一类高阶线性微分方程解的增长性（英文）

In this paper, we investigate the growth of solutions of the differential equations f~((k))+ A_(k-1)(z)f~((k-1))+ ··· + A_0(z)f = 0, where A_j(z)(j = 0, ···, k-1) are entire functions.When there exists some coefficient A_s(z)(s ∈ {1, ···, k-1}) being a nonzero solution of f'+P(z)f = 0, where P(z) is a polynomial with degree n(≥ 1) and A_0(z) satisfies σ(A_0) ≤1/2 or its Taylor expansion is Fabry gap, we obtain that every nonzero solution of such equations is of infinite order.     

一类高阶线性微分方程解的增长性

本文讨论一类一般的齐次和非齐次高阶线性微分方程解的增长性,证明了当整函数F,A_j,D_j和s≥1次多项式P_j(z)(j=0,1,…,k-1)满足某些条件时,方程(其中k≥2),f~(k) (A_(k-1)(z)e~(P_(k-1)(z)) D_(k-1)(z))f~((k-1)) … (A_0(z)e~(P_0(z)) D_0(z))f=F当F≡0时,所有非零解具无穷级；当F≠0时,至多除去一个有限级解f_0外,其余所有解均满足■(f)=λ(f)=σ(f)=∞且σ_2(f)≤max{s,σ(F)},从而推广了M．Frei,M．Ozawa,G．Gundersen,J．K．Langley,陈宗煊,李纯红等人的结果。     

Quasi-normal Family of Meromorphic Functions Whose Certain Type of Differential Polynomials Have No Zeros

Define the differential operators ?_n for n∈N inductively by ?_1 [f](z)=f(z) and ?_(n+1) [f](z)=f(z)?_n[f](z)+d/dz ?_n[f](z).For a positive integer k≥2 and a positive number δ,let F be the family of functions f meromorphic on domain D■C such that ?_k[f](z)≠0 and |Res(f,a)-j|≥δ for all j∈{0,1,…,k-1} and all simple poles a of f in D.Then F is quasi-normal on D of order 1.     

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   

一类高阶整函数系数微分方程解的增长级、下级与超级

该文研究了一类高阶整函数系数微分方程解的增长性,对方程f~(k)+A_(k-1)(z)e~(ak-1z).f~(k-1)+…+A_0(z)e~(a0z)f=0与方程f~(k)+(A_(k-1)(z)e~(ak-1z)+D_(k-1)(z))f~(k-1)+…+(A_0(z)e~(a0z)+D_0(z))f=0中a_j(0≤j≤k-1)幅角主值不全相等的情形,得到了解的增长级、下级与超级的精确估计.     

Coefficient Estimates for bounded Nonvanishing Functions

Let dnote the class of all functions f(z)=sum from n=0 to ∞(a_nz~n)analytic and satisfying O<|f(z)|<1 in|z|<1.Denote A_n=sup|a_n|.It is easy to prove that A_0=1 and A_1=2/e.In 1968,Krzyz provedA_2=2/e and conjectured that A_n=2/e for all n≥1 and the equality was attained only for functionse~(iα)F(e~(iβ)z~n),where F(z)=exp[(z-1)/(z+1)]=1/e+(2/e)z-(2/3e)z~3+….In 1977,Hummel,Scheinberg andZalcman proved A_3=2/e.     

一类二阶线性微分方程解的增长性

本文研究一类二阶齐次线性微分方程f"+A_1(z)e~(P(z))f'+A_0(z)e~(Q(z))f=0,解的增长性,其中P(z)=az~n,Q(z)=bz~n,ab≠0,a=cb(c1),A_j(z)(j=0,1)是非零多项式,证明了该方程的每个非零解满足σ(f)=∞并且σ_2(f)=n.     

On entire solutions of some type of nonlinear difference equations

In this article, the existence of finite order entire solutions of nonlinear difference equations f~n+ P_d(z, f) = p_1 e~(α1 z)+ p_2 e~(α2 z) are studied, where n ≥ 2 is an integer, Pd(z, f) is a difference polynomial in f of degree d(≤ n-2), p_1, p_2 are small meromorphic functions of ez, and α_1, α_2 are nonzero constants. Some necessary conditions are given to guarantee that the above equation has an entire solution of finite order. As its applications, we also find some type of nonlinear difference equations having no finite order entire solutions.     

On the Growth of Solutions of a Class of Higher Order Linear Differential Equations

In this paper, we investigate the growth of solutions of the differential equations f(k)+Ak?1(z)f(k?1)+· · ·+A0(z)f =0, where Aj(z)(j=0, · · · , k?1) are entire functions. When there exists some coe?cient As(z)(s ∈ {1, · · · , k?1}) being a nonzero solution of f00+P(z)f =0, where P(z) is a polynomial with degree n(≥1) and A0(z) satisfiesσ(A0)≤1/2 or its Taylor expansion is Fabry gap, we obtain that every nonzero solution of such equations is of infinite order.     

ON OSCILLATION THEOREMS. FOR HIGHER ORDER DIFFERENTIAL EQUATIONS

In this paper,we investigate the complex oscillation of the differential equation f^(k) Ak-1f^(k-1)… A1f^1 A0f=F(x) where Af(j=0,…,k-1)F 0 are finite order entire functions. And we give precise estimates of the exponent of convergence of the zero-sequence of solutions for the above equation under some additional hypotheses.     

单位圆上解析系数的高阶复微分方程

The growth of solutions of the following differential equation ■ is studied, where A_j(z) is analytic in the unit disc D = {z : |z| 1} for j = 0, 1,..., k-1. Some precise estimates of [p, q]-order of solutions of the equation are obtained by using a notion of new[p, q]-type on coefficients.     

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)<∞.     

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.     

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．     

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.     

无限级整函数在角域内的取值和增长性

借助熊庆来的无限级,将Nevanlinna建立的有限级整函数在角域内的取值和增长性的结果推广到无限级.作为应用,研究了高阶超越整函数系数微分方程f~((k))+A_k-2(z)f~((k-2))+…+A_1(x)f'+A_0(z)f=0解的径向振荡.     

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.     

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$a_{0}\neq b_{0}$.     

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.