Acta mathematica scientia, Series B >
THE SHARP ESTIMATE OF THE THIRD HOMOGENEOUS EXPANSION FOR A CLASS OF STARLIKE MAPPINGS OF ORDER ON THE#br# UNIT POLYDISK IN Cn
Received date: 2010-01-22
Revised date: 2011-01-26
Online published: 2012-03-20
Supported by
Project supported by National Natural Science Foundation of China (10971063, 11061015), Major Program of Zhejiang Provincial Natural Science Foundation of China (D7080080), and Guangdong Natural Science Foundation (06301315)
In this article, first, a sufficient condition for a starlike mapping of order f(x) defined on the unit ball in a complex Banach space is given. Second, the sharp estimate of the third homogeneous expansion for f is established as well, where f(z) =(f1(z), f2(z), · · · , fn(z))′ is a starlike mapping of order or a normalized biholomorphic starlike mapping defined on the unit polydisk in Cn, and D2fk(0)(z2)/2! = zk(∑nl=1aklzl), k =1, 2, · · · , n, here, akl = 1/2！∂2fk(0)/∂zk∂zl,k, l = 1, 2, · · · , n. Our result states that the Bieberbach conjecture in several complex variables (the case of the third homogeneous expansion for starlike mappings of order and biholomorphic starlike mappings) is partly proved.
LIU Xiao-Song , LIU Tai-Shun . THE SHARP ESTIMATE OF THE THIRD HOMOGENEOUS EXPANSION FOR A CLASS OF STARLIKE MAPPINGS OF ORDER ON THE#br# UNIT POLYDISK IN Cn[J]. Acta mathematica scientia, Series B, 2012 , 32(2) : 752 -764 . DOI: 10.1016/S0252-9602(12)60055-1
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