A Modification of the Roper-Suffridge Extension Operator for Some Holomorphic Mappings
A Modification of the Roper-Suffridge Extension Operator for Some Holomorphic Mappings
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发表时间：
2010
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通讯作者：
Wang Jianfei
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作者：
Wang Jianfei
设m∈N， m≥2。研究了F(z)=(F(z_1)+ F ' (z_1)P（z_0）给出的单位球上的Roper-Suffridge扩展算子的一个修正，[F ' (z_1)]~(1/m)z_0) '，其中F是单位圆盘D上的一个归一化生物全纯函数，P:C~（n-1）→C是一个m次齐次多项式，z_0=（z_2，…，z_n）。这个算子最初是由Muir在m=2时引入的。当P≡0且m=2时，算子化简为著名的Roper-Suffridge扩展算子。本文找到了‖P‖的一些条件，在这些条件下，算子分别保持有序α的几乎星形映射和有序α的星形映射。结果推广了许多最新的结果，该方法易于操作。特别是当f(z_1)=z_1, m=n=2时，该算子的经典形式为f(z)=(z_1+az_2~2,z_2) '，这对于构造多复变量几何函数理论中的一些具体例子具有重要意义。
Let m∈N and m≥2.The authors study a modification of the Roper-Suffridge extension operator on the unit ball given by F(z)=(f(z_1)+f′(z_1)P(z_0),[f′(z_1)]~(1/m)z_0)′, where f is a normalized biholomorphic function on the unit disc D,P:C~(n-1)→C is a homogeneous polynomial of degree m and z_0=(z_2,…,z_n).This operator was first introduced by Muir when m=2.In case P≡0 and m=2,the operator reduces to the well-known Roper-Suffridge extension operator.In this paper,some conditions for‖P‖are found under which the operator preserves almost starlike mappings of orderαand starlike mappings of orderα,respectively.The results generalize many recently new results,and the method is easier to handle.In particular,when f(z_1)=z_1 and m=n=2,this operator is of the classical form F(z)=(z_1+az_2~2,z_2)′which is important to construct some concrete examples in geometric function theory of several complex variables.