Acta mathematica scientia,Series B ›› 2014, Vol. 34 ›› Issue (6): 1761-1774.doi: 10.1016/S0252-9602(14)60121-1

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ROPER-SUFFRIDGE EXTENSION OPERATOR ON A REINHARDT DOMAIN

 LI Hong-Jun, FENG Shu-Xia*   

  1. School of Mathematics and Information Science, Henan University, Kaifeng 475004, China; Institute of Contemporary Mathematics, School of Mathematics and Information Science, Henan University, Kaifeng 475004, China
  • Received:2013-09-06 Revised:2014-03-06 Online:2014-11-20 Published:2014-11-20
  • Contact: FENG Shu-Xia,fengshx@henu.edu.cn E-mail:henanlihj@126.com;fengshx@henu.edu.cn
  • Supported by:

    Supported by the National Natural Science Foun-dation of China (11001074, 11061015, 11101124).

Abstract:

Let pj ∈ N and pj ≥ 1, j = 2, · · · , k, ≥ 2 be a fixed positive integer. We introduce a Roper-Suffridge extension operator on the following Reinhardt domain ΩN ={z = (z1, z2, … , zk)′ ∈C × Cn2 × … × Cnk : |z1|2 + ||z2||p22 + … + ||zk||pkk < 1} given by F(z) = (f(z1) + f ′(z1)∑kj=2Pj(zj), (f′(z1))1/p2 z2, … , (f′(z1))1/pk zk)′, where f is a normal-
ized biholomorphic function on the unit disc D, and for 2 ≤j ≤k, Pj : Cnj →C is a homogeneous polynomial of degree pj and zj = (zj1, … , zjnj )′ ∈ Cnj , nj ≥1, p≥1, ||zj ||j = (∑njl=1|zjl|pj )1/pj . In this paper, some conditions for Pj are found under which the operator preserves the properties of almost starlikeness of order, starlikeness of order and strongly starlikeness of order on ΩN, respectively.

Key words: Reinhardt domain, Roper-Suffridge operator, almost starlike of order α, star-like mapping of order α, strongly starlike mapping of order α

CLC Number: 

  • 32A10
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