整函数与其差分算子分担集合的唯一性问题

考虑整函数与其差分算子分担集合的唯一性问题.假设S={ω:ω~n+aw~(n-1)+b=0},m,n为两个正整数满足n2且n和n一m互素,a和b为两个非零复数使得方程ω~n+aw~n+b=0无重根.设f为满足λ(f)ρ(f)∞的非常数整函数,若f(z)和△_cf(z)CM分担集合S,则f(z+c)≡2f(z).这个结果改进了李效敏的定理.     

一类线性差分方程的亚纯解与一个亚纯函数分担3个值的唯一性

主要研究差分方程a_1(z)f(x+1)+a_0(z)f(z)=F(z)的一个有穷级超越亚纯解f(z)与亚纯函数g(z)分担0,1,∞CM时的唯一性问题(其中a_(z),a0(z),F(z)为非零多项式,且满足a_1(z)+a_0(z)■0),得到f(x)≡g(z),或f(z)+g(z)≡f(z)g(z),或存在一个多项式β(z)=az+b_0和一个常数a_0满足e~(a_0)≠e~(b_0),使得f(z)=(1-e~(β(x)))/(e~(β(x))(e~(a_o-b_0)-1))与g(z)=(1-e~(β(x)))/(1-e~(b_o-a_0)),其中a(≠0),b_0为常数.     

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.     

Entire functions sharing some values with their difference operators

We investigate a uniqueness topic of entire functions with finite order sharing some values with their difference operators and obtain one result that if△f(≠0)and entire function f share a,b CM(counting multiplicities),then△f≡f.This result partially confirms a conjecture due to Chen and Yi(2013).     

非线性微分—差分方程的整函数解

本文考虑了非线性微分—差分方程fn(z)+q(z)eQ(z)f(k)(z+c)=p1eα1z+p2eα2z与fn(z)+q(z)eQ(z)△cf=p1eλz+p2e-λz解的增长性,其中n≥1,k≥1是两个整数,q(z)是非零多项式,Q(z)是非常数多项式.c,λ,α1,α2,p1,p2为非零常数,α1≠α2.特别地,...     

On meromorphic solutions of a certain type of nonlinear differential equations

We consider transcendental meromorphic solutions with N(r,f) = S(r,f) of the following type of nonlinear differential equations:f~n + Pn-2(f) = p1(z)e~(α1(z)) +p2(z)e~(α2(z)),where n≥ 2 is an integer, Pn-2(f) is a differential polynomial in f of degree not greater than n-2 with small functions of f as its coefficients, p1(z), p2(z) are nonzero small functions of f, and α1(z), α2(z)are nonconstant entire functions. In particular, we give out the conditions for ensuring the existence of meromorphic solutions and their possible forms of the above equation. Our results extend and improve some known results obtained most recently.     

涉及导数与差分分担值的唯一性问题

本文研究了有穷级亚纯函数的导数及其差分分担值的唯一性问题,主要证明了以下结论:如果f′与△_cfCM分担a,b,∞,则f′≡△_cf.该结论解决了Qi等人在2018年提出的问题.     

整函数及其差分的唯一性

设f是一个有穷级的超越整函数,a,b,c是3个有穷复数,满足c≠0,a≠b,且n为正整数.如果a是f的Borel例外值,且Δ_c~nf(z)与f(z)IM分担b,则f(z)=a+Ae~(Bz),其中A,B为两个非零常数.     

ON THE NORMAL FAMILY OF MEROMORPHIC FUNCTIONS

In [1],W.K.Hayman has posed the following interesting problems: (A).Let be a family of holomorphic functions in the domain D.Let p be apositive integer,p≥3; and let a,b be two finite complex numbers,a≠0. If forany function f(z)∈ satisfies the condition f’(z)-a{f(z)}~P≠b,     

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)=∞.     

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$a_{0}\neq b_{0}$.     

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.     

