A reverse Denjoy theorem Ⅲ

Suppose that C 1 and C 2 are two simple curves joining 0 to ∞, non-intersecting in the finite plane except at 0 and enclosing a domain D which is such that, for all large r, the set {θ : re iθ∈ D} has measure at most 2α, where 0 α π. Suppose also that u is a non-constant subharmonic function in the plane such that u(z) = Φ(|z|) for all large z ∈ C 1 ∪ C 2 ∪～D, where Φ(|z|) is a convex, non-decreasing function of |z| and ～D is the complement of D. Let A D (r, u) = inf{u(z) : z ∈ D and |z| = r}. It is shown that if A D (r, u) = O(1) then lim inf r→∞ B(r, u)/r π/(2α) 0.     

The Schwarz-Pick lemma for planar harmonic mappings

The classical Schwarz-Pick lemma for holomorphic mappings is generalized to planar harmonic mappings of the unit disk D completely. (I) For any 0 < r < 1 and 0 ρ < 1, the author constructs a closed convex domain Er,ρ such that F((z,r))  eiαEr,ρ = {eiαz : z ∈ Er,ρ} holds for every z ∈ D, w = ρeiα and harmonic mapping F with F(D)D and F(z) = w, where △(z,r) is the pseudo-disk of center z and pseudo-radius r; conversely, for every z ∈ D, w = ρeiα and w ∈ eiαEr,ρ, there exists a harmonic mapping F such that ...     

Use Cesro Means To Describe Two Classes of Functions

Let H(D)be the collection of functions which are analytic in the unitdisc D.we call B_0={f∈H(D),(?)(1-|z|~2)|f’(z)|=0}litlle Bloch space.Letf∈H(D),0相似文献   

一道试题的背景研究及其推广

定理如果一个虚数的三次方是实数,那么,这个虚数必有形式Aw或Aw2,其中,w是1的立方虚根,A∈R且A≠0.证法1设z=r(cosθ isinθ),r∈R且r≠0,sinθ≠0,ω=-12 32i=cos23π i sin2π3,则z3=r3(cos3θ i sin3θ)∈R,∴sin3θ=0.3θ=kπ,θ=kπ3,k∈Z.1)当k=6n(n∈Z,下同)时,θ=2nπ,     

A NEW ROPER-SUFFRIDGE EXTENSION OPERATOR ON BOUNDED COMPLETE REINHARDT DOMAINS

In this paper, we construct a new Roper-Suffridge extension operator Φr n,β1,,βn(f)(z) = F(z) = ((rf(z1/r)/z1)β1z1,(rf(z1/r)/z1)β2z2,...,(rf(z1/r)/z1)βnzn)',where f is a normalized locally biholomorphic function on the unit disc D, r = sup{|z1| : z =(z1, ···, zn) ∈Ω}, β1∈ [0, 1], 0 ≤βk≤β1, k = 2, ···, n, then we prove it can preserve the property of spirallikeness of type β, almost starlikeness of order α and starlikeness of orderα on bounded complete Reinhardt domain Ω, respectively.     

扭曲退化同宿分支

§ 1　 HypothesesConsider the following system:z.=f(z) , (1 .1 )and its perturbed systemz.=f(z) +g(z,μ) (1 .2 )where z∈ Rm+n,μ∈ Rk,k≥ 3,0≤ |μ| 1 ,f,g∈ Cr,r≥ 4 ,g(z,0 ) =0 .For simplicity,we sup-pose thatf(p) =0 ,g(p,μ) =0 .Moreover,for(1 .1 ) we assume(H1 ) The stable manifold Wspand the unstable manifold Wupof z=p are m-dimension-al and n-dimensional,respectively.The linearization Df(p) atthe equilibrium z=p has realmultiple-2 eigenvaluesλ1 and -ρ1 ,such thatany remaining eige…     

The generalized Roper-Suffridge extension operator on bounded complete Reinhardt domains

