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

Sh. Chen ; S. Ponnusamy ; X. Wang

Annales Polonici Mathematici, Tome 105 (2012), p. 67-87 / Harvested from The Polish Digital Mathematics Library

Résumé

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.

Détail

Publié le : 2012-01-01

Zbl 1238.30016

EUDML-ID : urn:eudml:doc:280556