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

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 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).

Complex Symmetric Toeplitz Operators on the Unit Polydisk and the Unit Ball

2019 ◽  
Vol 40 (1) ◽  
pp. 35-44 ◽  
Author(s):  
Cao Jiang ◽  
Xingtang Dong ◽  
Zehua Zhou
Keyword(s):  
Unit Ball ◽  
Toeplitz Operators ◽  
Unit Polydisk ◽  
Complex Symmetric

scholarly journals On the Fekete and Szegö Problem for the Class of Starlike Mappings in Several Complex Variables

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Qing-Hua Xu ◽  
Tai-Shun Liu
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Unit Disk ◽  
Univalent Functions ◽  
Complex Banach Space ◽  
Complex Variables ◽  
Several Complex Variables ◽  
Unit Polydisk ◽  
Starlike Mappings

LetSbe the familiar class of normalized univalent functions in the unit disk. Fekete and Szegö proved the well-known resultmaxf∈S⁡a3-λa22=1+2e-2λ/(1-λ)forλ∈0, 1. We investigate the corresponding problem for the class of starlike mappings defined on the unit ball in a complex Banach space or on the unit polydisk inCn, which satisfies a certain condition.

A boundary Schwarz lemma on the classical domain of type I

2017 ◽  
Vol 60 (7) ◽  
pp. 1239-1258 ◽  
Author(s):  
TaiShun Liu ◽  
XiaoMin Tang
Keyword(s):  
Schwarz Lemma ◽  
Type I ◽  
Classical Domain ◽  
Boundary Schwarz Lemma

scholarly journals A boundary Schwarz lemma for pluriharmonic mappings between the unit polydiscs of any dimensions

Filomat ◽  
2020 ◽  
Vol 34 (9) ◽  
pp. 3151-3160
Author(s):  
Ziyan Huang ◽  
Di Zhao ◽  
Hongyi Li
Keyword(s):  
Harmonic Functions ◽  
Higher Dimensions ◽  
Schwarz Lemma ◽  
Boundary Schwarz Lemma ◽  
Bounded Harmonic Functions

In this paper, we present a boundary Schwarz lemma for pluriharmonic mappings between the unit polydiscs of any dimensions, which extends the classical Schwarz lemma for bounded harmonic functions to higher dimensions.

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