Landau's theorem for p-harmonic mappings in several variables

Sh. Chen, S. Ponnusamy, and X. Wang. "Landau's theorem for p-harmonic mappings in several variables." Annales Polonici Mathematici 103.1 (2012): 67-87. <http://eudml.org/doc/280556>.

@article{Sh2012,
abstract = {A 2p-times continuously differentiable complex-valued function f = u + iv in a domain D ⊆ ℂ is p-harmonic if f satisfies the p-harmonic equation $Δ^pf=0$, where p (≥ 1) is a positive integer and Δ represents the complex Laplacian operator. If Ω ⊂ ℂⁿ is a domain, then a function $f:Ω → ℂ^m$ is said to be p-harmonic in Ω if each component function $f_i$ (i∈ 1,...,m) of $f = (f₁,..., f_m)$ is p-harmonic with respect to each variable separately. In this paper, we prove Landau and Bloch’s theorem for a class of p-harmonic mappings f from the unit ball ⁿ into ℂⁿ with the form $f(z) = ∑_\{(k₁,..., kₙ) = (1,...,1)\}^\{(p,...,p)\} |z₁|^\{2(k₁-1)\} ⋯ |zₙ|^\{2(kₙ-1)\}G_\{p-k₁+1,...,p-kₙ+1\}(z)$, where each $G_\{p-k₁+1,..., p-kₙ+1\}$ is harmonic in ⁿ for $k_\{i\} ∈ \{1,...,p\}$ and i ∈ 1,. .., n.},
author = {Sh. Chen, S. Ponnusamy, X. Wang},
journal = {Annales Polonici Mathematici},
keywords = {harmonic mapping; -harmonic mapping; Landau's theorem},
language = {eng},
number = {1},
pages = {67-87},
title = {Landau's theorem for p-harmonic mappings in several variables},
url = {http://eudml.org/doc/280556},
volume = {103},
year = {2012},
}

TY - JOUR
AU - Sh. Chen
AU - S. Ponnusamy
AU - X. Wang
TI - Landau's theorem for p-harmonic mappings in several variables
JO - Annales Polonici Mathematici
PY - 2012
VL - 103
IS - 1
SP - 67
EP - 87
AB - A 2p-times continuously differentiable complex-valued function f = u + iv in a domain D ⊆ ℂ is p-harmonic if f satisfies the p-harmonic equation $Δ^pf=0$, where p (≥ 1) is a positive integer and Δ represents the complex Laplacian operator. If Ω ⊂ ℂⁿ is a domain, then a function $f:Ω → ℂ^m$ is said to be p-harmonic in Ω if each component function $f_i$ (i∈ 1,...,m) of $f = (f₁,..., f_m)$ is p-harmonic with respect to each variable separately. In this paper, we prove Landau and Bloch’s theorem for a class of p-harmonic mappings f from the unit ball ⁿ into ℂⁿ with the form $f(z) = ∑_{(k₁,..., kₙ) = (1,...,1)}^{(p,...,p)} |z₁|^{2(k₁-1)} ⋯ |zₙ|^{2(kₙ-1)}G_{p-k₁+1,...,p-kₙ+1}(z)$, where each $G_{p-k₁+1,..., p-kₙ+1}$ is harmonic in ⁿ for $k_{i} ∈ {1,...,p}$ and i ∈ 1,. .., n.
LA - eng
KW - harmonic mapping; -harmonic mapping; Landau's theorem
UR - http://eudml.org/doc/280556
ER -