Acta mathematica scientia,Series B ›› 2013, Vol. 33 ›› Issue (6): 1627-1638.doi: 10.1016/S0252-9602(13)60110-1

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MODIFIED ROPER-SUFFRIDGE OPERATOR FOR SOME SUBCLASSES OF STARLIKE MAPPINGS ON REINHARDT DOMAINS

 WANG Jian-Fei   

  1. Department of Mathematics and Physics, Information Engineering, Zhejiang Normal University, Jinhua 321004, China
  • Received:2012-05-21 Online:2013-11-20 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).

Abstract:

In this note, the author introduces some new subclasses of starlike mappings
S*Ωn, p2,···, pnβ, 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.

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

CLC Number: 

  • 32H02
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