Let pj ∈ N and pj≥ 1, j = 2, ···, k, k ≥ 2 be a fixed positive integer. We introduce a Roper-Suffridge extension operator on the following Reinhardt domain ΩN ={z =(z1, z′2, ···, z′k)′∈ C × Cn2×···× Cnk: |z1|2+ ||z2||p22+ ··· + ||zk ||pk k 1} given11 by F P′j(zj),(f(z1))p2 z′2, ···,(f′(z1))pk z′k)′, where f is a normaljized biholomorphic function k(z) =(f(z1) + f′(z1)=2 on the unit disc D, and for 2 ≤ j ≤ k, Pj : Cnj-→ C is a homogeneous polynomial of degree pj and zj =(zj1, ···, zjnj)′∈ Cnj, nj ≥ 1, pj ≥ 1,nj1||zj ||j =()pj. In this paper, some conditions for Pjare found under which the loperator p |zjl|pj=1reserves the properties of almost starlikeness of order α, starlikeness of order αand strongly starlikeness of order α on ΩN, respectively.
Let pj ∈ N and pj ≥ 1, j = 2, · · · , k, k ≥ 2 be a fixed positive integer. We introduce a Roper-Suffridge extension operator on the following Reinhardt domain ?N ={z = (z1, z′2, · · · , z′k)′ ∈ C × Cn2 × · · · × Cnk : |z1|2 +||z2||p22 +· · ·+||zk||pkk < 1} given by F(z) = (f(z1)+f′(z1) kP j=2 Pj(zj),(f′(z1))p12z′2,··· ,(f′(z1))p1kz′k)′, where f is a normal-ized biholomorphic function on the unit disc D, and for 2 ≤ j ≤ k, Pj : Cnj ?→ C is a homogeneous polynomial of degree pj and zj = (zj1, · · · , zjnj )′ ∈ Cnj , nj ≥ 1, pj ≥ 1,||zj||j = ( njP l=1|zjl|pj ) p1j . In this paper, some conditions for Pj are found under which the operator preserves the properties of almost starlikeness of order α, starlikeness of order αand strongly starlikeness of order αon?N , respectively.
