关键词: biholomorphic mappings, Roper-Suffridge extension operator, Reinhardt do-mains, Starlike mappings, homogeneous polynomial of degree m

Abstract:

In this note, the author introduces some new subclasses of starlike mappings
S^*_Ωn, _p2,_···, _pn( β, A, B)
={f ∈H(Ω) : | i tan β+ (1 − i tanβ ) 2/ρ(z) ∂ρ/∂z (z)J⁻¹_f (z)f(z) − 1 − AB/1 − B² |< B − A/1 − B²},
on Reinhardt domains _Ωn, _p2,_···, _pn= {z ∈ Cⁿ : |z₁|²+∑ⁿ_j=2|z_j |^pj < 1}, where −1 ≤ A < B <1, q = min{p₂, · · · , p_n} ≥ 1, l = max{p₂, · · · , p_n} ≥ 2 and β ∈ (−π/2 , π/2 ). Some different conditions for P are established such that these classes are preserved under the following modified Roper-Suffridge operator

F(_{z) =（f(z1) + f′(z1)Pm(z0), (f′(z1)^1/m z0）′,}

where f is a normalized biholomorphic function on the unit disc D, z = (z₁, z₀) ∈Ω_{n, p2,···, pn}, z₀ = (z₂, · · · , z_n) ∈Cⁿ⁻¹. Another condition for P is also obtained such that the above generalized Roper-Suffridge operator preserves an almost spirallike function of type and order . These results generalize the modified Roper-Suffridge extension oper-ator from the unit ball to Reinhardt domains. Notice that when p₂ = p₃ = · · · = p_n = 2, our results reduce to the recent results of Feng and Yu.

Key words: biholomorphic mappings, Roper-Suffridge extension operator, Reinhardt do-mains, Starlike mappings, homogeneous polynomial of degree m