The generalized Roper-Suffridge extension operator Ф(f) on the bounded complete Reinhardt domain Ω in Cn with n ≥ 2 is defined by Φrn,β2,γ2,…,βn,γn(f)(z)=(rf(z1/r),(rf(z1/r)/z1)β2(f'(z1/r))γ2z2,…,(rf(z1/r)/z1)βn(f'(z1/r)γnzn) for (z1,z2,…,zn) ∈Ω, where r = r(Ω) = sup{|z1| (z1,z2,…,zn) ∈ Ω},0 ≤ γj ≤ 1 -βj,0 ≤ βj ≤ 1,and we choose the branch of the power functions such that (f(z1)/z1)βj |z1=0 = 1 and (f′(z1))γj |z1=0 =1,j = 2,…,n. In this paper, we prove that the operator Фrn,β2,γ2,…,βn,γn(f) is from the subset of S*α(U) to S*α(Ω)(0 ≤ α ＜ 1) on Ω and the operator Фrn,β2,γ2,…, βn,γn(f) preserves the starlikeness of order α or the spirallikeness of type β on Dp for some suitable constantsβj,γj,pj, where Dp ={(z1,z2,…,zn) ∈ Cn ∑nj=1|zj|pj ＜ 1} (pj ＞ 0, j = 1,2,…,n), U is the unit disc in the complex plane C, and Sα* (Ω) is the class of all normalized starlike mappings of order α on Ω. We also obtain that Φrn,β2,γ2,…,γn(f) ∈ S*α(Dp) if and only if f ∈ S*a(U) for 0 ≤ α ＜ 1 and some suitable constants βj,γj,pj.     

ON THE NEVANLINNA DIRECTION OF ALGEBROID FUNCTIONS

In this paper, we prove that for an algebroid function w(z), the singular direction arg z =φ0 , satisfying that for arbitrary ε(0 < ε < π 2 ) and any given a ∈ C, lim r → + ∞ n(r,φ0-ε,φ0 +ε,w=a)/ log r = +∞ holds with at most 2v possible exceptional values of a, is the Nevanlinna direction of w(z).     

复数多项式H（z）模最值的一种解析求法

设z为复数,且|z|=1,对于实系数复多项式为h(z)=h0 h1z h2z2 … hnzn,h0·hn≠0,为求|h(z)|max与|h(z)|min,令f(z)=h(z)h(z-1)=r0 nj=1 rj (zj z-j),其中r0=nk=0 h2k,rj=nk=0 hk·hk j (hk=0,k＞n时),由|z|=1可设z=cosθ isinθ,θ∈［0,2π］,由欧拉公式知z=eiθ.于是有|h(z)|=h(eiθ)=|h(eiθ)·h(e-iθ)|12=|f(eiθ)|12=|f(z)|12,所以f(z)=f(eiθ)=r0 nj=1 rj(eijθ e-ijθ)=r0 nj=1 2rjcosjθ,其中cosjθ可表示成cosθ的函数,因此f(eiθ)也可表示成cosθ的一元函数,即f(eiθ)=r0 2r1cos…     

主元法解一道北大自主招生试题

题目设x,y,z∈R＋且x2（1/2）＋y2＋z=1,求xy＋2xz的最大值.这是2010年北京大学自主招生试题,是一道含有三变元的条件最值问题,本题难度较大,很难找到解题入口,本文用主元法给出两种解法与大家分享.解法1依题意,设x=rcosθ,y=rsinθ,θ∈（0,π2）,r∈（0,1）,则x2（1/2）＋y2＋z=1为r＋z=1,所以z=1-r.设w=xy＋2xz,则w=r2sinθcosθ＋2r（1-r）cosθ,     

Uniform meyer solution to the three dimensional cauchy problem for laplace equation

We consider the three dimensional Cauchy problem for the Laplace equation uxx(x,y,z)+ uyy(x,y,z)+ uzz(x,y,z) = 0, x ∈ R,y ∈ R,0 z ≤ 1, u(x,y,0) = g(x,y), x ∈ R,y ∈ R, uz(x,y,0) = 0, x ∈ R,y ∈ R, where the data is given at z = 0 and a solution is sought in the region x,y ∈ R,0 z 1. The problem is ill-posed, the solution (if it exists) doesn't depend continuously on the initial data. Using Galerkin method and Meyer wavelets, we get the uniform stable wavelet approximate solution. Furthermore, we shall give a recipe for choosing the coarse level resolution.     

一种灵活的混合GMRES算法

1　引　　言考虑线性方程组Ax =b (1 .1 )其中 A∈RN× N是非奇异的 .求解方程组 (1 .1 )的很多迭代方法都可归类于多项式法 ,即满足x(n) =x(0 ) +qn- 1 (A) r(0 ) ,degqn- 1 ≤ n -1这里 x(n) ,n≥ 0为第 n步迭代解 ,r(n) =b-Ax(n) 是对应的迭代残量 .等价地 ,r(n) =pn(A) r(0 ) ,degpn≤ n;pn(0 ) =1 (1 .2 )其中 pn(z) =1 -zqn- 1 (z)称为残量多项式 .或有r(n) -r(0 ) ∈ AKn(r(0 ) ,A)其中 Kn(v,A)≡span{ Aiv} n- 1 i=0 是对应于 v,A的 Krylov子空间 .对于非对称问题 ,可以用正交性条件r(n)⊥ AKn(r(0 ) ,A)来确定 (1 .2 )中的…     

