Let G be a finite domain in the complex plane with K-quasicon formal boundary, z 0 be an arbitrary fixed point in G and p>0. Let π(z) be the conformal mapping from G onto the disk with radius r 0>0 and centered...Let G be a finite domain in the complex plane with K-quasicon formal boundary, z 0 be an arbitrary fixed point in G and p>0. Let π(z) be the conformal mapping from G onto the disk with radius r 0>0 and centered at the origin 0, normalized by ?(z 0) = 0 and ?(z 0) = 1. Let us set $\varphi _p \left( z \right): = \int_{x_0 }^x {\left[ {\phi \left( \zeta \right)} \right]^{2/8} } d\zeta $ , and let π n,p (z) be the generalized Bieberbach polynomial of degree n for the pair (G,z 0) that minimizes the integral $\iint\limits_c {\left| {\varphi _p \left( z \right) - P_x^1 (z)} \right|^p d0_x }$ in the class $\mathop \prod \limits_n $ of all polynomials of degree ≤ n and satisfying the conditions P n (z 0) = 0 and P′ n (z 0) = 1. In this work we prove the uniform convergence of the generalized Bieberbach polynomials π n,p (z) to ? p (z) on $\bar G$ in case of $p > 2 - \frac{{K^2 + 1}}{{2K^4 }}$ .展开更多
In this paper, the sharp estimates of all homogeneous expansions for a subclass of starlike mappings on the unit ball in complex Banach spaces are first established. Meanwhile, the sharp estimates of all homogeneous e...In this paper, the sharp estimates of all homogeneous expansions for a subclass of starlike mappings on the unit ball in complex Banach spaces are first established. Meanwhile, the sharp estimates of all homogeneous expansions for the above generalized mappings on the unit polydisk in Cnare also obtained. Our results show that a weak version of the Bieberbach conjecture in several complex variables is proved, and the obtained conclusions reduce to the classical results in one complex variable.展开更多
Let G C C be a simply connected domain whose boundary L := G is a Jordan curve and 0 ∈ G. Let w = φ(z) be the conformal mapping of G onto the disk B(0, r0) := {w : |w| 〈 r0), satisfying φ0(0) = 0, φ'...Let G C C be a simply connected domain whose boundary L := G is a Jordan curve and 0 ∈ G. Let w = φ(z) be the conformal mapping of G onto the disk B(0, r0) := {w : |w| 〈 r0), satisfying φ0(0) = 0, φ't(0) = 1. We consider the following extremal problem for p 〉 0:∫∫G|φ'(z)-P'n(z)|Pdσz→min in the class of all polynomials Pn(z) of degree not exceeding n with Pn(0) = 0, P'n (0)=- 1. The solution to this extremal problem is called the p-Bieberbach polynomial of degree n for the pair (G, 0). We study the uniform convergence of the p-Bieberbach polynomials Bn,p(z) to the φ(z) on G^- with interior and exterior zero angles determined depending on the properties of boundary arcs and the degree of their "touch".展开更多
This paper discusses discrete elementary subgroups in the Mobius groupM■.With a help of geometry,it is proved that these groups are isomorphic to either agroup extension of Z by a finite subgroup of SO(n) or an exten...This paper discusses discrete elementary subgroups in the Mobius groupM■.With a help of geometry,it is proved that these groups are isomorphic to either agroup extension of Z by a finite subgroup of SO(n) or an extension of a finite group by afree Abelian group of rank k≤n.展开更多
In this paper, we establish the Fekete and Szego inequality for a class of holomorphic functions in the unit disk, and then we extend this result to a class of holomorphic mappings on the unit ball in a complex Banach...In this paper, we establish the Fekete and Szego inequality for a class of holomorphic functions in the unit disk, and then we extend this result to a class of holomorphic mappings on the unit ball in a complex Banach space or on the unit polydisk in Cn.展开更多
文摘Let G be a finite domain in the complex plane with K-quasicon formal boundary, z 0 be an arbitrary fixed point in G and p>0. Let π(z) be the conformal mapping from G onto the disk with radius r 0>0 and centered at the origin 0, normalized by ?(z 0) = 0 and ?(z 0) = 1. Let us set $\varphi _p \left( z \right): = \int_{x_0 }^x {\left[ {\phi \left( \zeta \right)} \right]^{2/8} } d\zeta $ , and let π n,p (z) be the generalized Bieberbach polynomial of degree n for the pair (G,z 0) that minimizes the integral $\iint\limits_c {\left| {\varphi _p \left( z \right) - P_x^1 (z)} \right|^p d0_x }$ in the class $\mathop \prod \limits_n $ of all polynomials of degree ≤ n and satisfying the conditions P n (z 0) = 0 and P′ n (z 0) = 1. In this work we prove the uniform convergence of the generalized Bieberbach polynomials π n,p (z) to ? p (z) on $\bar G$ in case of $p > 2 - \frac{{K^2 + 1}}{{2K^4 }}$ .
基金supported by Key Program of National Natural Science Foundation of China(Grant No.11031008)National Natural Science Foundation of China(Grant No.11061015)
文摘In this paper, the sharp estimates of all homogeneous expansions for a subclass of starlike mappings on the unit ball in complex Banach spaces are first established. Meanwhile, the sharp estimates of all homogeneous expansions for the above generalized mappings on the unit polydisk in Cnare also obtained. Our results show that a weak version of the Bieberbach conjecture in several complex variables is proved, and the obtained conclusions reduce to the classical results in one complex variable.
文摘Let G C C be a simply connected domain whose boundary L := G is a Jordan curve and 0 ∈ G. Let w = φ(z) be the conformal mapping of G onto the disk B(0, r0) := {w : |w| 〈 r0), satisfying φ0(0) = 0, φ't(0) = 1. We consider the following extremal problem for p 〉 0:∫∫G|φ'(z)-P'n(z)|Pdσz→min in the class of all polynomials Pn(z) of degree not exceeding n with Pn(0) = 0, P'n (0)=- 1. The solution to this extremal problem is called the p-Bieberbach polynomial of degree n for the pair (G, 0). We study the uniform convergence of the p-Bieberbach polynomials Bn,p(z) to the φ(z) on G^- with interior and exterior zero angles determined depending on the properties of boundary arcs and the degree of their "touch".
基金Project partially supported by the National Natural Science Foundation of China and a grant of FEYUT of SEDC of China.
文摘This paper discusses discrete elementary subgroups in the Mobius groupM■.With a help of geometry,it is proved that these groups are isomorphic to either agroup extension of Z by a finite subgroup of SO(n) or an extension of a finite group by afree Abelian group of rank k≤n.
基金supported by National Natural Science Foundation of China(Grant Nos.11561030,11261022 and 11471111)the Jiangxi Provincial Natural Science Foundation of China(Grant Nos.20152ACB20002 and 20161BAB201019)Natural Science Foundation of Department of Education of Jiangxi Province of China(Grant No.GJJ150301)
文摘In this paper, we establish the Fekete and Szego inequality for a class of holomorphic functions in the unit disk, and then we extend this result to a class of holomorphic mappings on the unit ball in a complex Banach space or on the unit polydisk in Cn.