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

Authors:

Institution:

1. Department of Mathematics, Fuzhou University, Fuzhou 350002, China
2. Department of Mathematics, South China Normal University, Guangzhou 510631, China

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. 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)

Keywords:

Roper-Suffridge extension operator biholomorphic starlike mapping complete Reinhardt domain

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