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 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.

The Schwarz Lemma at the Boundary of the Egg Domain Bp1,p2 in ℂn

2015 ◽  
Vol 58 (2) ◽  
pp. 381-392 ◽  
Author(s):  
Xiaomin Tang ◽  
Taishun Liu
Keyword(s):  
Schwarz Lemma ◽  
Kobayashi Metric ◽  
Classical Boundary ◽  
New Type ◽  
Boundary Schwarz Lemma ◽  
The Schwarz Lemma

AbstractLet be an egg domain in ℂn. In this paper, we first characterize the Kobayashi metric on and then establish a new type of classical boundary Schwarz lemma at for holomorphic self-mappings of ), where .

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).

scholarly journals An improved Schwarz Lemma at the boundary

2018 ◽  
Vol 16 (1) ◽  
pp. 1140-1144
Author(s):  
Peter R. Mercer
Keyword(s):  
Unit Disk ◽  
Analytic Functions ◽  
Schwarz Inequality ◽  
Schwarz Lemma

AbstractWe obtain an new boundary Schwarz inequality, for analytic functions mapping the unit disk to itself. The result contains and improves a number of known estimates.

scholarly journals Landau’s theorem for slice regular functions on the quaternionic unit ball

2017 ◽  
Vol 28 (03) ◽  
pp. 1750017 ◽  
Author(s):  
Cinzia Bisi ◽  
Caterina Stoppato
Keyword(s):  
Unit Ball ◽  
Unit Disk ◽  
Analogous Result ◽  
Open Unit ◽  
Open Unit Ball ◽  
Slice Regular Functions ◽  
New Approach ◽  
The Real ◽  
Regular Functions ◽  
Real Algebra

During the development of the theory of slice regular functions over the real algebra of quaternions [Formula: see text] in the last decade, some natural questions arose about slice regular functions on the open unit ball [Formula: see text] in [Formula: see text]. This work establishes several new results in this context. Along with some useful estimates for slice regular self-maps of [Formula: see text] fixing the origin, it establishes two variants of the quaternionic Schwarz–Pick lemma, specialized to maps [Formula: see text] that are not injective. These results allow a full generalization to quaternions of two theorems proven by Landau for holomorphic self-maps [Formula: see text] of the complex unit disk with [Formula: see text]. Landau had computed, in terms of [Formula: see text], a radius [Formula: see text] such that [Formula: see text] is injective at least in the disk [Formula: see text] and such that the inclusion [Formula: see text] holds. The analogous result proven here for slice regular functions [Formula: see text] allows a new approach to the study of Bloch–Landau-type properties of slice regular functions [Formula: see text].

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