Parabolic Starlike Mappings of the Unit Ball B^N

Let f be a locally univalent function on the unit disk u. we consider the normalized extensions of ff to the euclidean unit ball b^n⊆c^n given by$$phi_{n,gamma}(f)(z)=left(f(z_1),(f#039 (z_1))^gammahat{z}right),$$ where $gammain[0,1/2]$, $z=(z_1,hat{z})in b^n$ and$$psi_{n,beta}(f)(z)=left(f(z_1),(frac{f(z_1)}{z_1})^betahat{z}right),$$in which β∈[0,1], f(z1)≠0f and z=(z1,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.