Meromorphic Function Sharing Sets with Its Difference Operator or Shifts

Let f be a nonconstant meromorphic function, c ∈ C, and let ■be a meromorphic function. If f(z) and P(z, f(z)) share the sets {a(z),-a(z)},{0} CM almost and share {∞} IM almost, where P(z, f(z)) is defined as(1.1), then f(z) ≡±P(z, f(z)) or f(z)P(z, f(z)) ≡±a~2(z). This extends the results due to Chen and Chen(2013), Liu(2009) and Yi(1987).     

限制零点个数的亚纯函数的正规定则

设k,n(≥k+1)是两个正整数,a(≠0),b是两个有穷复数,F为区域D内的一族亚纯函数.如果对于任意的f∈F,f的零点重级大于等于k+1,并且在D内满足f+a[L(f)]~n-b至多有n-k-1个判别的零点,那么F在D内正规·这里L(f)=f~((k))(z)+a_1f~((k-1))(z)+…+a_(k-1)f'(z)+a_kf(z),其中a_1(z),a_2(z),…,a_k(z)是区域D上的全纯函数.     

$\alpha$阶星象亚纯函数

令$S(p)$表示单位圆盘$\mathbb{D}$上在$p\in(0,1)$处有一个简单极点的单叶亚纯函数全体.令$\alpha\in[0,1)$,我们用$\Sigma^{*}(p,\omega_{0},\alpha)$表示$f\in S(p)$使得$\hat{\mathbb{C}}\setminus f(\mathbb{D})$是关于不动点$\omega_{0}\neq0$, $\infty$星象的$\alphga$阶区域的函数全体.在本文中,$f\in\Sigma^{*}(p,\omega_{0},\alpha)$的一些解析刻画条件和系数估计被考虑.     

DISTRIBUTION OF THE（0，∞）ACCUMULATIVE LINES OF MEROMORPHIC FUNCTIONS

Suppose that f(z)is a meromorphic function of order λ（0<λ<+∞）and of lower order μ in the plane.Let ρ be a positive number such that μ≤ρ≤λ.(1)If f^(l)(z)(0≤l<+∞）has p(1≤p<+∞）finite nonzero deficient valnes αi(i=1,…，p)with deficiencies δ（αi,f^(l)),then f(z)has a (0,∞）accumulative line of order ≥ρin any angular domain whose vertex is at the origin and whose magnitude is larger than max(π/ρ，2π-4/ρ ∑i=1^p arcsin √δ（αi,f^(l))/2).(2)If f(z) has only p(0<p<+∞）（0,∞）,accumulative lines of order≥ρ：arg z=θk(0≤θ1<θ2<…<θp<2π，θp+1=θ1+2π）,then λ≤π/ω，where ω=min I≤k≤p(θk+1-θk),provided that f^(l)(z)(0≤l<+∞）has a finite nonzero deficient value.     

Uniqueness of meromorphic functions concerning sharing two small functions with their derivatives

In this paper, we study the uniqueness of meromorphic functions that share two small functions with their derivatives. We prove the following result: Let $f$ be a nonconstant meromorphic function such that $\mathop {\overline{\lim}}\limits_{r\to\infty} \frac{\bar{N}(r,f)}{T(r,f)}<\frac{3}{128}$, and let $a$, $b$ be two distinct small functions of $f$ with $a\not\equiv\infty$ and $b\not\equiv\infty$. If $f$ and $f"$ share $a$ and $b$ IM, then $f\equiv f"$.     

涉及重级零点的亚纯函数的拟正规定则

设F为区域D内的只有重级零点的亚纯函数族,H(z)为区域D内的非常数亚纯函数,且存在v∈N,使得对于任意的a∈C,n(D,1/H(z)-a)≤v.如果对于任意的f∈F,f′(z)≠H′(z),那么F在区域D内v阶拟正规.     

分担一个亚纯函数的亚纯函数族的正规定则

Let F be a family of meromorphic functions in D,and let Ψ(≠0) be a meromorphic function in D all of whose poles are simple.Suppose that,for each f ∈F,f≠0 in D.If for each pair of functions {f,g}(?) F,f' and g' share Ψ in D,then F is normal in D.