Approximation properties of univalent mappings on the unit ball in Cn

Let n≥2. In this paper, we obtain approximation properties of various families of normalized univalent mappings f on the Euclidean unit ball Bn in Cn by automorphisms of Cn whose restrictions to Bn have the same geometric property of f. First, we obtain approximation properties of spirallike, convex and g-starlike mappings f on Bn by automorphisms of Cn whose restrictions to Bn have the same geometric property of f, respectively. Next, for a nonresonant operator A with m(A)>0, we obtain an approximation property of mappings which have A-parametric representation by automorphisms of Cn whose restrictions to Bn have A-parametric representation. Certain questions will be also mentioned. Finally, we obtain an approximation property by automorphisms of Cn for a subset of SIn0(Bn) consisting of mappings f which satisfy the condition ‖Df(z)−In‖<1, z∈Bn. Related results will be also obtained.