C~n中单位多圆柱上一类B型α次准凸映射齐次展开式各项的精确估计

本文给出Cn中单位多圆柱上一类B型α次准凸映射f(z)齐次展开式各项的精确估计,其中f(z)=(f1(z),f2(z),...,fn(z))T是k折对称映射(或z=0是f(z)-z的k+1阶零点),且满足sup∥z∥=1,∥w∥=1∥Dmf(0)(zm-1,w)∥=sup∥z∥=1∥Dmf(0)(zm)∥,m=2,3,...所得到的估计包含已有文献的许多结论.     

一类分担有理函数的亚纯函数的唯一性

利用复分析的值分布理论研究了亚纯函数的唯一性,给出了下面的结果.设q(z)为k次有理函数,f(z)和g(z)是两个超越亚纯函数,fg与q没有共同的极点.n是正整数且n≥max{11,k+1}.如果f~n(z)f′(z),g~n(z)g′(z)分担有理函数q(z)CM,则f(z)=c_1e~(c∫q(z)dz),g(z)=c_2e~(-c∫q(z)dz),这里c_1,c_2和c是三个常数且满足(c_1c_2)~(n+1)c~2=-1;或者f(z)≡tg(z),其中t是一个常数满足t~(n+1)=1.     

THE FEKETE-SZEG PROBLEM FOR CLOSE-TO-CONVEX FUNCTIONS WITH RESPECT TO THE KOEBE FUNCTION

An analytic function f in the unit disk D := {z ∈ C : |z| 1}, standardly normalized, is called close-to-convex with respect to the Koebe function k(z) := z/(1-z)2, z ∈ D, if there exists δ∈(-π/2, π/2) such that Re eiδ(1- z)2f′(z) 0, z ∈ D.For the class C(k) of all close-to-convex functions with respect to k, related to the class of functions convex in the positive direction of the imaginary axis, the Fekete-Szeg¨o problem is studied.     

分担值与具有重零点的亚纯函数正规族

主要证明了:设k≥2是一个正整数,M是一个正数,c是一个非零有穷复数.F是区域D内的一族亚纯函数,其中每个函数的零点的重数至少是k.若对于F中的任意函数f,f(z)=0f^((k))(x)=0,f((k))(x)=0,f^((k))(z)=c■|f((k))(z)=c■|f^((k+1))(z)|≥M,则F在D内正规,其中c≠0是必需的.     

THE FEKETE-SZEG ¨O PROBLEM FOR CLOSE-TO-CONVEX FUNCTIONS WITH RESPECT TO THE KOEBE FUNCTION

An analytic function f in the unit disk D ：= {z ∈ C ： ｜z｜ 〈 1}, standardly normalized, is called close-to-convex with respect to the Koebe function k（z） ：= z/（1-z）2, z ∈ D, if there exists δ ∈ （-π/2,π/2） such that Re {eiδ（1-z）2f′（z）} 〉 0, z ∈ D. For the class C（k） of all close-to-convex functions with respect to k, related to the class of functions convex in the positive direction of the imaginary axis, the Fekete-Szego problem is studied.     

同时求解f(x)零点的一种迭代解法

1　引　　言在许多实际问题中 ,常常会遇到求解非线性方程 f( x) =0的根 ,或称为求函数 f( x)的零点 .此时 f( x) =( x-α) μg( x) ,且 g(α)≠ 0 ,μ为大于零的常数 ,称为零点α的根指数 .当 f( x)为 n次多项式 ,设 δ(l)k =-f( z(l)k ) /f′( z(l)k ) ,牛顿修正量迭代解法为z(l+1 )k =z(l)k +δ(l)k /( 1 +δ(l)k ni=1 ,i≠ k1z(l)k -z(l)i) ,　　 k =1 ,2 ,… ,n,　 l =0 ,1 ,2 ,… ( 1 )当所有根为单根时 ,迭代法收敛 ,且收敛阶为 3阶 (见 [1 ] ,[2 ] ,[3 ] ,[4 ] ) .当 f ( x)为 n次多项式 ,所有互不相同的根为 r1 ,r2 ,… ,rm,对应…     

解析函数与Lipschitz条件

研究了解析函数与Lipschitz条件,得到了如下两个结果:(i)设D是一平面区域,f(z)在D中解析,00,对任意z∈D有|f′(z)|≤md(z,D)k-1,则f∈Lipk(D)且‖f‖k≤cmk,其中c=c(D)是仅与D有关的常数.     

