C^n中一类星形映射子族的增长定理及推广的Roper-Suffridge算子

On the Growth Theorem and the Roper-Suffridge Extension Operator for a Class of Starlike Mappings in C^n

在有界星形圆形域上定义了一个新的星形映射子族,它包含了α阶星形映射族和α阶强星形映射族作为两个特殊子类.给出了此类星形映射子族的增长定理和掩盖定理.另外,还证明了Reinhardt域Ω_(n,p_2,…,p_n)上此星形映射子族在Roper-Suffridge算子F(z)=(f(z_1),((f(z_1))/(z_1))^(β_2)(f'(z_1))^(γ_2)z_2,…,((f(z_1))/(z_1))^(β_n)(f'(z_1))^(γ_n)z_n)'作用下保持不变,其中Ω_(n,p_2,…,p_n)={z∈C^n:|z_1|~2+|z_2|^(p_2)+…+|z_n|^(p_n)<1},p_j≥1,β_j∈[0,1],γ_j∈[0,1/(p_j)]满足β_j+γ_j≤1,所取的单值解析分支使得((f(z_1))/(z_1))^(β_j)|_(z_1=0)=1,(f'(z_1))^(γ_j)|_(z_1=0)=1,j=2,…,n.这些结果不仅包含了许多已有的结果,而且得到了新的结论.

Abstract The author introduces a new subclass of starlike mappings on bounded starlike circular domains, which contains the starlike mappings of order α and the strong starlike mappings of order α as two special classes. The growth and the covering theorems of the subclass of starlike mappings are established. Next, it is proved that the new class is preserved under the following generalized Roper-Suffridge operator： F（z）=（f（z1）,（f（z1）/z1）^β2（f′（z1））^γ2 z2,…,（f（z1）/z1）^βn（f′（z1））^γn zn）′on Reinhardt domains Ωn,p2,…,pn={z∈C^n：｜z1｜^2＋｜z2｜^p2＋…＋｜z_n｜pn〈1} where pj≥1,βj∈[0,1],γj∈[0,1/pj] such that pj≥1,βj∈[0,1],γj∈[0,1/pj]and the branches are chosen such that（f（z1/z1））^βj｜_z1=0 =1,（f′（z1）^γj｜z1=0=1,j=2,…,n.These results enable us to generalizemany known results and also lead to some new results.