平面上具有有界Fréchet导数的调和映照单叶半径的精确估计

给定单位圆盘D={z||z|1}上调和映照f(z)=h(z)+g(z),其中h(z)和g(z)为D上的解析函数,满足f(0)=0,λf(0)=1,ΛfΛ.通过引入复参数λ,|λ|=1,本文研究调和映照Fλ(z)=h(z)+λg(z)和解析函数Gλ(z)=h(z)+λg(z)的性质,得到Fλ(z)和Gλ(z)单叶半径的精确估计.作为应用,本文得到单位圆盘D上某些K-拟正则调和映照Bloch常数的更好估计,改进和推广由Chen等人所得的相应结果.     

积分算子下调和映照的单叶半径估计

研究单位圆盘上的调和映照在不同条件下积分算子I_f(z)的单叶半径问题,得到在满足不同条件下的Landau型常数,其结论是渐进精确的;其次在调和映照f(z)有界的情况下,研究积分算子I_f(z)的有界性,其结论也是渐进精确的.     

单位圆到水平条形无界区域的调和拟共形映照

研究单位圆盘到水平条形无界区域在原点满足一定规范条件的单叶保向调和映照的解析特征.推导出该类单叶调和映照的解析表示法.得到单位圆盘到水平条形无界区域在原点满足一定规范条件的单叶保向调和映照f(z)成为调和拟共形映照的充分必要条件,对该类调和拟共形映照的系数作出精确估计.作为应用,证明了该类调和拟共形映照的像在欧氏度量下的长度和面积与原像在非欧度量下的偏差定理.本文的结果改进和推广了由Hengartner和Schober所得的相应结论.     

微分算子作用下平面调和映照的单叶半径和 Bloch 常数估计

利用单位圆$D=\{z\mid | z |<1\}$上单叶调和映照的稳定性特征, 研究平面调和映照$f=h+\ov g$在微分算子 $L=z \frac {\partial }{\partial z}-\ov {z}\frac {\partial}{\partial {\ov z}}$作用下调和映照的单叶半径和 Bloch 常数估计, 得到一些精确性结论, 并改进了近期由刘名生和刘志文所得的相应结果.     

调和映照的星象半径

基于对单叶调和函数系数估计的猜想,对定义在单位圆盘上的调和映照类的星象半径进行研究.首先研究系数在满足一定条件下的调和映照类的星象半径,得到其精确的估计,其次研究两类调和函数的卷积的星象半径,所得到的结论也是精确的.     

调和映照的双Lipschitz性质

设w(z)为单位圆盘U到约当区域Ω?C上的调和映照.给出w(z)具有Lipschitz性质的等价条件.进一步地,若Ω为有界凸区域,对其边界函数给出一个较弱的条件,使得w=P[f](z)为调和拟共形映照.     

平面有界调和函数的B1och常数估计

得到了一个平面有界调和函数系数的精确估计式,由此改进了平面有界调和映照的Bloch常数估计,并相应地改进了双调和映照的单叶半径估计.这些结果是Grigoryan,Huang和Abdulhadi等所得结论的推广.     

一类近于凸调和映照的凸像半径与Pre-Schwarz导数的估计

对任意给定的α∈[0,1),对单位圆盘D上规范化的保向调和映照类H的一个近于凸子类P~0(α)={f=h+g∈H:R{h′(z)-α}|g′(z)|,z∈D,g′(0)=0}的性质进行了研究,如P~0(α)类的凸像和星象半径估计、偏差定理、像域面积的估计、拟共形性,其中得到的凸像和星象半径估计值改进了文献[8-9]中相应结果.此外,对包含P~0(α)的稳定单叶调和映照类(SHU)的Pre-Schwarz导数进行了考虑,得到了精确的上界估计.     

