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).

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.

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 Some Properties and Regions of Variability of Affine Harmonic Mappings and Affine Biharmonic Mappings

2009 ◽  
Vol 2009 ◽  
pp. 1-14
Author(s):  
Sh. Chen ◽  
S. Ponnusamy ◽  
X. Wang
Keyword(s):  
Analytic Functions ◽  
Schwarz Lemma ◽  
Harmonic Mappings

We first obtain the relations of local univalency, convexity, and linear connectedness between analytic functions and their corresponding affine harmonic mappings. In addition, the paper deals with the regions of variability of values of affine harmonic and biharmonic mappings. The regions (their boundaries) are determined explicitly and the proofs rely on Schwarz lemma or subordination.

scholarly journals ON HARMONIC BLOCH SPACES IN THE UNIT BALL OF ℂn

2011 ◽  
Vol 84 (1) ◽  
pp. 67-78 ◽  
Author(s):  
SH. CHEN ◽  
X. WANG
Keyword(s):  
Unit Ball ◽  
Lipschitz Functions ◽  
Harmonic Mappings ◽  
Bloch Spaces ◽  
Bloch Constant ◽  
Vector Valued

AbstractIn this paper, our main aim is to discuss the properties of harmonic mappings in the unit ball 𝔹n. First, we characterize the harmonic Bloch spaces and the little harmonic Bloch spaces from 𝔹n to ℂ in terms of weighted Lipschitz functions. Then we prove the existence of a Landau–Bloch constant for a class of vector-valued harmonic Bloch mappings from 𝔹n to ℂn.

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