全纯映射子族上改进的Roper-Suffridge算子

A Modification of the Roper-Suffridge Extension Operator for Some Holomorphic Mappings

设m∈N且m≥2,f是单位圆盘D上的正规化双全纯函数,P:C^(n-1)→C是m次齐次多项式.研究了单位球上改进的Roper-Suffridge算子F(z)=(f(z_1)+f′(z_1)P(z_0),[f′(z_1)]^(1/m)z_0)′,其中z_0=(z_2,z_3,…,z_n).当m=2时,该算子由Muir引入.当P≡0和m=2时,此算子就是著名的Roper-Suffridge算子.给出了‖P‖分别满足不同条件时,改进的算子分别保持α次的殆星和α次的星形映射性质.此结果推广了最近许多已有结果,而且方法更易于操作.特别地,当f(z_1)=z_1和m=n=2时,这就导出了多复变几何函数论中构造具体实例的经典形式F(z)=(z_1+az_2~2,z_2)′.

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′（z1）P（z_0）,[f′（z_1）]（1/m）z0）′, 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=（z2,…,zn）.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（z1）=z1 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.