Let B n be the unit ball in C n, we study ε-starlike mappings on B n. The upper bounds of second order item coefficients of homogeneous expansion for ε-starlike mappings are obtained.
Bohr's type inequalities are studied in this paper: if f is a holomorphic mapping from the unit ball B^n to B^n, f(0)=p, then we have sum from k=0 to∞|Dφ_P(P)[D^kf(0)(z^k)]|／k!||Dφ_P(P)||＜1 for|z|＜max{1／2+|...Bohr's type inequalities are studied in this paper: if f is a holomorphic mapping from the unit ball B^n to B^n, f(0)=p, then we have sum from k=0 to∞|Dφ_P(P)[D^kf(0)(z^k)]|／k!||Dφ_P(P)||＜1 for|z|＜max{1／2+|P|,(1-|p|)／2^(1／2)andφ_P∈Aut(B^n) such thatφ_(p)=0. As corollaries of the above estimate, we obtain some sharp Bohr's type modulus inequalities. In particular, when n=1 and |P|→1, then our theorem reduces to a classical result of Bohr.展开更多
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
文摘Let B n be the unit ball in C n, we study ε-starlike mappings on B n. The upper bounds of second order item coefficients of homogeneous expansion for ε-starlike mappings are obtained.
基金Supported by the NNSF of China(10571164)Supported by Specialized Research Fund for the Doctoral Program of Higher Education(SRFDP)(2050358052)Supported by the NSF of Zhejiang Province(Y606197)
文摘Bohr's type inequalities are studied in this paper: if f is a holomorphic mapping from the unit ball B^n to B^n, f(0)=p, then we have sum from k=0 to∞|Dφ_P(P)[D^kf(0)(z^k)]|／k!||Dφ_P(P)||＜1 for|z|＜max{1／2+|P|,(1-|p|)／2^(1／2)andφ_P∈Aut(B^n) such thatφ_(p)=0. As corollaries of the above estimate, we obtain some sharp Bohr's type modulus inequalities. In particular, when n=1 and |P|→1, then our theorem reduces to a classical result of Bohr.
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.
