Acta mathematica scientia, Series B >
ROPER-SUFFRIDGE EXTENSION OPERATOR ON A REINHARDT DOMAIN
Received date: 2013-09-06
Revised date: 2014-03-06
Online published: 2014-11-20
Supported by
Supported by the National Natural Science Foun-dation of China (11001074, 11061015, 11101124).
Let pj ∈ N and pj ≥ 1, j = 2, · · · , k, k ≥ 2 be a fixed positive integer. We introduce a Roper-Suffridge extension operator on the following Reinhardt domain ΩN ={z = (z1, z′2, … , z′k)′ ∈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 z′2, … , (f′(z1))1/pk z′k)′, 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, pj ≥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.
LI Hong-Jun , FENG Shu-Xia . ROPER-SUFFRIDGE EXTENSION OPERATOR ON A REINHARDT DOMAIN[J]. Acta mathematica scientia, Series B, 2014 , 34(6) : 1761 -1774 . DOI: 10.1016/S0252-9602(14)60121-1
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