星象积分算子与 Bazilevi函数族

<正> 一、引言我们要讨论在单位圆内解析的某些单叶函数族内部进行的几种运算.单位圆内部的区域|z|<1记作 U.假设 f(z)在 U 内是单叶的解析函数,并且 f(0)=0,f’(0)=1,这种函数的全体记为 S.如果 S 中的函数 w=f(z)映照 U 成为关于原点的星形区域,则称 f(z)为星象函数,其全体记为 S~*.f(z)∈S~*的充要条件是ρ≥0,使     

关于稳定调和映照的一点注记

§1 引言设 f 是从紧致 Riemann 流形 M 到 Riemann 流形 N 的一个光滑映照.映照 f 的能量积分定义为E(f)=1/2 integral from M‖df‖~2dV_M.如果映照 f 是能量泛函 E 的一个临界点,则称 f 为从 M 到 N 的调和映照.调和映照f 称为稳定的如果其二阶变分非负.设 S~n 表示 n 维欧氏球面.我们知道不存在从任意紧致 Riemann 流形到 S~n 或从 S~n 到任意 Riemann 流形的非常值稳定调和映照(n≥3).文献[3]、[4]、[5]和[6]进一     

亚纯函数在角域内的波莱耳方向

Suppose that f(z) is a meromorphic function of order λ(0<λ≤∞) and of lower order μ(0≤μ<∞) in the plane. Let ρ(μ≤ρ≤λ) be a finite positive number. B: arg z=θ₀(0≤θ₀ <2π) is called a Borel direction of order ρ of f(z), if for any complex number a, the equality holds, except at most for some a belonging to a set of linear measure zero. For the exceptional values a, we have ρ(θ₀, a)>ρ, except two possible values. With the above hypotheses on f(z), λ, μ and ρ, We have the following lemmas. Lemma 1. There exists a sequence of positive numbers (r_n) such that(?)=∞ and that Lemma 2. If f(z) has a deficient value a₀ with deficiency δ(a₀, f), then we have where (r_n) is the sequence defined in the Lemma 1 and when a_0=∞, we have to replace(?)by (?) in the left hand side of (*). Lemma 3. Suppose that B_1 : arg z =θ₁ and B₂ : arg z=θ₂ (0≤θ₁<θ₂<2π+θ₁) are two half straight lines from the origin and there are no Borel directions of order≥ρ(ρ>1/2) of f(z) in θ₁₀, the inequality holds as n is sufficiently large, where K₁ is a positive number not depending on n andεand when a₀=∞, it is necessary to replace we have θ₂-θ₁≤π/ρ. Theorem 1. Suppose that f(z) is a meromorphic function of order λ (1/2<λ≤+∞) and of lower order μ(0≤μ<+∞) in the plane. Let p be a number such that μ≤ρ≤λ and that 1/2<ρ<+∞If f~((k))(z) has p(1≤P<+∞) deficient values a_i (i=1,2,…,p) with deficiencies δ(a_i,f^(k)), then f(z) has a Borel direction of order ≥ρ in any angular domain, the magnitude of which is larger than It is convenient to consider Julia directions as Borel directions of order zero.Under this assumption, We have the following. Theorem 2. Suppose that f(z) is a meromorphic function of order λ and of finite lower order μ in the plane and that ρ(μ≤ρ≤λ) is a finite number. If p denotes the number of deficient values of f(z) and q denotes the number of Borel directions of order ≥p of f(z), then we have p≤q.     

The boundary distortion and extremal problems in certain classes of univalent functions

We consider the class S₁(τ), 0<τ<1, of functions f(z)=rz+a₂z²+... that are regular and univalent in the unit disk U and have |f(z)|<1. We obtain sharp estimates for the 1-measure of the sets {θ: |f(e^iθ)|=1}. As a corollary, for the familiar class S we find Kolmogorov-type estimates for the sets {θ: |f(e^iθ)|>M}, M>1, and prove inequalities for the harmonic measure, which are similar to those by Carleman-Milloux and Baernstein. We also consider problems on distortion of fixed systems of boundary arcs in the classes of functions that are regular (or meromorphic) and univalent in the disk or circular annulus. Bibliography: 23 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 204, 1993, pp. 115–142. Translated by A. Yu. solynin.     

Analytic iteration and differential equations

In this paper we study some mapping properties of analytic iterationsW(a, z). Our purpose is to establish a sufficient condition forW(a, z) to be conformal and univalent inz forz ∈D, whereD is a given domain and for sufficiently small |a |. To this end we consider the differential equationϱW(a, z)/ϱa=L[W(a, z)] with the conditionW(O, z)=z. A sufficient condition for the solutionW(a, z) of this system to be conformal and univalent inD for |a |<a ₀ (for somea ₀>0), and to satisfy the iteration equation, is established. This paper is based on a part of the author’s thesis towards the D.Sc. degree, under the guidance of Professor E. Jabotinsky at the Technion Israel Institute of Technology, Haifa.     

