角域内代数体函数的唯一性定理

Let W (z) and M(z) be v-valued and k-valued algebroidal functions respectively,(θ) be a b-cluster line of order ∞ (or ρ(r)) of W (z) (or M(z)).It is shown that W (z) ≡ M(z) provided E(a j ,W (z)) = E(a j ,M(z)) (j = 1,...,2v + 2k + 1) holds in the angular domain Ω(θ- δ,θ + δ),where b,a j (j = 1,...,2v + 2k + 1) are complex constants.The same results are obtained for the case that (θ) is a Borel direction of order ∞ (or ρ(r)) of W (z) (or M(z)).     

THE BOREL DIRECTIONS OF ALGEBROIDAL FUNCTION AND ITS COEFFICIENT FUNCTIONS

In this article, the relationship between the Borel direction of algebroidal function and its coefficient functions is studied for the first time. To begin with, several theorems of algebroidal functions in unit disk are proved. By these theorems, some interesting conclusions are obtained.     

Normal Functions Concerning Shared Values

In this paper we discuss normal functions concerning shared values. We obtain the follow result. Let F be a family of meromorphic functions in the unit disc A, and a be a nonzero finite complex number. If for any f ∈F, the zeros of f are of multiplicity, f and f′ share a, then there exists a positive number M such that for any f∈F1（1-｜z｜^2） ｜f′（z）｜/1＋｜f（z）｜^2≤M.     

TWO ENTIRE FUNCTIONS SHARING FOUR SMALL FUNCTIONS IN THE SENSE OF －↑Ek（β，f）=－↑E）（β，g）

this paper proves a result that if two entire functions f(z) and g(z) share four small functions aj(z) (j=1,2,3,4) in the sense of -↑Ek(aj,f)=-↑E)(aj,g),(j=1,2,3,4)(k≥11),then there exists f(z)≡g(z).     

含零点弧的双解析函数在极点处的性质

In this paper, the properties of bianalytic functions w（z） = z^-Ф1（z） ＋Ф2（z） with zero arc at the pole z = 0 are discussed. Some conditions under which there exists an arc γ, an end of which is z = 0, such that w（z） =0 for arbitary z ∈γ／{0} are given. Secondly, that the limit set of w（z） is a circle or line as z → 0 is proved in this case. Finally, two numerical examples are given to illustrate our results.     

ALGEBROIDAL FUNCTION AND ITS DERIVED FUNCTION IN UNIT CIRCULAR DISC

In this article,the authors define the derived function of an algeboidal function in the unit disc,prove it is an algabriodal function,and study the order of algebroidal function and that of its derived function in unit circular disc.     

Properties on solutions of some q-difference equations

We consider the properties on solutions of some q-difference equations of the form ∑ n j=0 aj（z）f（qj z）=an＋1（z）, where a0（z）,..., an＋1（z） are meromorphic functions, a0（z）an（z） ≠ 0 and q ∈ C such that 0 〈 ｜q｜ ≤ 1. We give estimates on the upper bound for the length of the gap in the power series of entire solutions of （＊） when the coefficients a0（z）,..., an＋1（z） are polynomials and 0 〈 ｜q｜ 〈 1. For some special cases, we give estimates of growth of f（z）. And we also show that the case 0 〈 ｜q｜ 〈 1 is different from the case ｜q｜=1.     

单位圆内代数体函数的唯一性(英文)

In this paper, discussed are the problems about uniqueness of algebroidal functions in the unit disc with share-values in a sector domain instead of the whole disk. Results are obtained extending some uniqueness theorems of meromorphic functions.     

ON THE ORDER OF SUMMABILITY OF THE FOURIER INVERSION FORMULA

In this article we show that the order of the point value, in the sense of Lojasiewicz, of a tempered distribution and the order of summability of the pointwise Fourier inversion formula are closely related. Assuming that the order of the point values and certain order of growth at infinity are given for a tempered distribution, we estimate the order of summability of the Fourier inversion formula. For Fourier series, and in other cases, it is shown that if the distribution has a distributional point value of order k, then its Fourier series is e.v. Cesaro summable to the distributional point value of order k＋1. Conversely, we also show that if the pointwise Fourier inversion formula is e.v. Cesaro summable of order k, then the distribution is the （k ＋ 1）-th derivative of a locally integrable function, and the distribution has a distributional point value of order k ＋ 2. We also establish connections between orders of summability and local behavior for other Fourier inversion problems.     

Complex and p-Adic Meromorphic Functions f′P′( f ),g′P′(g) Sharing a Small Function

Let K be a complete algebraically closed p-adic field of characteristic zero.We apply results in algebraic geometry and a new Nevanlinna theorem for p-adic meromorphic functions in order to prove results of uniqueness in value sharing problems, both on K and on C. Let P be a polynomial of uniqueness for meromorphic functions in K or C or in an open disk. Let f, g be two transcendental meromorphic functions in the whole field K or in C or meromorphic functions in an open disk of K that are not quotients of bounded analytic functions. We show that if f′P′( f) and g′P′(g) share a small function α counting multiplicity, then f = g, provided that the multiplicity order of zeros of P′satisfy certain inequalities. A breakthrough in this paper consists of replacing inequalities n ≥ k+2 or n ≥ k+3 used in previous papers by Hypothesis(G). In the p-adic context, another consists of giving a lower bound for a sum of q counting functions of zeros with(q-1) times the characteristic function of the considered meromorphic function.     

