Homeomorphic extension of strongly spirallike mappings in ℂ n

In this paper we consider a proper subclass ŜAn of the full class of spirallike mappings on the Euclidean unit ball Bn in ℂn with respect to a given linear operator A. We use the method of subordination chains to obtain an upper growth result for ŜAn, and we obtain various examples of mappings in the same class of normalized biholomorphic mappings on the unit ball Bn in ℂn. We also prove that the class ŜAn is compact, while the full class of spirallike mappings with respect to a linear operator need not be compact in dimension n ⩾ 2, even when the operator is diagonal. This is one of the motivations for considering the class ŜAn. Finally we prove that if f is a quasiregular strongly spirallike mapping on Bn such that |[Df(z)]−1Af(z)| is uniformly bounded on Bn, then f extends to a homeomorphism of ℝ2n onto itself. In addition, if A+A* = 2aIn for some a > 0, this extension is also quasiconformal on ℝ2n.