复微分算子下调和映照的单叶半径

设f(z)为定义在单位圆盘D上的调和映照,定义复微分算子L:=z(?)/((?)z)-z(?)/((?)z).该文在f满足系数条件(1.7)下,得到L(f)的单叶半径ρ_0如(1.9)式.进而当f为调和K-拟共形映照时,得到L(f)的单叶半径ρ_K.     

具有线性连结像域的局部单叶调和映照

研究了单位圆上具有像域线性连结性的局部单叶调和函数成为调和拟共形映照的充要条件,确定了一类具有线性连结像域的单叶调和函数的单叶调和稳定性参数区域,推广了Chuaqui和Hernández的相应结果.     

单叶调和映照的反函数

设是在一个单连通区域上的单叶调和映照，我们证明了反函数ｚ＝ｆ－１（）也是调和映照的充要条件是ｆ为下面三类函数之一：（ｉ）单叶共形映照；（ｉｉ）仿射交换映照；（ｉｉｉ）具有形式ｆ（ｚ）＝Ａ［ａｚ＋β＋ｌｏｇ（１－ｅ－ａｚ－β）－ｌｏｇ（１－ｅ－ａｚ－β）］＋Ｂ的调和映照，其中Ａ，Ｂ，α和β是常数且满足条件Ｒ（ａｚ＋β）＞０，Ｚ∈Ｄ．     

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.     

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 \(0<p<1/3\).     

A Proof of a Conjecture on the Convolution of Harmonic Mappings and Some Related Problems

Ukrainian Mathematical Journal - Recently, Kumar, et al. proposed a conjecture concerning the convolution of a generalized right half-plane mapping with a vertical strip mapping. They verified this...     

A Counterexample to Uniqueness in the Riemann Mapping Theorem for Univalent Harmonic Mappings

Let f be an orientation-preserving univalent harmonic mappingof the unit disk U. Then , where h and g are analytic in U. Furthermore, f satisfies theequation in U, where a(z)=g'(z)/h'(z), and |a(z)| < 1 in U. The functiona(z) is the analytic dilatation of f. In [2], Hengartner and Schober proved the following versionof the Riemann mapping theorem for univalent harmonic mappings.1991 Mathematics Subject Classification 31A05, 31A20.     

Univalent harmonic mappings of annuli and a conjecture of J. C. C. Nitsche

Letw=f(z) be a univalent harmonic mapping of the annulus {ρ≤|z|≤1} onto the annulus {σ≤|w|≤1}. It is shown thatσ≤1/(1+(ρ ²/2)(logρ)²).     

调和单叶函数族的一些结果

借用著名的Ruscheweyh导数,引入了一类单叶保向调和函数族. 通过建立极值理论,得到了关联该族的最优系数边界、最优增长定理和最优偏差定理. 同时,给出了该族与先前已有调和函数族之间的转换半径. 最后,讨论了基于该族的修正哈达玛乘积结果.     

On the Springer''s Conjecture of the Coefficients of Univalent Meromorphic Functions

Let $\sigma$ denote the family of univalent functions $\[F(z) = z + \sum\limits_{n = 1}^\infty {\frac{{{b_n}}}{{{z^n}}}} \]$ in l< |z| <\infty if G(w) is the inverse of a function $F(z) \in \sigma ^'$, the expansion of G(w) in some neighborhood of w=\infty is $\[G(w) = w - \sum\limits_{n = 1}^\infty {\frac{{{B_n}}}{{{w^n}}}} \]$ It is well known that |B_1|\leq 1 for any F(z) \in \sigma ^'. Springer^[1] proved that | B_3| \leq 1 and conjectured that $\[|{B_{2n - 1}}| \le \frac{{(2n - 2)!}}{{n!(n - 1)!}}{\rm{ }}(n = 3,4, \cdots )\]$ (1) Kubota^[2] proved (1) for n=3, 4, 5. Schober^[3] proved (1) for n = 6, 7. Ren Fuyao[4,5] has verified (1) for n=6, 7, 8. In this article we are going to verify (1) for n=9.     