限定Borel点的亚纯函数

设E是任意一个非空的闭实数集(mod 2π),ρ(θ)是E上一个上半连续的有界正值函数(0<ρ(θ)相似文献   

On Weighted Estimates of Solutions for Nonlinear Higher Order Elliptic Problems

In the domain D = z m F it is considered a degenerate nonlinear higher order elliptic equation such that the corresponding energetic space is W 1, q (D, w q ) 7 W m , p ( D , w p ), mp < q < n , and w q , w p are weighted functions from some Muckenhoupt classes. A behavior of a solution u ( x ) is studied for the equation under consideration with the boundary value condition , where and f ( x ) = 1 for x ] F . The pointwise estimate for u ( x ) is proved in terms of the weighted higher order capacity of the set F and the distance from the point x to the set F .     

开拓劳宾生和戈鲁辛的几个定理

<正> 1.设 p 次对称函数(?)在单位圆|z|<1中是正则的单叶的,此种函数的全体成一函数族 S_p.当p=1时,简讯 S_1为 S.设ω=f(z)∈S_p 映照|z|<1于 W 面上时,其像关于原点成星形,此种 f(z)成 S_p 之一子族S_p.设 f(z)∈S_p,     

关于p叶近于凸性的充分条件

Let Ap(n)(p, n∈N={1,2,...}) denote the class of functions of the form f(z)=zp+ap+nzp+n+... which are analytic in the unit disc E={z:|z|＜1}. By using the method of differential subordinations we give some sufficient conditions for a function f(z)∈Ap(n) to be a certain subclass Rp(n,k) of p-valently close-to-convexity functions.     

函数集的完全性

1.:敲g(x)篇〔一二,二]上之非降的有界缝差两数,业具有性鬓(K)s‘二一0,一。(:);f--:.,。g。尹(:)!d:一郁匕,(‘一”,”;dg)篇在〔一二,司上定羲业且满足修件:,一{户,(柳dg(·)}青<一,>l的可测蝮值函数族{f(幻}.封龄一徊乙“(一二,侧d刃中之子族凌B(幻},若由f(劣)(乙,(一二,二:dg),夕>1生+上夕q=1,及f--:ha”“’“““’一0纷{B(x)}之任何B(哟成立必滇致f(幻在〔一二,司上规乎虚虚等焚零则释{B(x)}在乙“(一二,侧dg)中完全. 函数族的完全性是舆函数横造的一些简题很有阴保的.徙【l]我们知道{e‘”}豁。是在乙,(一二,州dg),,>1,中完全的,…     

Über das Anwachsen der Lösungen linearer Differentialgleichungen mit rationalen Koeffizienten in Winkelräumen

Let z=∞ be the only irregular singular point of the linear differential equation $$(D)w^{(n)} + P_{n - 1} (z)w^{(n - 1)} + \cdots + P_o (z)w = 0$$ with rational coefficients p_j(z). If w is a multivalent and irregular solution of (D), we define the index I(τr,W), τ ∈ (0,1), of a branch W of w in the plane cut along a half ray. If r→∞, I(τr,W) possesses a finite number of aymptotic directions being exactly the asymptotic directions of the points z with maximal possible modulus of W(z). It follows that each branch W is of mean type σ(W)=|d|/λ in each sector containing an asymptotic direction of I. The possible values of the order of growth λ=λ(W)<∞ and the constant d are given by the PUISEUX-diagram of (D).     

关于辐角估计的注记

设N为在单位圆盘U内解析且满足p(0)=1的函数的全体.本文将给出p(z)在圆|z|=r<1的边界上的辐角极值点处的一个性质,所得结果推广了前人关于辐角估计的结论,并给出这一结果的一些应用.     

线性可移算子与可调和的随机过程

<正> §1.引言T.Onoyama 利用了 Weyl-Stone-Titchmarsh 的特征函数(Eigenfunction)展开公式,对具有二阶矩的实的连续机过程(本文中所用的极限,系指在均方意义下的极限)求得了下列隨机函数方程     

交换的实W~*-代数

本文指出，交换的（复）Ｗ 代数的大部分结果对于实情形仍然成立，自然有周期为２同胚的差别．此外，不同于复情形，可分实Ｈｉｌｂｅｒｔ空间中不包含任何极小投影的交换实ＶＮ代数同构于Ｌ∞（［０，１］），Ｌ∞ｒ（［０，１］）或Ｌ∞（［０，１］）Ｌ∞ｒ（［０，１］）．