Wachstumsordnung und Index der Lösungen linearer Differentialgleichungen mit rationalen Koeffizienten

Let z=∞ be an irregular singular point of the differential equation wⁿ+p_n?1(z)w^(n?1)+...+p₀(z)w=0 with rational coefficients. The functions of the canonical set of solutions relative to z=∞ are of the form $$w(z) = z^\rho \cdot \sum { d_m (z) (\log z)^m , } \rho \varepsilon \mathbb{C}$$ with univalent functions d_m(z) in a neighbourhood of z=∞. Let λ(w)=max {λ(d_m)} denote the maximal order of growth of an irregular solution relative to z=∞, then it is shown that there exists a branch of w in the plane cut along a half ray, which attains the maximal order λ(w). An important tool for the proof is the index of the branches of w.     

The Distortion Theorems for Harmonic Mappings with Negative Coefficient Analytic Parts

Some sharp estimates for coefficients, distortion and the growth order are obtained for harmonic mappings $f ∈ TL^α_H$which are locally univalent harmonic mappings in the unit disk $\mathbb{D}:=\{z:|z| < 1\}$. Moreover, denoting the subclass $TS^α_H$ of the normalized univalent harmonic mappings, we also estimate the growth of $|f|,$ $f ∈ TS^α_H,$ and their covering theorems.     

由卷积和从属关系刻画的单叶调和函数某些子类

Let SH be the class of functions f = h + ˉg that are harmonic univalent and sensepreserving in the open unit disk U = {z ∈ C : |z| 1} for which f(0) = f′(0)-1 = 0. In the present paper, we introduce some new subclasses of SH consisting of univalent and sensepreserving functions defined by convolution and subordination. Sufficient coefficient conditions,distortion bounds, extreme points and convolution properties for functions of these classes are obtained. Also, we discuss the radii of starlikeness and convexity.     

The Distortion Theorems for Harmonic Mappings with Negative Coefficient Analytic Parts

Some sharp estimates for coefficients, distortion and the growth order are obtained for harmonic mappings $f \in TL^{\alpha}_H,$ which are locally univalent harmonic mappings in the unit disk $\mathbb{D}:=\{z:|z|<1\}.$ Moreover, denoting the subclass $TS^{\alpha}_H$ of the normalized univalent harmonic mappings, we also estimate the growth of $|f|,$ $f \in TS^α_H,$ and their covering theorems.     

On the Analytic Part of Univalent Harmonic Mappings

In this article we obtain two sharp results concerning the analytic part of harmonic mappings \(f=h+\overline{g}\) from the class \(\mathcal {S}^0_H(\mathcal {S})\) which was recently introduced by Ponnusamy and Sairam Kaliraj. For example, we get the sharp estimate for \(|\arg h'(z)|\) in the case when \(|z| \le 1/\sqrt{2}\) and obtain the sharp radius of convexity for h. Our approach is applicable to a more general situation. Finally, we determine simple condition on the analytic part of univalent harmonic mappings so that it is in \(H_p\) spaces for \(0<p<1/3\).     

The value regions of initial coefficients in a certain class of meromorphic functions

Let M_k,λ(0≤λ≤1, k≥2) be the class of functions f(z)=1/z+a_o+a₁z+... that are regular and locally univalent for 0<⩛z⩛<1 and satisfy the condition where J_λ(z)=λ(1+zf″(z)/f'(z))+(1-λ)zf'(z)/f(z). In the class M_k,λ we consider sorne coefficient problems and problems concerning distortion theorems. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 212, 1994, pp. 91–96. Translated by N. Yu. Netsvetaev.     

关于方程f″+Af′+ Bf=0解的增长性,其中系数A是一个二阶线性微分方程的解

设A(z)是方程f″+P(z)f=0的非零解,其中P(z)是n次多项式,B(z)是一个超越整函数且满足ρ(B)≤1/2,那么方程f″+Af′+Bf =0的每一个非零解都是无穷级.并且方程f″+A(z)f=0两个线性无关解乘积的零点序列收敛指数为无穷.