A Refined Schwarz Lemma for the Spectral Nevanlinna-Pick Problem

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.

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Consequences of the Schwarz Lemma

Grundlehren der mathematischen Wissenschaften - Function Theory in the Unit Ball of ℂn ◽

10.1007/978-1-4613-8098-6_8 ◽

1980 ◽

pp. 161-184

Author(s):

Walter Rudin

Keyword(s):

Schwarz Lemma ◽

The Schwarz Lemma

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The Schwarz Lemma for Super-Conformal Maps

Hermitian–Grassmannian Submanifolds - Springer Proceedings in Mathematics & Statistics ◽

10.1007/978-981-10-5556-0_6 ◽

2017 ◽

pp. 59-68

Author(s):

Katsuhiro Moriya

Keyword(s):

Schwarz Lemma ◽

Conformal Maps ◽

The Schwarz Lemma

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A Schwarz lemma for hyperbolic harmonic mappings in the unit ball

MATHEMATICA SCANDINAVICA ◽

10.7146/math.scand.a-128528 ◽

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).

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