Landau's theorem for p-harmonic mappings in several variables
Annales Polonici Mathematici, Tome 103 (2012) no. 1, pp. 67-87.

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A 2p-times continuously differentiable complex-valued function f=u+iv in a domain DC is p-harmonic if f satisfies the p-harmonic equation Δpf=0, where p (1) is a positive integer and Δ represents the complex Laplacian operator. If ΩCn is a domain, then a function f:ΩCm is said to be p-harmonic in Ω if each component function fi (i{1,,m}) of f=(f1,,fm) 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 Bn into Cn with the form $$f(z)=\sum_{(k_{1},\ldots, k_{n})=(1,\ldots,1)}^{(p,\ldots,p)}|z_{1}|^{2(k_{1}-1)} \cdots|z_{n}|^{2(k_{n}-1)}G_{p-k_{1}+1,\ldots, p-k_{n}+1}(z), $$ where each Gpk1+1,,pkn+1 is harmonic in Bn for ki{1,,p} and i{1,,n}.
DOI : 10.4064/ap103-1-6
Mots-clés : p times continuously differentiable complex valued function domain subseteq mathbb p harmonic satisfies p harmonic equation vardelta where geq positive integer vardelta represents complex laplacian operator varomega subset mathbb domain function varomega rightarrow mathbb said p harmonic varomega each component function ldots ldots p harmonic respect each variable separately paper prove landau blochs theorem class p harmonic mappings unit ball mathbb mathbb form sum ldots ldots ldots cdots p k ldots p k where each p k ldots p k harmonic mathbb ldots ldots

Sh. Chen 1 ; S. Ponnusamy 2 ; X. Wang 1

1 Department of Mathematics Hunan Normal University Changsha, Hunan 410081 People's Republic of China
2 Department of Mathematics Indian Institute of Technology Madras Chennai 600 036, India
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Sh. Chen; S. Ponnusamy; X. Wang. Landau's theorem for
  $p$-harmonic mappings in several  variables. Annales Polonici Mathematici, Tome 103 (2012) no. 1, pp. 67-87. doi : 10.4064/ap103-1-6. https://geodesic-test.mathdoc.fr/articles/10.4064/ap103-1-6/

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