SHARP ESTIMATES OF ALL HOMOGENEOUS EXPANSIONS FOR A SUBCLASS OF QUASI-CONVEX MAPPINGS OF TYPE B AND ORDER α IN SEVERAL COMPLEX VARIABLES

In this article, first, the sharp estimates of all homogeneous expansions for a subclass of quasi-convex mappings of type B and order B on the unit ball in complex Ba- nach spaces are given. Second, the sharp estimates of all homogeneous expansions for the above generalized mappings on the unit polydisk in （in are also established. In particular, the sharp estimates of all homogeneous expansions for a subclass of quasi-convex mappings （include quasi-convex mappings of type A and quasi-convex mappings of type B） in several complex variables are get accordingly. Our results state that a weak version of the Bieber- bach conjecture for quasi-convex mappings of type B and order a in several complex variables is proved, and the derived conclusions are the generalization of the classical results in one complex variable.

A SUBCLASS OF QUASI-CONVEX MAPPINGS ON A REINHARDT DOMAIN IN C^n

Let D p 1,p 2,⋯,p n={z∈C n:∑l=1 n|z l|p l<1},p l>1,l=1,2,⋯,n.In this article,we first establish the sharp estimates of the main coefficients for a subclass of quasi-convex mappings(including quasi-convex mappings of type A and quasi-convex mappings of type B)on D p 1,p 2,⋯,p n under some weak additional assumptions.Meanwhile,we also establish the sharp distortion theorems for the above mappings.The results that we obtain reduce to the corresponding classical results in one dimension.

AN INEQUALITY OF HOMOGENEOUS EXPANSION FOR BIHOLOMORPHIC QUASI-CONVEX MAPPINGS ON THE UNIT POLYDISK AND ITS APPLICATION

In this article,the authors obtain an inequality of homogeneous expansion for f,where f is a quasi-convex mapping(including quasi-convex mapping of type A and quasi-convex mapping of type B)defined on the open unit polydisk in C n .Meanwhile,the authors also investigate its application.

SHARP ESTIMATES OF QUASI-CONVEX MAPPINGS OF TYPE B AND ORDER α

In this paper, we first establish several sharp inequalities of homogeneous expansion for biholomorphic quasi-convex mappings of type B and order α on the unit ball E in a complex Banach space X by applying the method and technique of complex analysis. Then, as the application of these sharp inequalities, we derive the sharp estimate of third homogeneous expansions for the above mappings defined on the unit polydisk U^n in C^n.

THE GROWTH THEOREM FOR QUASI-CONVEX MAPPINGS IN HILBERT SPACES

In this paper, the authors extend the definition of quasi-convex mappings and obtain the corresponding growth theorem-on the unit ball of a complex Hilbert space X.

ON REFINEMENT OF THE COEFFICIENT INEQUALITIES FOR A SUBCLASS OF QUASI-CONVEX MAPPINGS IN SEVERAL COMPLEX VARIABLES

Let K be the familiar class of normalized convex functions in the unit disk.In[14],Keogh and Merkes proved that for a function f(z)=z+∑k=2∞a k z k in the class K,|a 3−λa 22|≤max{13,|λ−1|},λ∈C.The above estimate is sharp for eachλ.In this article,we establish the corresponding inequality for a normalized convex function f on U such that z=0 is a zero of order k+1 of f(z)−z,and then we extend this result to higher dimensions.These results generalize some known results.

Annulus and Disk Complex Is Contractible and Quasi-convex

The annulus and disk complex is defined and researched. Especially, we prove that this complex is contractible and quasi-convex in the curve complex.

Fekete and Szego problem for a subclass of quasi-convex mappings in several complex variables

Let K be the familiar class of normalized convex functions in the unit disk. Keogh and Merkes proved the well-known result that maxf∈A ｜a3 - λa22｜ ≤ max{1/3, ｜λ - 1}, ,λ ∈ C, and the estimate is sharp for each ∈. We investigate the corresponding problem for a subclass of quasi-convex mappings of type B defined on the unit ball in a complex Banach space or on the unit polydisk in Cn. The proofs of these results use some restrictive assumptions, which in the case of one complex variable are automatically satisfied.

On quasi-convex mappings of order α in the unit ball of a complex Banach space

In this paper, a class of biholomorphic mappings named quasi-convex mapping of order α in the unit ball of a complex Banach space is introduced. When the Banach space is confined to Cn, we obtain the relation between this class of mappings and the convex mappings.Furthermore, the growth and covering theorems of this class of mappings are given on the unit ball of a complex Banach space Ⅹ. Finally, we get the second order terms coefficient estimations of the homogeneous expansion of quasi-convex mapping of order α defined on the polydisc in Cn and on the unit ball in a complex Banach space, respectively.

