Acta mathematica scientia, Series B >
MODIFIED ROPER-SUFFRIDGE OPERATOR FOR SOME SUBCLASSES OF STARLIKE MAPPINGS ON REINHARDT DOMAINS
Received date: 2012-05-21
Online published: 2013-11-20
Supported by
This work was supported by the National Natural Science Foundation of China (11001246, 11101139) and and Zhejiang Innovation Project (T200905).
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−1f (z)f(z) − 1 − AB/1 − B2 |< B − A/1 − B2},
on Reinhardt domains Ωn, p2,···, pn= {z ∈ Cn : |z1|2+∑nj=2|zj |pj < 1}, where −1 ≤ A < B <1, q = min{p2, · · · , pn} ≥ 1, l = max{p2, · · · , pn} ≥ 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) ∈Ωn, p2,···, pn, z0 = (z2, · · · , zn) ∈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 = · · · = pn = 2, our results reduce to the recent results of Feng and Yu.
WANG Jian-Fei . MODIFIED ROPER-SUFFRIDGE OPERATOR FOR SOME SUBCLASSES OF STARLIKE MAPPINGS ON REINHARDT DOMAINS[J]. Acta mathematica scientia, Series B, 2013 , 33(6) : 1627 -1638 . DOI: 10.1016/S0252-9602(13)60110-1
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