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

Abstract

The generalized Roper-Suffridge extension operator Φ(f) on the bounded complete Reinhardt domain Ω in C ⁿ with n ⩾2 is defined by

$$\Phi _{n, \beta _2 , \gamma _2 , ..., \beta _n , \gamma _n }^r (f)(z) = \left( {rf\left( {\frac{{z_1 }}{r}} \right), \left( {\frac{{rf(\tfrac{{z_1 }}{r})}}{{z_1 }}} \right)^{\beta _2 } \left( {f'\left( {\frac{{z_1 }}{r}} \right)} \right)^{\gamma _2 } z_2 ,...,\left( {\frac{{rf(\tfrac{{z_1 }}{r})}}{{z_1 }}} \right)^{\beta _n } \left( {f'\left( {\frac{{z_1 }}{r}} \right)} \right)^{\gamma _n } z_n } \right)$$

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 $(\tfrac{{f(z_1 )}}{{z_1 }})^{\beta _j } |_{z_1 = 0} = 1$ and $(f'(z_1 ))^{\gamma _j } |_{z_1 = 0} = 1$, j = 2, ..., n. In this paper, we prove that the operator $\Phi _{n, \beta _2 , \gamma _2 , ..., \beta _n , \gamma _n }^r$ (f) is from the subset of S ^*_α (U) to S ^*_α (Ω) (0 ≼ α < 1) on Ω and the operator $\Phi _{n, \beta _2 , \gamma _2 , ..., \beta _n , \gamma _n }^r$ (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: $\sum\nolimits_{j = 1}^n {\left| {z_j } \right|^{p_j } }$ < 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 $\Phi _{n, \beta _2 , \gamma _2 , ..., \beta _n , \gamma _n }^r$ (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 .