$C$上和$\Delta$内某类高阶线性微分方程解的增长性

本文分别在复平面$\mathbb{C}$上和单位圆$\Delta$内考虑方程$$f^{(k)}+A_{k-1}(z)f^{(k-1)}+\cdots+A_1(z)f''+A_0(z)f=0$$的解的增长性与其系数的增长性之间的关系.当$A_0(z)$或某个$A_j(z)(0相似文献   

非齐次线性微分方程解的增长性

研究了非齐次线性微分方程f^{(k)}+A_{k-1}(z)f^{(k-1)}+...+A_{s}(z)f^{(s)}+...+A_{0}(z)f=F(z) 解的增长性,其中A_{j}(j=0,1,\cdots,k-1)及F是整函数. 在A_{s}比其他系数有较快增 长的情况下,得到了上述非齐次微分方程在一定条件下的超越整函数解的超级的精确估计.     

高阶线性微分方程解的零点分布及Zygmund-型空间

本文主要考虑以下两个问题: (1) 建立非齐次线性微分方程$$f''+A_2(z)f''+A_1(z)f''+A_0(z)f=A_3(z),$$ 系数增长性与解的零点的几何分布的相互关系, 其中 $A_0(z),\ldots, A_3(z)$为单位圆内的解析函数; (2) 找到一些使方程$$f^{(k)}+A_{k-1}(z)f^{(k-1)}+\cdots+A_1(z)f''+A_0(z)f=0,$$ 所有解属于Zygmund-型空间的充分条件. 我们得到的结果推广了Heittokangas, Gr\"{o}hn, Korhoneon 和 R\"{a}tty\"{a}的部分结果.     

亚纯函数的导函数分担集合的唯一性

研究了两个亚纯函数的导函数分担两个值集的唯一性问题. 证明了对于集合$S$和$T$, 如果$\overline{E}_{f^{(k)}}(S)=\overline{E}_{g^{(k)}}(T)$, 则存在非零常数$A$, 使得$f^{(k)}=Ag^{{(k)}}$. 特别地, 当$k=0$时, 有$f=Ag$.     

一类高阶周期微分方程解的性质

本文研究了高阶线性微分方程$$f^{(k)}(z)+A_{k-2}(z)f^{(k-2)}(z)+\cdots+A_0(z)f(z)=0,\eqno(*)$$解的线性相关性,其中$A_j(z)(j=0,2,\ldots,k-2)$是常数, $A_1$为非常数的的整周期函数,周期为$2\pi i$,且是$e^z$的有理函数.在一定条件下,我们给出了方程(*)解的表示.     

某类$q$差分方程亚纯解的性质

对于一个有穷非零复数$q$, 若下列$q$差分方程存在一个非常数亚纯解$f$, $$f(qz)f(\frac{z}{q})=R(z,f(z))=\frac{P(z,f(z))}{Q(z,f(z))}=\frac{\sum_{j=0}^{\tilde{p}}a_j(z)f^{j}(z)}{\sum_{k=0}^{\tilde{q}}b_k(z)f^{k}(z)},\eqno(\dag)$$ 其中 $\tilde{p}$和$\tilde{q}$是非负整数, $a_j$ ($0\leq j\leq \tilde{p}$)和$b_k$ ($0\leq k\leq \tilde{q}$)是关于$z$的多项式满足$a_{\tilde{p}}\not\equiv 0$和$b_{\tilde{q}}\not\equiv 0$使得$P(z,f(z))$和$Q(z,f(z))$是关于$f(z)$互素的多项式, 且$m=\tilde{p}-\tilde{q}\geq 3$. 则在$|q|=1$时得到方程$(\dag)$不存在亚纯解, 在$m\geq 3$和$|q|\neq 1$时得到方程$(\dag)$解$f$的下级的下界估计.     

The Lebesgue Constants and Gibbs Phenomenon for Non-negative (f , dn) Summability Method

The ( f,d_n) -summability method is defined as follows^[1,4]: Let f be a nonconstant function, analytic in |z | < R for R > l, and let {d_n} be a sequence of complex numbers,such that for all n,$d_n \ne -f(1)$.Suppose that the elements of the metrix A = (a_nk) are given by the relations $a_00=1,a_0k=0(k \geq 1)$ $[\prod\limits_{j = 1}^n {\frac{{f(z) + {d_j}}}{{f(1) + {d_j}}} = \sum\limits_{k = 0}^\infty {{a_{nk}}{z^k}} } \]$ A sequence {S_n} is said to be ( f, d_n), —summable to s, if \sigma_n = \sum\limits_{k=0}^\infty \arrow s as n \arrow \infty. The ( f, d_n) —summability method is said to be non-negative if for all n, d_n> 0 and the Maclaurin coefficients of f are real and non-negative. The Lebesgue constants for the ( f,d_n)-method are defined by $L_n(A)=2/\pi \int_0^\pi /2 {\frac{|\sum\limits_{k=0}^\infty {a_nk sin(2k+1)t|}{sint}dt}$ In this parer we prove the following two theorems.     