M.Riesz定理的注记

设u(z)是单位圆内的实值调和函数 ,若 p_平均Mp(r ,u) =12π∫2π0｜u(reiθ) ｜pdθ1 p <∞ ,则称u(z) ∈hp( 1

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Stable recovery of low-rank matrix via nonconvex Schatten p-minimization

In this paper, a sufficient condition is obtained to ensure the stable recovery(ε≠ 0) or exact recovery(ε = 0) of all r-rank matrices X ∈ Rm×nfrom b = A(X) + z via nonconvex Schatten p-minimization for anyδ4r∈ [3～(1/2))2, 1). Moreover, we determine the range of parameter p with any given δ4r∈ [(3～(1/2))/22, 1). In fact, for any given δ4r∈ [3～(1/2))2, 1), p ∈(0, 2(1- δ4r)] suffices for the stable recovery or exact recovery of all r-rank matrices.     

Uniform convergence of the p-Bieberbach polynomials in domains with zero angles

Let G  C be a simply connected domain whose boundary L := αG is a Jordan curve and 0 ∈ G.Let w = ψ(z) be the conformal mapping of G onto the disk B(0, r0) := {w : |w| r0}, satisfying ψ(0) = 0,ψ′(0) = 1. We consider the following extremal problem for p 0:∫∫G|ψ′(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 ψ(z) on G with interior and exterior zero angles determined depending on the properties of boundary arcs and the degree of their "touch".     

Some properties for certain classes of univalent functions defined by differential inequalities

Let A be the space of functions analytic in the unit disk D = {z:|z| 1}.Let U denote the set of all functions f ∈ A satisfying the conditions f(0) = f'(0)-1 = 0 and|f'(z)(z/f(z))~2-1|1(|z|1).Also,let Ω denote the set of all functions f ∈ A satisfying the conditions f(0) = f'(0)-1 = 0and|zf'(z)-f(z)|1/2(|z|1).In this article,we discuss the properties of U and Ω.     

Area Estimates of Images of Disks under Analytic Functions

Let D_r := {z = x + iy ∈ C : |z| r}, r ≤ 1. For a normalized analytic function f in the unit disk D := D1, estimating the Dirichlet integral Δ(r, f) =∫∫_(D_r)|f'(z)|~2 dxdy, z = x + iy, is an important classical problem in complex analysis. Geometrically, Δ(r, f) represents the area of the image of D_r under f counting multiplicities. In this paper, our main ob jective is to estimate areas of images of D_r under non-vanishing analytic functions of the form(z/f)~μ, μ 0, in principal powers,when f ranges over certain classes of analytic and univalent functions in D.     

变量核分数次积分在Hardy空间的有界性

§ 1　 Introduction and main resultsLet Sn- 1 be the unitsphere in Rn(n≥ 2 ) equipped with normalized Lebesgue measure dσ= dσ(z′) .We say that a functionΩ(x,z) defined on Rn× Rnbelongs to L∞ (Rn)× Lr(Sn- 1 )(r≥ 1 ) ,ifΩ(x,z) satisfies the following two conditions,(i) for any x,z∈Rnandλ>0 ,there hasΩ(x,λz) =Ω(x,z) ;(ii)‖Ω‖L∞(Rn)× Lr(Sn- 1) :=supx∈ Rn∫Sn- 1|Ω(x,z′) | rdσ(z′) 1 / r<∞ .For 0 <α相似文献   

SPECTRAL RADIUSES OF THE GALTON-WATSON BRANCHING PROCESSES

§1IntroductionSuppose that(X,Px)is a Markov chain on a countable(or finite)state space E.Givenany x,y∈E,we say that y can be reached from x and write x y,if there is n≥1suchthat Px(Xn=y)>0.If x y and y z,then x z.The markov chain X is said to beirreducible if any two states can be reached from each other.(See[1]or[2]).If X isirreducible,then there is a number r,with0≤r≤1,such that lim supn→∞[Px(Xn=y)]n1=r forany x,y∈E.The number r is called the spectral radius of X(refer to[3]).…     

On the Extension Operator in Banach Spaces

In [1], Roper and Suffridge introduced an extension operator. This operator is defined for normalized locally univalent function f on the unit disc U = {z∈C：｜z｜〈1} in C by Фn（f）（z）=（f（z1）,√ f′（z1）z0）, where z = （z1,z0） belongs to the unit ball Bn in Cn, z1∈U, z0 = （z2,...,zn） ∈ C^n-1.