On growth and covering theorems of quasi-convex mappings in the unit ball of a complex Banach space

A class of biholomorphic mappings named 'quasi-convex mapping' is introduced in the unitball of a complex Banach space. It is proved that this class of mappings is a proper subset of the class ofstarlike mappings and contains the class of convex mappings properly, and it has the same growth and coveringtheorems as the convex mappings. Furthermore, when the Banach space is confined to Cn, the 'quasi-convexmapping' is exactly the 'quasi-convex mapping of type A' introduced by K. A. Roper and T. J. Suffridge.

Banach空间中一类微分逆变分不等式解的存在性、唯一性及稳定性

在实自反Banach空间中证明了一类微分逆变分不等式解的存在性、唯一性及稳定性.首先,通过KKM定理证明了逆变分不等式解的存在性,进一步在映射满足单调plus假设下得到逆变分不等式解的唯一性;其次,将微分逆变分不等式问题转化为一个发展方程,利用算子半群的相关知识得到微分逆变分不等式mild解的存在性及其唯一性;再次,考虑了微分逆变分不等式受参数扰动时,微分逆变分不等式的mild解的稳定性.最后,举出一个数值例子,绘制轨线图像,验证定理结论.

Distortion Theorems for Normalized Biholomorphic Quasi-convex Mappings

In this paper, we will use the Schwarz lemma at the boundary to character the distortion theorems of determinant at the extreme points and distortion theorems of matrix on the complex tangent space at the extreme points for normalized locally biholomorphic quasi-convex mappings in the unit ball Bn respectively.

Quasi-convex subsets in Alexandrov spaces with lower curvature bound

We introduce quasi-convex subsets in Alexandrov spaces with lower curvature bound,which include not only all closed convex subsets without boundary but also all extremal subsets.Moreover,we explore several essential properties of such kind of subsets including a generalized Liberman theorem.It turns out that the quasi-convex subset is a nice and fundamental concept to illustrate the similarities and differences between Riemannian manifolds and Alexandrov spaces with lower curvature bound.

半拟可微拟凸规划的KKT型最优性条件

最优性条件在优化问题中起着重要的作用,它为优化算法的研究提供了重要的理论依据。众所周知,凸规划方面最优性条件已比较完善。然而,由于拟凸函数性质的特殊性,对于拟凸规划问题解的Karush-Kuhn-Tucker(KKT)类型最优性条件的研究相对较少。本文利用半拟可微刻画了拟凸规划的最优性条件,同时研究了可行集法锥与带半拟可微性质的约束函数之间的关系,并证明了上述两个结果与Greenberg-Pierskalla次微分的关系。

协同拟凸函数的Riemann-Liouville分数阶积分不等式

基于Riemann-Liouville分数阶积分,对协同拟凸函数的Hermite-Hadamard分数阶积分不等式进行了研究。剖析Riemann-Liouville分数阶积分的运算特点,深入探究SAR?KAYA给出的Riemann-Liouville分数阶积分恒等式。在该Riemann-Liouville分数阶积分恒等式的基础上,利用二元函数的单调性和协同拟凸性,巧妙应用三角不等式和H?lder不等式等经典不等式,建立了若干个协同拟凸函数的Hermite-Hadamard分数阶积分不等式。

The Sharp Estimates of all Homogeneous Expansions for a Class of Quasi-convex Mappings on the Unit Polydisk in C^n

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.

The Estimates of All Homogeneous Expansions for a Subclass of ε Quasi-convex Mappings in Several Complex Variables

The authors obtain the estimates of all homogeneous expansions for a subclass of ε quasi-convex mappings on the unit ball in complex Banach spaces. Moreover, the estimates of all homogeneous expansions for the above generalized mappings on the unit polydisk in Cnare also obtained. Especially, the above estimates are only sharp for a subclass of starlike mappings, quasi-convex mappings and quasi-convex mappings of type A. The results are the generalization of many known results.

参数向量Ky Fan不等式与对偶问题解映射Lipschitz连续性最优条件

在赋范线性空间中研究参数向量Ky Fan不等式与对偶问题解映射的Lipschitz连续性。提出了参数向量Ky Fan不等式与对偶问题及其有效解的概念,引入了向量函数伪单调性和强拟凸(凹)性,借助分析方法获得了参数向量Ky Fan不等式与对偶问题解映射的Lipschitz连续性最优条件,并举例加以说明。

一类带映射差的非凸向量优化问题解的稳定性

在实际生活中,问题数据常常受到干扰,求原问题解时,常常利用近似问题解去逼近原问题解。使用这种方法进行求解时,原问题解集的稳定性是一个重要的前提条件。本文考虑一类带映射差的非凸向量优化问题,研究了近似问题数据收敛于原问题数据时,通过对映射差的两个映射凸性和收敛性的限制,获得了带映射差的非凸向量优化问题在Painlevé-Kuratowski收敛性意义下有效解的稳定性结果。

广义复阶α型Bazilevic函数的一些性质

通过引入一个由拟从属关系定义的广义复阶α型Bazilevic函数的新子类,应用微分方程求解技巧和解析函数的基本不等式,研究此函数类的积分表达式、前两项系数估计和Fekete-Szego问题,得到了全新的结果.