Acta mathematica scientia, Series B >
A NEW ROPER-SUFFRIDGE EXTENSION OPERATOR ON BOUNDED COMPLETE REINHARDT DOMAINS
Received date: 2013-05-28
Revised date: 2014-05-15
Online published: 2014-11-20
Supported by
Supported by the NSFC (11271359).
In this paper, we construct a new Roper-Suffridge extension operator
Φrn, β1,… , βn(f)(z) = F(z) = ((rf( z1/r /z1 )β1 z1, (rf( z1/r /z1 )β2 z2, … , (rf( z1/r /z1 )βnzn)′,
where f is a normalized locally biholomorphic function on the unit disc D, r = sup{|z1| : z =(z1, … , zn) ∈Ω}, β1 ∈ [0, 1], 0≤ βk ≤ β1, k = 2, … , n, then we prove it can preserve the property of spirallikeness of type , almost starlikeness of order and starlikeness of order α on bounded complete Reinhardt domain Ω, respectively.
LIU Hao , XIA Hong-Chuan . A NEW ROPER-SUFFRIDGE EXTENSION OPERATOR ON BOUNDED COMPLETE REINHARDT DOMAINS[J]. Acta mathematica scientia, Series B, 2014 , 34(6) : 1835 -1844 . DOI: 10.1016/S0252-9602(14)60127-2
[1] Roper K A, Suffridge T J. Convex mappings on the unit ball of Cn. J Anal Math, 1995, 65: 333–347
[2] Graham I, Kohr G. Univalent mappings associated with the Roper-Suffridge extension operator. J d´Analyse Math, 2000, 81: 331–342
[3] Graham I, Hamada H, Kohr G, Suffridge T J. Extension operators for locally univalent mappings. Michigan Math J, 2002, 50: 37–55
[4] Gong S, Liu T S. The generalized Roper-Suffridge extension operator. J Math Anal Appl, 2003, 284: 425–434
[5] Liu X S, Liu T S. The generalized Roper-Suffridge extension operator for locally biholomorphic mappings. Chin Quart J Math, 2003, 18(3): 221–229
[6] Yan C Y. The Roper-Suffridge operator and Loewner chain [D]. Kaifeng: Henan University, 2008
[7] Liu X S, Feng S X. A remark on the generalized Roper-Suffridge extension operator for spirallike mappings of type and order . Chin Quart J Math, 2009, 24(2): 310–316
[8] Zhu Y C, Liu M S. The generalized Roper-Suffridge extension operator on Reinhardt domain Dp. Tai J Math, 2010, 14: 359–372
[9] Feng S X, Liu T S. The generalized Roper-Suffridge extension operator. Acta Math Sci, 2008, 28(1): 63–80
[10] Wang J F. Modified Roper-Suffridge operator for some sunclasses of starlike mappings on Reinhardt do-mains. Acta Math Sci, 2013, 33(6): 1627–1638
[11] Feng S X. Some families of biholomorphic mappings in several complex variables[D]. Heifei: University of Science and Technology of China, 2004
[12] Liu M S, Zhu Y C. The generalized Roper-Suffridge extension operator on bounded complete Reinhardt domains. Sci China Ser A, 2007, 50(12): 1193–1206
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