ANNALES POLONICI MATHEMATICI 103.1 (2012) Landau’s theorem for p-harmonic mappings in several variables by Sh. Chen (Changsha), S. Ponnusamy (Chennai) and X. Wang (Changsha) Abstract. A2p-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 p f = 0, where p (≥ 1) is a positive integer and Δ represents the complex Laplacian operator. If Ω ⊂ C n is a domain, then a function f : Ω → C m 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 B n into C n with the form f (z)= (p,...,p) X (k 1 ,...,kn)=(1,...,1) |z1| 2(k 1 −1) ···|zn| 2(kn−1) G p−k 1 +1,...,p−kn+1 (z), where each G p−k 1 +1,...,p−kn+1 is harmonic in B n for ki ∈{1,...,p} and i ∈{1,...,n}. 1. Introduction and main results. A2p times continuously differen- tiable complex-valued function f = u + iv in a domain D ⊆ C is p-harmonic if f satisfies the p-harmonic equation Δ p f = 0, where Δ p f = Δ(Δ p−1 f )= Δ ··· Δ    p times f, and Δ represents the complex Laplacian operator Δ =4 ∂ 2 ∂z∂ z := ∂ 2 ∂x 2 + ∂ 2 ∂y 2 , where z = x + iy ∈ C. If this holds for p = 1, then f is (planar) harmonic, and if it holds for p = 2 then f is (planar) biharmonic. If f is harmonic in a simple connected domain D, then f = h + g, where h and g are analytic in D, and are called the analytic and co-analytic parts of f , respectively. See [AA, AAK1, AAK2, CPW1, CPW2, CPW4, CPW7, CSh, Du, He, Sh] for further discussions on harmonic mappings and biharmonic mappings. More 2010 Mathematics Subject Classification : Primary 30C65; Secondary 31B05, 30C10. Key words and phrases : harmonic mapping, p-harmonic mapping, Landau’s theorem. DOI: 10.4064/ap103-1-6 [67] c  Instytut Matematyczny PAN, 2012