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

Sh. Chen¹ ; S. Ponnusamy² ; X. Wang¹

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

Annales Polonici Mathematici, Tome 103 (2012) no. 1, pp. 67-87

Résumé

A $2p$-times continuously differentiable complex-valued function $f=u+iv$ in a domain $D\subseteq\mathbb{C}$ is $p$-harmonic if $f$ satisfies the $p$-harmonic equation $\varDelta^pf=0$, where $p$ $(\geq 1)$ is a positive integer and $\varDelta$ represents the complex Laplacian operator. If $\varOmega\subset\mathbb{C}^{n}$ is a domain, then a function $f:\,\varOmega\rightarrow\mathbb{C}^m$ is said to be $p$-harmonic in $\varOmega$ if each component function $f_i$ ($i\in \{1, \ldots, m\}$) of $f=(f_1,\ldots, f_m)$ 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 $\mathbb{B}^{n}$ into $\mathbb{C}^{n}$ 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 $G_{p-k_{1}+1,\ldots, p-k_{n}+1}$ is harmonic in $\mathbb{B}^{n}$ for $k_{i}\in\{1,\ldots,p\}$ and $i\in\{1, \ldots, n\}$.

Détail

DOI : 10.4064/ap103-1-6

Keywords: 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