The generalized Roper-Suffridge extension operator on bounded complete Reinhardt domains

The generalized Roper-Suffridge extension operator Φ(f) on the bounded complete Reinhardt domain Ω in C ⁿ with n ⩾2 is defined by for (z ₁, z ₂, ..., z _n ) ∈ Ω, where r = r(Ω) = sup{|z ₁|: (z ₁, z ₂, ..., z _n ) ∈ Ω}, 0 ≼ γ _j ≼ 1 − β _j , 0 ≼ β _j ≼1, and we choose the brach of the power functions such that and , j = 2, ..., n. In this paper, we prove that the operator (f) is from the subset of S _α ^* (U) to S _α ^* (Ω) (0 ≼ α < 1) on Ω and the operator (f) preserves the starlikeness of order α or the spirallikeness of type β on D _p for some suitable constants β _j , γ _j , p _j , where D _p = {(z ₁, z ₂, ..., z _n ) ∈ C ⁿ : < 1} (p _j > 0, j = 1,2, ..., n), U is the unit disc in the complex plane C, and S _α ^* (Ω) is the class of all normalized starlike mappings of order α on Ω. We also obtain that (f) ∈ S _α ^* (D _p ) if and only if f ∈ f ∈ S _α ^* (U) for 0 ≼ α < 1 and some suitable constants β _j , γ _j , p _j . This work was partially supported by the National Natural Science Foundation of China (Grant No. 10471048), the Research Fund for the Doctoral Program of Higher Education (Grant No. 20050574002), the Natural Science Foundation of Fujian Province of China (Grant No. Z0511013) and the Education Commission Foundation of Fujian Province of China (Grant No. JB04038)