Roper-Suffridge extension operator on a reinhardt domain

Abstract

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||2p2+…+||zk||kpk<1} given by F(z)=(f(z1)+f′(z1)∑j=2kPj(zj),(f′(z1))1p2z′2,…,(f′(z1)1pkz′k)′,, where f is a normalized 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=(∑l=1nj|zjl|pj)1pj.. 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 α and strongly starlikeness of order α on ΩN, respectively.