亚纯函数与其差分的唯一性

研究了亚纯函数与其差分算子分担多项式的唯一性问题,证明了:设f是一个有穷级非常数亚纯函数,p(z)(■0)是一个多项式.如果f,△_cf与△_c~2f CM分担∞,p(z),则f≡△_cf或f(z)=e~(Az+B)+b,其中p(z)≡b≠0,A≠0满足e~(Ac)=1.本文结果是对Chang, Fang(Chang J M, Fang M L. Uniqueness of entire functions and fixed points [J]. Kodai Math J, 2002, 25(1):309-320.)结果的差分模拟,并且完整回答了Chen, Chen(Chen B Q, Chen Z X, Li S. Uniqueness theorems on entire functions and their difference operators or shifts [J]. Abstr Appl Anal, 2012,Art. ID 906893, 8 pp.)的问题.     

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.     

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.     

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.     

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.     

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).     

$\alpha$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.     

On Hermite-Hadamard-type characterizations of higher-order differential inequalities

Let $I$ be an open interval of $\mathbb{R}$ and $f: I\to \mathbb{R}$. It is well-known that $f$ is convex in $I$ if and only if, for all $x,y\in I$ with $x相似文献   

亚纯函数微分多项式f^kQ[f]+P[f]的零点分布

研究了超越亚纯函数$f$的微分多项式$f^kQ[f]+P[f]$的零点分布. 给出了以下结果:对于满足$\delta(\infty,f)\geq1-\alpha>0$ ($\alpha$为常数, $0\leq \alpha<1$ )的超越亚纯函数$f(z)$, 若$T(r,f)=O((\log r)^2)$,则微分多项式$f^kQ[f]+P[f]$ ($Q[f]\not\equiv 0,\ P[f] \not\equiv 0$)在 可数个圆盘并集之外有无穷多个零点,其中$k>\frac{1+\Gamma_{P}+\gamma_{P}+\alpha(1+\Gamma_Q+\Gamma_{P}-\gamma_{P})} {1-\alpha }$, $\Gamma_{Q}$是$Q[f]$的权, $\Gamma_{P}$和$\gamma_{P}$是$P[f]$的权和次数.     

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)<∞.     

ON SOME CONSTANTS OF QUASICONFORMAL DEFORMATION AND ZYGMUND CLASS

A real-valued function f(x) on Ж belongs to Zygmund class A.(Ж) ff its Zygmund norm ‖f‖x=inf,|f(x+t)-2f(x)+f(x-t)/t|is finite. It is proved that when f∈A*(Ж), there exists an extension F(z) of f to H={Imz&gt;0} such that ‖Э^-F‖∞≤√—1+53^2/72‖f‖z.It is also proved that if f(0)=f(1)=0, thenmax,x∈[0,1]|f(x)|≤1/3‖f‖x.     

由$q$-积分算子定义的某些亚纯函数类的Fekete-Szeg\"{o}问题

本文首先引入满足如下条件$$-\frac{qzD_{q}f(z)}{f(z)}\prec \varphi (z)$$和$$\frac{-(1-\frac{\alpha }{q})qzD_{q}f(z)+\alpha qzD_{q}[zD_{q}f(z)]}{(1-\frac{\alpha}{q})f(z)-\alpha zD_{q}f(z)}\prec \varphi (z)~(\alpha \in\mathbb{C}\backslash (0,1],\ 0相似文献   

Hardy-Sobolev类上的Bohr型不等式

Let β 〉 0 and Sβ ：= {z ∈ C ： ｜Imz｜ 〈β} be a strip in the complex plane. For an integer r ≥ 0, let H∞^Г,β denote those real-valued functions f on R, which are analytic in Sβ and satisfy the restriction ｜f^（r）（z）｜ ≤ 1, z ∈ Sβ. For σ 〉 0, denote by Bσ the class of functions f which have spectra in （-2πσ, 2πσ）. And let Bσ^⊥ be the class of functions f which have no spectrum in （-2πσ, 2πσ）. We prove an inequality of Bohr type
‖f‖∞≤π/√λ∧σ^r∑k=0^∞（-1）^k（r＋1）/（2k＋1）^rsinh（（2k＋1）2σβ）,f∈H∞^r,β∩B1/σ,
where λ∈（0,1）,∧and ∧′are the complete elliptic integrals of the first kind for the moduli λ and λ′=√1- λ^2,respectively,and λ satisfies
4∧β/π∧′=1/σ.
The constant in the above inequality is exact.     

