Bounds for Jacobian of harmonic injective mappings in n-dimensional space

Using normal family arguments, we show that the degree of the first nonzero homogenous polynomial in the expansion of \(n\) dimensional Euclidean harmonic \(K\)-quasiconformal mapping around an internal point is odd, and that such a map from the unit ball onto a bounded convex domain, with \(K< 3^{n-1}\), is co-Lipschitz. Also some generalizations of this result are given, as well as a generalization of Heinz's lemma for harmonic quasiconformal maps in \(\mathbb R^n\) and related results.