ROPER-SUFFRIDGE EXTENSION OPERATOR ON A REINHARDT DOMAIN

doi:10.1016/S0252-9602(14)60121-1

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*

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

Abstract:

Let p_j ∈ N and p_j≥ 1, j = 2, · · · , k, k ≥ 2 be a fixed positive integer. We introduce a Roper-Suffridge extension operator on the following Reinhardt domain Ω_N ={z = (z₁, z′₂, … , z′_k)′ ∈C × Cⁿ² × … × C^nk : |z₁|² + ||z²||^p2₂ + … + ||z_k||^pk_k < 1} given by F(z) = (f(z₁) + f ′(z₁)∑^k_j₌₂P_j(z_j), (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, P_j: C^nj→C is a homogeneous polynomial of degree p_jand z_j = (zj1, … , z_jnj )′ ∈ C^nj, n_j ≥1, p_j≥1, ||z_j ||_j = (∑^nj_l₌₁|z_jl|^pj)^1/pj . In this paper, some conditions for P_j are found under which the operator preserves the properties of almost starlikeness of order, starlikeness of order and strongly starlikeness of order on Ω_N, 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

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.

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ROPER-SUFFRIDGE EXTENSION OPERATOR ON A REINHARDT DOMAIN

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