Parabolic starlike mappings of the unit ball B n

Abstract. Let f be a locally univalent function on the unit disk U. We consider the normalized extensions of f to the Euclidean unit ball Bn ⊆Cn given by Φn,γ(f)(z) =(f(z1),(f′(z1))γ ˆ z),where γ ∈ [0,1/2], z = (z1, ˆ z) ∈ Bn and Ψn,β(f)(z) =(f(z1),(f(z1) z1)β ˆ z), in which β ∈ [0,1], f(z1) ̸= 0 and z = (z1, ˆ z) ∈ Bn. In the case γ = 1/2, the function Φn,γ(f) reduces to the well known Roper-Suffridge extension operator. By using different methods, we prove that if f is parabolic starlike mapping on U then Φn,γ(f) and Ψn,β(f) are parabolic starlike mappings on Bn.