Schwarz Lemma at the Boundary of the Unit Ball in $$\mathbb {C}^n$$ C n and Its Applications

2014 ◽  
Vol 25 (3) ◽  
pp. 1890-1914 ◽  
Author(s):  
Taishun Liu ◽  
Jianfei Wang ◽  
Xiaomin Tang
Keyword(s):  
Unit Ball ◽  
Schwarz Lemma

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 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 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).

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

A Schwarz Lemma for Harmonic Functions in the Real Unit Ball

2019 ◽  
Vol 39 (5) ◽  
pp. 1339-1344
Author(s):  
Shaoyu Dai ◽  
Huaihui Chen
Keyword(s):  
Unit Ball ◽  
Harmonic Functions ◽  
Schwarz Lemma ◽  
The Real

scholarly journals Schwarz lemma and boundary Schwarz lemma for pluriharmonic mappings

Filomat ◽  
2018 ◽  
Vol 32 (15) ◽  
pp. 5385-5402 ◽  
Author(s):  
Jian-Feng Zhu
Keyword(s):  
Unit Ball ◽  
Unit Disk ◽  
Schwarz Lemma ◽  
Classical Boundary ◽  
Boundary Schwarz Lemma

In this paper, we first improve the boundary Schwarz lemma for holomorphic self-mappings of the unit ball Bn, and then we establish the boundary Schwarz lemma for harmonic self-mappings of the unit disk D and pluriharmonic self-mappings of Bn. The results are sharp and coincides with the classical boundary Schwarz lemma when n = 1.

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

On Dirichlet type spaces on the unit ball of Cn

2011 ◽  
Vol 65 (2) ◽  
Author(s):  
Małgorzata Michalska
Keyword(s):  
Unit Ball ◽  
Dirichlet Type ◽  
Type Spaces ◽  
Dirichlet Type Spaces

Holomorphically embedded discs with rapidly growing area

1999 ◽  
Vol 129 (2) ◽  
pp. 343-349
Author(s):  
Josip Globevnik
Keyword(s):  
Unit Ball ◽  
Open Unit ◽  
Open Unit Ball

It is shown that if V is a closed submanifold of the open unit ball of ℂ2 biholomorphically equivalent to a disc, then the area of V ∩ r can grow arbitrarily rapidly as r ↗ 1. It is also shown that if V is a closed submanifold of ℂ2 biholomorphically equivalent to a disc, then the area of V ∩ r can grow arbitrarily rapidly as r ↗ ∞.

Polynomial extreme elements of the closed unit ball ofH∞(B)

1994 ◽  
Vol 25 (1) ◽  
pp. 49-56 ◽  
Author(s):  
John N. McDonald
Keyword(s):  
Unit Ball ◽  
Closed Unit Ball
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