Convex subordination chains and injective mappings in Cn

In this paper we continue the work related to convex subordination chains in C and Cn, and prove that if is a holomorphic mapping on the Euclidean unit ball Bn in Cn such that , a:[0,1]→[0,∞) is a function of class C2 on (0,1) and continuous on [0,1], such that a(1)=0, a(t)>0, ta′(t)>−1/2 for t∈(0,1), and if a(⋅) satisfies a differential equation on (0,1), then f(z,t)=a(t2)Df(tz)(tz)+f(tz) is a convex subordination chain over (0,1] and the mapping F(z)=a(‖z‖2)Df(z)(z)+f(z) is injective on Bn. We also present certain coefficient bounds which provide sufficient conditions for univalence, quasiregularity and starlikeness for the chain f(z,t). Finally we give some examples of convex subordination chains over (0,1].