Inequalities on the inner radius of univalency and the norm of pre-Schwarzian derivative

In this paper, we get a lower bound of inner radius of univalency of Schwarzian derivative by means of the norm of pre-Schwarzian derivative. Furthermore, we apply the theory of Universal Teichmuller Space to explain its geometric meaning which shows the relationship between the inner radius in Universal Teichmuller Space embedded by Schwarzian derivative and the norm defined in Universal Teichmuller Space embedded by pre-Schwarzian derivative.     

区域的对数导数单叶性内径

在万有Teichmller空间的对数导数嵌入模型T1(△)中,我们证明了存在无穷多个点[h]∈LT1(△),h(△)相互不Mbius等价,它们到边界的距离均为1,而在万有Teichmller空间的Schwarz导数嵌入模型T(△)中,只有一个点Sid具有类似性质.论文还得到了万有Teichmller空间两类嵌入模型的测地线的一些新的性质.     

The Inner Radius of Univalency of the Rhom busDom ain

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On Nehari disks and the inner radius

Let D be a simply connected plane domain and B the unit disk. The inner radius of D, , is defined by . Here S _f is the Schwarzian derivative of f, the hyperbolic density on D and . Domains for which the value of is known include disks, angular sectors and regular polygons, as well as certain classes of rectangles and equiangular hexagons. All of the mentioned domains except non-convex angular sectors have an interesting property in common, namely that , where h maps B conformally onto D. Because of the importance of this property for computing , we say that D is a Nehari disk if holds.?This paper is devoted to the problem of characterizing Nehari disks. We give a necessary and sufficient condition for a domain to be a Nehari disk provided it is a regulated domain with convex corners. Received: March 7, 1996; revised version: July 15, 1999     

单叶性内径研究的新方法

用与Leila Miller-Van Wieren的方法不同的方法对一类六边形H进行了研究,得到了此类六边形H的单叶性内径的计算公式.同时证明了此类六边形H是Nehari圆.     

On accuracy of approximation of the spectral radius by the Gelfand formula

The famous Gelfand formula ρ(A)=limsup_n→∞A^n1/n for the spectral radius of a matrix is of great importance in various mathematical constructions. Unfortunately, the range of applicability of this formula is substantially restricted by a lack of estimates for the rate of convergence of the quantities A^n1/n to ρ(A). In the paper this deficiency is made up to some extent. By using the Bochi inequalities we establish explicit computable estimates for the rate of convergence of the quantities A^n1/n to ρ(A). The obtained estimates are then extended for evaluation of the joint spectral radius of matrix sets.     

On the norm and spectral radius of Hermitian elements

Let be a complex unital Banach algebra. An element a ∈ is said to be Hermitian if ‖exp(ita)‖ = 1 for all t ∈ ℝ. In the case of the algebra of bounded linear operators in a Hilbert space, this Hermitian property agrees with the ordinary self-adjointness. If a ∈ is Hermitian, then ‖a‖ = |a|, where |a| denotes the spectral radius of a. A function F: ℝ → ℂ is called a universal symbol if ‖F(a)‖ = | F(a)| for every and all Hermitian a ∈ . We characterize universal symbols in terms of positive-definite functions.     

Maps preserving numerical radius or cross norms of products of self-adjoint operators

Let H be a complex Hilbert space with dimH ≥3, Bs（H） the （real） Jordan algebra of all self-adjoint operators on H. Every surjective map Ф ： Bs（H）→13s（H） preserving numerical radius of operator products （respectively, Jordan triple products） is characterized. A characterization of surjective maps on Bs （H） preserving a cross operator norm of operator products （resp. Jordan triple products of operators） is also given.     

Uniformly locally univalent harmonic mappings associated with the pre-Schwarzian norm

In this paper, we consider the class of uniformly locally univalent harmonic mappings in the unit disk and build a relationship between its pre-Schwarzian norm and uniformly hyperbolic radius. Also, we establish eight ways of characterizing uniformly locally univalent sense-preserving harmonic mappings. We also present some sharp distortions and growth estimates and investigate their connections with Hardy spaces. Finally, we study subordination principles of norm estimates.     

Norm of pre-Schwarzian derivative for the class of k-uniformly convex and k-starlike functions

A quasiconformal extension for the class of k-uniformly convex functions, denoted , and for the class of k-starlike functions, denoted is provided. Also, estimation of the norm of pre-Schwarzian derivative in is given.     

On an upper bound of the spectral radius of graphs

We give some upper bounds for the spectral radius of bipartite graph and graph, which improve the result in Hong’s Paper [Y. Hong, J.-L. Shu, K. Fang, A sharp upper bound of the spectral radius of graphs, J. Combin. Theory Ser. B 81 (2001) 177-183].     

万有Teichmuller空间各分支边界点的一些几何性质

在对数导数意义下,万有Teichmuller空间T1可表示为无穷多个互不相交的连通分支的并集.本文研究了该模型各分支的几何性质,给出了为e-iθ/（1-e-iθz）为L和Le的公共边界点,且在‖·‖1的意义下,证明了L,L0,Lθ两两公共边界点之间的距离均为2.     

On the spectral radius of cacti with k-pendant vertices

A graph G is a cactus if any two of its cycles have at most one common vertex. In this article, we determine graphs with the largest spectral radius among all the cacti with n vertices and k-pendant vertices.     

On the spectral radius of a nonnegative centrosymmetric matrix

In this paper, we discuss the spectral radius of nonnegative centrosymmetric matrices. By using the centrosymmetric structure, we establish some estimations of the spectral radius.     

Characterization of the stability radius via bifurcation techniques

Robustness of stability of linear time-invariant systems using the relationship between the structured complex stability radius and a parametrized algebraic Riccati equation is analysed. Our approach is based on the observation that the algebraic Riccati equation can be viewed as a bifurcation problem. It is proved that the stability radius is, under certain assumptions, associated with a turning point of the bifurcation problem given by the parametrized algebraic Riccati equation. As a byproduct, the stability radius can be computed via path following. Some numerical examples are presented.     

On the numerical radius of Lipschitz operators in Banach spaces

We study the numerical radius of Lipschitz operators on Banach spaces via the Lipschitz numerical index, which is an analogue of the numerical index in Banach space theory. We give a characterization of the numerical radius and obtain a necessary and sufficient condition for Banach spaces to have Lipschitz numerical index 1. As an application, we show that real lush spaces and C  -rich subspaces have Lipschitz numerical index 1. Moreover, using the Gâteaux differentiability of Lipschitz operators, we characterize the Lipschitz numerical index of separable Banach spaces with the RNP. Finally, we prove that the Lipschitz numerical index has the stability properties for the _c0$c_0$-, _l1$l_1$-, and _l∞$l_{\infty }$-sums of spaces and vector-valued function spaces. From this, we show that the C(K)$C(K)$ spaces, _L1(μ)$L_1(\mu )$-spaces and _L∞(ν)$L_{\infty }(\nu )$-spaces have Lipschitz numerical index 1.