An Existence Result for a Class of Chemically Reacting Systems with Sign-Changing Weights

We prove the existence of positive solutions for the system$$\begin{align*}\begin{cases}-\Delta_{p} u =\lambda a(x){f(v)}{u^{-\alpha}},\qquad x\in \Omega,\\-\Delta_{q} v = \lambda b(x){g(u)}{v^{-\beta}},\qquad x\in \Omega,\\u = v =0, \qquad x\in\partial \Omega,\end{cases}\end{align*}$$where $\Delta_{r}z={\rm div}(|\nabla z|^{r-2}\nabla z)$, for $r>1$ denotes the r-Laplacian operator and $\lambda$ is a positive parameter, $\Omega$ is a bounded domain in $\mathbb{R}^{n}$, $n\geq1$ with sufficiently smooth boundary and $\alpha, \beta \in (0,1).$ Here $ a(x)$ and $ b(x)$ are $C^{1}$ sign-changingfunctions that maybe negative near the boundary and $f,g $ are $C^{1}$ nondecreasing functions, such that $f, g :\ [0,\infty)\to [0,\infty);$ $f(s)>0,$ $g(s)>0$ for $s> 0$, $\lim_{s\to\infty}g(s)=\infty$ and$$\lim_{s\to\infty}\frac{f(Mg(s)^{\frac{1}{q-1}})}{s^{p-1+\alpha}}=0,\qquad \forall M>0.$$We discuss the existence of positive weak solutions when $f$, $g$, $a(x)$ and $b(x)$ satisfy certain additional conditions. We employ the method of sub-supersolution to obtain our results.     

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.     

亚纯函数正规族与分担值

设 $k, m$ 是两个正整数, $a\ ( \ne 0)$是有穷复数. $\mathcal{F}$ 是区域 $D$ 内的一族亚纯函数, $f\in\mathcal{F}$ 的零点重数至少为 $k$, $P$ 是多项式,次数或者 ${\rm deg}\, P\geq3$ 或者 ${\rm deg}\, P=2$ 且 $P$ 只有一个不同的零点.若对于 $\mathcal{F}$ 中的任意两个函数 $f$ 和 $g$, $P(f){({f^{(k)}})^m}$ 与 $P(g){({g^{(k)}})^m}$ 在 $D$ 内 IM 分担 $a$, 则 $\mathcal{F}$ 在 $D$ 内正规.     

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.     

多复变整函数涉及全导数的唯一性定理

结合金路提出的多复变上整函数全导数的概念,得到了如下定理:对于n维复欧式空间C~n上两个非常数整函数f和g,以及一个正整数k,如果δ(0,f)+δ(0,g)1,D~kf=1D~kg=1,那么f≡g.这一结论是仪洪勋和杨重骏的定理的推广.     

Function Spaces in Lipschitz Domains and Optimal Rates of Convergence for Sampling

Assume that we want to recover $f : \Omega \to {\bf C}$ in the $L_r$-quasi-norm ($0 < r \le \infty$) by a linear sampling method $$ S_n f = \sum_{j=1}^n f(x^j) h_j , $$ where $h_j \in L_r(\Omega )$ and $x^j \in \Omega$ and $\Omega \subset {\bf R}^d$ is an arbitrary bounded Lipschitz domain. We assume that $f$ is from the unit ball of a Besov space $B^s_{pq} (\Omega)$ or of a Triebel--Lizorkin space $F^s_{pq} (\Omega)$ with parameters such that the space is compactly embedded into $C(\overline{\Omega})$. We prove that the optimal rate of convergence of linear sampling methods is $$ n^{ -{s}/{d} + ({1}/{p}-{1}/{r})_+} , $$ nonlinear methods do not yield a better rate. To prove this we use a result from Wendland (2001) as well as results concerning the spaces $B^s_{pq} (\Omega) $ and $F^s_{pq}(\Omega)$. Actually, it is another aim of this paper to complement the existing literature about the function spaces $B^s_{pq} (\Omega)$ and $F^s_{pq} (\Omega)$ for bounded Lipschitz domains $\Omega \subset {\bf R}^d$. In this sense, the paper is also a continuation of a paper by Triebel (2002).