Periodic points and normal families concerning multiplicity

In 1992,Yang Lo posed the following problem:let F be a family of entire functions,let D be a domain in C,and let k 2 be a positive integer.If,for every f∈F,both f and its iteration f~khave no fixed points in D,is F normal in D?This problem was solved by Ess′en and Wu in 1998,and then solved for meromorphic functions by Chang and Fang in 2005.In this paper,we study the problem in which f and f~(k ) have fixed points.We give positive answers for holomorphic and meromorphic functions.(I)Let F be a family of holomorphic functions in a domain D and let k 2 be a positive integer.If,for each f∈F,all zeros of f(z)-z are multiple and f~khas at most k distinct fixed points in D,then F is normal in D.Examples show that the conditions"all zeros of f(z)-z are multiple"and"f~k having at most k distinct fixed points in D"are the best possible.(II)Let F be a family of meromorphic functions in a domain D,and let k 2 and l be two positive integers satisfying l 4 for k=2 and l 3 for k 3.If,for each f∈F,all zeros of f(z)-z have a multiplicity at least l and f~khas at most one fixed point in D,then F is normal in D.Examples show that the conditions"l 3for k 3"and"f~k having at most one fixed point in D"are the best possible.     

关于近于凸函数类的某些子类

§1 引言设 f(z)在单位圆盘 E={z∶|z|<1}内解析,f(0)=1-f′(0)=0,其全体记作 A.用S~*,S~*(β)(β≤1),K 与 C 表示 A 的子类,类中函数在 E 内分别是星象的(关于原点),β级星象的,凸象的与近于凸的.函数 f(z)∈A 是β(β≤1)级预星象的(prestarxlike)当且仅当z/((1-z)~(2(1-β)))*f(z)∈S~*(β),若β<1;Re(f(z))/z>1/2(z∈E),若β=1,这里运算*表示两解析函数的 Hadamard 乘积(卷积).β级预星象函数类记作 R(β).显物 R(0)=K,R(1/2)=S~*(1/2).给定实数λ>-1,用 D~λ(z)=z/((1-z)~(λ+1))*f(z)定义算子 D~λ,这里 f(z)∈A.设 α≥0,0≤β<1,k 为正整数,又设解析函数 h(z)在 E 内是凸象单叶的,h(0)=1,Reh(z)>β     

分担值与正规族

设F是平面区域D上的亚纯函数族,a,b是两个有穷非零复数.如果(A)f∈F,f(z)=a(=)f(k)(z)=a,f(k)(z)=b(=)f(k+1)(z)=b,且f-a的零点重数至少为k(k≥3),那么函数族F在D内正规;当k=2时,在条件a≠4b的情况下,同样有函数族F在D内正规.     

Nonclassical weighted estimates of certain Calderónzygmund operators on the plane

Let F be a compact subset of ? let μ be a Borel measure on F, and let ρ(z) be the distance of z to F. Denote $$A_K (f)(z) = \int\limits_F {K(\varsigma ,z)f(\varsigma ) dm(\varsigma ), z \in \mathbb{C}\backslash F}$$ where K (ζ, z) is either (ζ-z)² or (|ζ-z|(ζ-z))^-1 and m is the Lebesgue measure. Let ψ be a monotone nondecreasing positive function on (0, ∞) and let Φ(z)=Ψ(ρ(z))ρ(z), z ε ?/F. Under some additional assumptions on μ, it is proved that A_K is bounded from L² (μ) to L² (Φm) if and only if $$\int\limits_0^{ + 1} {\tfrac{{\psi (t)}}{t} + \int\limits_1^\infty {\tfrac{{\psi (t)}}{{t^2 }}dt< \infty } }$$ Thus, no interference of values of K of various signs is observed in such a situation. Bibliography: 4 titles.     