一类高阶线性微分方程解的增长性

本文研究了在Aj(z),aj(j=0,1,…,k-1)满足一些条件下方程f(k)+Ak-1(z)eak-1f(k-1)+…+A0(z)ea0zf=0解的超级和在Aj(z),Pj(j)(j=0,1,…,k-1)满足一些条件下方程f(k)+Ak-1(z)ePk-1(z)f(k-1)+…+Aj(z)eajzf(j)+…+A0(z)eP0(z)f=0解的级。     

一类复线性微分差分方程亚纯解的唯一性

本文主要研究一类复线性微分差分方程超越亚纯解的唯一性.特别地,假设$f(z)$为复线性微分差分方程: $W_{1}(z)f''(z+1)+W_{2}(z)f(z)=W_{3}(z)$的一个有穷级超越亚纯解,其中$W_{1}(z)$, $W_{2}(z)$, $W_{3}(z)$为增长级小于1的非零亚纯函数并且满足$W_{1}(z)+W_{2}(z)\not\equiv 0$.若$f(z)$与亚纯函数$g(z)$, $CM$分担0,1,$\infty$,则$f(z)\equiv g(z)$或$f(z)+g(z)\equiv f(z)g(z)$或$f^{2}(z)(g(z)-1)^2+g^{2}(z)(f(z)-1)^2=g(z)f(z)(g(z)f(z)-1)$或存在一个多项式$\varphi(z)=az+b_{0}$使得$f(z)=\frac{1-e^{\varphi(z)}}{e^{\varphi(z)}(e^{a_{0}-b_{0}}-1)}$与$g(z)=\frac{1-e^{\varphi(z)}}{1-e^{b_{0}-a_{0}}}$,其中$a(\neq 0)$, $a_{0}$ $b_{0}$均为常数且$a_{0}\neq b_{0}$.     

Properties of Complex Oscillation of Solutions of a Class of Higher Order Linear Differential Equations

Let A(z) be an entire function with μ(A) 1/2 such that the equation f~((k))+A(z)f = 0, where k ≥ 2, has a solution f with λ(f) μ(A), and suppose that A_1 = A+h,where h■0 is an entire function with ρ(h) μ(A). Then g~((k))+ A_1(z)g = 0 does not have a solution g with λ(g) ∞.     

一类整函数系数微分方程解的复振荡

本文研究了微分方程f~(k) A_((k-1))f~((k-1)) … A_0f=F(k≥2)解的增长级和零点收敛指数,其中A_j=B_je~(P_j),j=0,1,…,k-1,B_j(z)为整函数,P_j(z)为多项式,σ(B_j)＜degP_j．     

一类高阶线性微分方程解的复振荡

In this paper,we shall use Nevanlinna theory of meromorphic functions to investigate the complex oscillation theory of solutions of some higher order linear differential equation.Suppose that A is a transcendental entire function with ρ(A)<1/2.Suppose that k≥2 and f(k)+A(z)f=0 has a solution f with λ(f)<ρ(A),and suppose that A1=A+h,where h≡0 is an entire function with ρ(h)<ρ(A).Then g(k)+A1(z)g=0 does not have a solution g with λ(g)<∞.     

系数的级相同的高阶微分方程解的增长性

In this paper,we consider the growth of solutions of some homogeneous and nonhomogeneous higher order differential equations.It is proved that under some conditions for entire functions F,A_(ji) and polynomials P_j(z),Q_j(z)(j=0,1,…,k-1;i=1,2)with degree n≥1,the equation f~(k)+(A_(k-1,1)(z)e~(p_(k-1)(z))+A_(k-1,2)(z)e~(Q_(k-1(z)))/~f~(k-1)+…+(A_(0,1)(z)e~(P_o(z))+A_(0,2)(z)e~(Q_0(z)))f=F,where k≥2,satisfies the properties:When F ≡0,all the non-zero solutions are of infinite order;when F=0,there exists at most one exceptional solution fo with finite order,and all other solutions satisfy λ(f)=λ(f)=σ(f)=∞.     

THE ZEROS AND ORDER OF MEROMORPHIC SOLUTIONS OF f^（k）＋Bf=H（z）

1.IntroductionandResultsConsidernon-homogeneouslineardifferentialequationsoftheform1.Lameprovedin[7]TheoremA.LetB(z),PO(z),PI(z)*06epolynomialssuchthatdegB=n21,degPO=p<(n k)/kandH=PI(z)epo('),then(a)IfdegPI相似文献   

高阶线性微分方程解的角域增长性

假设A0,A1,…,Ak-1在某个角域内解析,讨论高阶线性微分方程,f(k) Ak-1f (k-1) … A1f' A0f=0在特定角域内解的增长性和渐近性,改进了一些结果.     

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.     

涉及分担集的整函数的唯一性

In this paper,uniqueness of entire function related to shared set is studied.Let f be a non-constant entire function and k be a positive integer,d be a finite complex number.There exists a set S with 3 elements such that if f and its derivative f（k）satisfy E（S,f）= E（S,f（k））,and the zeros of f（z）-d are of multiplicity ≥ k ＋ 1,then f = f（k）.     

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■ff∈F,f(z)=a■f~((k))(z)=a,ff~((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.