ n *

2007 ◽  
Vol 28 (6) ◽  
pp. 677-684 ◽  
Author(s):  
Jianfei Wang ◽  
Taishun Liu
Keyword(s):  
Unit Ball ◽  
Holomorphic Mappings ◽  
Bloch Constant

scholarly journals Schwarz-Pick Estimates for Holomorphic Mappings with Values in Homogeneous Ball

2012 ◽  
Vol 2012 ◽  
pp. 1-9
Author(s):  
Jianfei Wang
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Arbitrary Order ◽  
Complex Banach Space ◽  
Holomorphic Mappings ◽  
Partial Derivatives ◽  
Carathéodory Metric ◽  
Homogeneous Ball ◽  
Derivatives Of

LetBXbe the unit ball in a complex Banach spaceX. AssumeBXis homogeneous. The generalization of the Schwarz-Pick estimates of partial derivatives of arbitrary order is established for holomorphic mappings from the unit ballBntoBXassociated with the Carathéodory metric, which extend the corresponding Chen and Liu, Dai et al. results.

A boundary Schwarz lemma for pluriharmonic mappings from the unit polydisk to the unit ball

2018 ◽  
Vol 38 (3) ◽  
pp. 926-934
Author(s):  
Ling LI ◽  
Hongyi LI ◽  
Di ZHAO
Keyword(s):  
Unit Ball ◽  
Schwarz Lemma ◽  
Unit Polydisk ◽  
Boundary Schwarz Lemma

Holomorphic Mappings of the Hyperbolic Space into the Complex Euclidean Space and the Bloch Theorem

1975 ◽  
Vol 27 (2) ◽  
pp. 446-458 ◽  
Author(s):  
Kyong T. Hahn
Keyword(s):  
Euclidean Space ◽  
Unit Ball ◽  
Hyperbolic Space ◽  
Holomorphic Mapping ◽  
Continuous Functions ◽  
Holomorphic Mappings ◽  
Euclidean Metric ◽  
The Family ◽  
Complex Euclidean Space ◽  
Bounded Holomorphic

This paper is to study various properties of holomorphic mappings defined on the unit ball B in the complex euclidean space Cn with ranges in the space Cm. Furnishing B with the standard invariant Kähler metric and Cm with the ordinary euclidean metric, we define, for each holomorphic mapping f : B → Cm, a pair of non-negative continuous functions qf and Qf on B ; see § 2 for the definition.Let (Ω), Ω > 0, be the family of holomorphic mappings f : B → Cn such that Qf(z) ≦ Ω for all z ∈ B. (Ω) contains the family (M) of bounded holomorphic mappings as a proper subfamily for a suitable M > 0.

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