正规族与复合亚纯函数

本文研究了亚纯函数族涉及复合有理函数与分担亚纯函数的正规性. 证明了一个正规定则:设 α(z) 和 F 分别是区域 D 上的亚纯函数与亚纯函数族, R(z) 是一个次数不低于 3 的有理函数.如果对族 F 中函数 f(z) 和 g(z), R○f(z) 和 R○g(z) 分担 α(z) IM,并且下述 条件之一成立:
(1) 对任何 z₀ ∈ D, R(z)-α(z₀) 有至少三个不同的零点或极点;
(2) 存在 z₀ ∈ D 使得 R(z)-α(z₀):=(z-β₀)^pH(z) 至多有两个零点(或极点) β₀,同时 k ≠ l|p|,其中 l 和 k 分别是 f(z)-β₀ 和 α(z)-α(z₀) 在 z₀ 处的零点重数, H(z) 是满足 H(β₀) ≠ 0, ∞ 的有理函数, α(z) 非常数并满足 α(z₀) ∈ C ∪{∞}.
那么 F 在 D 内正规.特别地,这个结果是著名的 Montel 正规定则的一种推广.     

涉及分担函数的正规性

设k为一个正整数,a(z)(■0,∞)为区域D的亚纯函数,F是区域D内的一族亚纯函数,其零点的重级至少为k.若对于任意f∈F,f(z)=0f~((k))(z)=a(z)?0|f~((k+1))(z)-a′(z)||a(z)|,则F在D内正规.     

论Szegǒ的定理

设f(z)=Z+a₂z²+…∈S.Szegǒ证明:S_n(z)=z+a₂z²+…+a_nzⁿ(n=2,3…)在|z|<1/4内单叶。ρ₀=1/4最好的,我们证明了更强的结果: 定理:若f(z)∈s.则s_n(z)(n=2,3…)在|z|<1/4内关于原点成星形。 当f∈S^*时为吴卓人所得。     

与分担全纯函数相关的正规族

设F是在区域D内的一族亚纯函数,其零点重级至少为k,k是一个正整数,a(z)(≠0)在区域D内全纯.若对于任意的f∈F,有(1)f(z)与a(z)没有公共的零点;(2)f(z)=0f(k)(z)=a(z)■0|f~((k+1))(z)-a'(x)||a(z)|,则F在D内正规.     

涉及分担值的正规定则

设F是区域D上的一个亚纯函数族,k(≥2)是一个正整数,b是一个非零复数,M是一个正数.若对任意给定的f∈F,f的零点重数至少为k,且f(z)=0=|f~((k))(z)|≤M.如果对任意给定的函数f,g∈F,L(f)与L(g)的零点都为重零点,且L(f)与L(g)在区域D内分担b,则F在区域D内正规.     

涉及分担函数的正规定则

设k为正整数,M为正数;F为区域D内的亚纯函数族,且其零点重级至少为k;h为D内的亚纯函数(h(z)≠0,∞),且h(z)的极点重级至多为k.若对任意给定的函数f∈F,f与f~((k))分担0,且f~((k))(z)-h(z)=0?|f(z)|≥M,则F在D内正规.     

亚纯函数族涉及微分单项式分担移动靶的正规定则

设a(z)是一个没有零点的整函数,k≥3是个整数,F是区域D上的亚纯函数族,对每一个f∈F至少有k重零点和2重极点.若对每一对f,g∈F有ff(k)与gg(k)IM分担a(z),则F在区域D内正规.     

涉及正规族与分担值的Hayman 问题

设n, k (n ≥ k + 3) 是两个正整数, a (≠0), b 是两个有穷复数, F 是区域D 内的一族亚纯函数,其中族中每个函数的零点都至少是k 重. 若对于F 中的任意两个函数f, g, f^(k)-afⁿ 与g^(k)-agⁿ 在D 内分担b, 则F 在D 内正规. 两个例子说明函数族中的每个函数的零点都至少是k 重以及n ≥ k+3是最佳的.     

关于亚纯函数正规族

本文对杨乐、张广厚建立的亚纯函数正规定则,作了在限制条件较少情况下定则仍成立的证明,文中主要证明了下面的定理1。 设{f(z)}为区域D内亚纯函数族,其中每个函数f(z)的极点之级≥s(≥1),f(z)取p(≥1)个互为判别的有穷值a_i(i=1,…,p)的点之级分别≥m_i(≥2),f^(k)(z) 取q(≥1)个互为判别的非零有穷值b_j(j=1,…,q)的点之级分别≥nj(≥2)。若 则亚纯函数族{f(z)}在区域D内正规。     

亚纯函数的弱斥性不动点与拟正规族

本文研究亚纯函数的弱斥性不动点与拟正规族的关系,得到了以下结果: 设F是区域D内的亚纯函数族,q是一个非负整数。如果对任意f∈F,存在自然数k=k(f)>1,使得f的k次迭代f^k在D内最多只有q个弱斥性不动点,则F是D内阶至多为max{4,q+2}的拟正规族。     

RECENT PROGRESS ON SPHERICAL HARMONIC APPROXIMATION MADE BY BNU RESEARCH GROUP ——In memory of Professor Sun Yongsheng

As early as in 1990, Professor Sun Yongsheng, suggested his students at Beijing Normal University to consider research problems on the unit sphere. Under his guidance and encouragement his students started the research on spherical harmonic analysis and approximation. In this paper, we incompletely introduce the main achievements in this area obtained by our group and relative researchers during recent 5 years （2001-2005）. The main topics are： convergence of Cesaro summability, a.e. and strong summability of Fourier-Laplace series; smoothness and K-functionals; Kolmogorov and linear widths.     

Instructions for Authors

<正>Submission Authors must use LaTeX for typewriting,and visit our website www.actamath.com to submit your paper.Our address is Editorial Office of Acta Mathematica Sinica,Academy of Mathematics and Systems Science,Chinese Academy of Sciences,Beijing 100190,P.R.China.     

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Boundedness for commutators of multipliers on Hardy type spaces

In this paper, we study the commutators generalized by multipliers and a BMO function. Under some assumptions, we establish its boundedness properties from certain atomic Hardy space Hb^p（R^n） into the Lebesgue space L^p with p 〈 1.     

A unified approach to certain problems of best local approximation

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Towards Excellence in Science Chinese Academy of Sciences

<正>May 26,2014,Beijing Science is a human enterprise in the pursuit of knowledge.The scientific revolution that occurred in the 17th Century initiated the advances of modern science.The scientific knowledge system created by     

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<正>August 10-14,2015Beijing,ChinaThe International Congress on Industrial and Applied Mathematics(ICIAM)is the premier international congress in the field of applied mathematics held every four years under the auspices of the International Council for Industrial and Applied Mathematics.From August 10 to 14,2015,mathematicians,scientists     

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