The Schwarzian Derivative of Harmonic Mappings in the Plane

In this paper, the authors introduce a definition of the Schwarzian derivative of any locally univalent harmonic mapping defined on a simply connected domain in the complex plane. Using the new definition, the authors prove that any harmonic mapping f which maps the unit disk onto a convex domain has Schwarzian norm ■≤ 6. Furthermore, any locally univalent harmonic mapping f which maps the unit disk onto an arbitrary regular n-gon has Schwarzian norm■.     

调和函数类的一些界的估计结果北大核心CSCD

研究了单位开圆盘内保向复值的调和函数类,其中函数的解析部分满足一定的从属条件.完整的给出了该函数类Bloch常数的界、系数估计、增长和偏差不等式的界.进一步,通过指定一些关键参数,证明其pre-Schwarzian导数的范数也是有界的.     

On a Problem of the Constructive Theory of Harmonic Mappings

The problem of irremovable error appears in finite difference realization of the Winslow approach in the constructive theory of harmonic mappings. As an example, we consider the well-known Roache–Steinberg problem and demonstrate a new approach, which allows us to construct harmonic mappings of complicated domains effectively and with high precision. This possibility is given by the analytic-numerical method of multipoles with exponential convergence rate. It guarantees effective construction of a harmonic mapping with precision controlled by an a posteriori estimate in a uniform norm with respect to the domain.     

关于全纯映照模的Schwarz引理一点注记

记DC为单位圆盘,B~k C~k为开欧氏单位球,Ω是C~k(或C)中的域.记H_n(D,Ω)为满足一定条件的全纯映照族(或函数族)的全体.作者证明了若,∈Hn(D,D),则|f′(z)|≤(n|z|~(n-1))/(1-|z|~(2n))(1-|f|(z|~2),z∈DD同时,对Hn(D,B~k)中映照的模也得到类似的结果.该结论推广了Pavlovic的相应结果.     

Properties of the Mappings That Are Close to the Harmonic Mappings. II

We continue studying the mappings that are close to the harmonic mappings (-quasiharmonic mappings with small). This study originates with the previous articles of the author. The results of the article include a theorem on connection between the notion of -quasiharmonic mapping and the solutions to Beltrami systems, an analog to the arithmetic mean property of harmonic functions for -quasiharmonic mappings, a theorem on stability in the Poisson formula for harmonic mappings in the ball, and a theorem on the local smoothing of -quasiharmonic mappings with small which preserves proximity to the harmonic mappings.     

An Extension Theorem and Subclasses of Univalent Mappings in Several Complex Variables

Let B be the unit ball in C ⁿ and let U be the unit disc in C. The aim of this work is to construct a family of operators Ψ _n,α that provide a way to extend a locally univalent function ? ? H(U) to a locally univalent mapping F_α ? H(B), where α ? (0,1]. If ? is normalized univalent, then F_α can be imbedded in a Loewner chain. Also if ? ? S?, then F_α is starlike. We show that if ? belongs to a class of univalent functions which satisfy growth and distortion results, then the mapping F_α satisfies similar growth and distortion results. Also we study the concept of linear-invariant families as it relates to families generated by the operator Ψ _n,0, and we obtain in this way another example of a L.I.F. that has minimum order (n + 1)/2 and is not a subset of the normalized convex mappings in the unit ball of C ⁿ (for n ≤ 2.)     

关于正映射锥上不动点指标的一个猜想

本文在一些附加条件下证明了Dancer E.N.提出的一个关于正映射锥上不 动点指标计算的猜想,并举例说明其应用.     

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 Overfull Conjecture and the Conformability Conjecture

In this paper we show that under some fairly general conditions the Overfull Conjecture about the chromatic index of a graph G implies the Conformability Conjecture about the total chromatic number of G. We also show that if G has even order and high maximum degree, then G is conformable unless the deficiency is very small.