Convex subordination chains and injective mappings in

In this paper we continue the work related to convex subordination chains in C and C n , and prove that if f ( z ) = z + ∑ k = 2 ∞ A k ( z k ) is a holomorphic mapping on the Euclidean unit ball B n in C n such that ∑ k = 2 ∞ k 2 ‖ A k ‖ ⩽ 1 , a : [ 0 , 1 ] → [ 0 , ∞ ) is a function of class C 2 on ( 0 , 1 ) and continuous on [ 0 , 1 ] , such that a ( 1 ) = 0 , a ( t ) > 0 , t a ′ ( t ) > − 1 / 2 for t ∈ ( 0 , 1 ) , and if a ( ⋅ ) satisfies a differential equation on ( 0 , 1 ) , then f ( z , t ) = a ( t 2 ) D f ( t z ) ( t z ) + f ( t z ) is a convex subordination chain over ( 0 , 1 ] and the mapping F ( z ) = a ( ‖ z ‖ 2 ) D f ( z ) ( z ) + f ( z ) is injective on B n . 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 ] .