典型實照函數的最小?妥钚∑

<正> 1.如果函數f(z)在包含實軸土某一區間的區域B中是正則的,f(z)在此實軸區間上取實值.在區域B的其餘地方f(z)與(z)同符號;即     

Non-real Zeros of Derivatives of Real Entire Functions and the Pólya-Wiman Conjectures

A function f is in the class $ V_2p $ iff $ f(z) = e^{-az^{2p+2}}g(z) $ where a S 0 and g is a constant multiple of a real entire function of genus h 2 p + 1 with only real zeros. The class $ U_2p $ is defined as follows: $ U_0 = V_0 $ , $ U_{2p} = V_{2p}-V_{2p-2} $ . Functions in the class $ U_{2p}^{*} $ are represented as $ g(z) = c(z)f(z) $ where $ f\in U_{2p} $ and c is a real polynomial with no real zeros. Every real entire function g , of finite order with at most finitely many non-real zeros satisfies $ g\in U_{2p}^{*} $ for a unique p . We show the exact number of non-real zeros of f" , for $ f\in U_{2p} $ , in terms of the number of non-real zeros of f' and a geometrical condition on the components of Im Q ( z ) > 0, where $ \displaystyle Q(z) = z-({f(z)}/{f'(z)}) $ . Further, for a subclass of $ f\in U_{2p} $ , we show necessary and sufficient conditions for f" to have exactly 2 p non-real zeros. For a subclass of $ U_{2p}^{*} $ we show that if f' has only real zeros, then f" has exactly 2 p non-real zeros. For $ f\in U_{2p}^{*} $ we show that 2 p is a lower bound for the number of non-real zeros of $ f^{(k)} $ for k S 2.     

高阶亚纯函数系数微分方程的解及其导数的不动点

研究了高阶线性微分方程f~(k)+A_(k-1)(z)f~(k-1)+…+A_1(z)f′+A_0(z)f=0的非零解f,及其一阶、二阶导数,f~(i)(i=1,2)的不动点性质,这里A_j(z)(j=0,1,…k-1)为亚纯函数,得到了若δ(∞,A_0)>0,且满足max{i(A1),i(A2),…,i(A_(k-1))}相似文献   

典型實照的奇函數

<正> 1.設函數f(z)在單位圓|z|<1是正則的,f(0)=0,f’(0)=1;且在此單位圓內,當(z)≠0時,記此種典型實照函數的全體為T_r. 戈魯淨證明T_r中的函數f(z)在單位圓|z|<1內,可以用司帝皆積分     

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

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

Universal sums of three quadratic polynomials

Let a,b,c,d,e and f be integers with a≥ c≥ e> 0,b>-a and b≡a(mod 2),d>-c and d≡c(mod 2),f>-e and f≡e(mod 2).Suppose that b≥d if a=c,and d≥f if c=e.When b(a-b),d(c-d) and f(e-f) are not all zero,we prove that if each n∈N={0,1,2,...} can be written as x(ax+b)/2+y(cy+d)/2+z(ez+f)/2 with x,y,z∈N then the tuple(a,b,c,d,e,f) must be on our list of 473 candidates,and show that 56 of them meet our purpose.When b∈[0,a),d∈[0,c) and f∈[0,e),we investigate the universal tuples(a,b,c,d,e,f) over Z for which any n∈N can be written as x(ax+b)/2+y(cy+d)/2+z(ez+f)/2 with x,y,z∈Z,and show that there are totally 12,082 such candidates some of which are proved to be universal tuples over Z.For example,we show that any n∈N can be written as x(x+1)/2+y(3y+1)/2+z(5z+1)/2 with x,y,z∈Z,and conjecture that each n∈N can be written as x(x+1)/2+y(3y+1)/2+z(5z+1)/2 with x,y,z∈N.     

一族特殊的星像函數

<正> 設函數w=f(z)在單位圓|z|<1中是正則的.f(0)=0,f′(0)=1.假如f(z)是單葉的,那末w=f(z)映照|z|<1於w平面上的單葉的像D_f.記這種單葉函數的全體為S.若D_f以原點w=0為星形中心,就稱f(z)是|z|<1中的星     

关于CM分担四个公共小函数的亚纯函数结论的改进

关于CM分担四个公共小函数的亚纯函数结论，我们在考虑重值的条件下，改进了李平和杨重骏^[2]的结论：设f(z)与g(z)为非常数亚纯函数，aj(z)(j=1,2,….4)为f(z)与g(z)的四个判别的小函数，若f(z)与g(z)满足Ek)(aj,f)=Ek)(aj,g),(j=1,2,3,4)且k(≥15）是正整数，则f(z)是g(z)的拟分式线性变换。即:存在f(z)与g(z)的四个小函数a(z),b(z),c(z),d(z),使得f=ag b/cg d(ad-bc≠0），（亦称Quasi-Mobuys变换）。