In this paper,the sharp estimates of all homogeneous expansions for f are established,where f(z) = (f1(z),f2(z),··· ,fn(z)) is a k-fold symmetric quasi-convex mapping defined on the unit polydisk in Cn ...In this paper,the sharp estimates of all homogeneous expansions for f are established,where f(z) = (f1(z),f2(z),··· ,fn(z)) is a k-fold symmetric quasi-convex mapping defined on the unit polydisk in Cn and Dtk+1fp(0)(ztk+1) (tk + 1)! = n l1,l2,···,ltk+1=1 |apl1l2···ltk+1|ei θpl1+θpl2+···+θpltk+1t k+1 zl1zl2 ··· zltk+1,p = 1,2,··· ,n.Here i = √?1,θplq ∈ (-π,π] (q = 1,2,··· ,tk + 1),l1,l2,··· ,ltk+1 = 1,2,··· ,n,t = 1,2,···.Moreover,as corollaries,the sharp upper bounds of growth theorem and distortion theorem for a k-fold symmetric quasi-convex mapping are established as well.These results show that in the case of quasi-convex mappings,Bieberbach conjecture in several complex variables is partly proved,and many known results are generalized.展开更多
基金Project supported by the National Natural Science Foundation of China (Nos. 10971063, 11061015)the Major Program of Zhejiang Provincial Natural Science Foundation of China (No. D7080080) the Guangdong Provincial Natural Science Foundation of China (No. 06301315)
文摘In this paper,the sharp estimates of all homogeneous expansions for f are established,where f(z) = (f1(z),f2(z),··· ,fn(z)) is a k-fold symmetric quasi-convex mapping defined on the unit polydisk in Cn and Dtk+1fp(0)(ztk+1) (tk + 1)! = n l1,l2,···,ltk+1=1 |apl1l2···ltk+1|ei θpl1+θpl2+···+θpltk+1t k+1 zl1zl2 ··· zltk+1,p = 1,2,··· ,n.Here i = √?1,θplq ∈ (-π,π] (q = 1,2,··· ,tk + 1),l1,l2,··· ,ltk+1 = 1,2,··· ,n,t = 1,2,···.Moreover,as corollaries,the sharp upper bounds of growth theorem and distortion theorem for a k-fold symmetric quasi-convex mapping are established as well.These results show that in the case of quasi-convex mappings,Bieberbach conjecture in several complex variables is partly proved,and many known results are generalized.
