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.     

在Freud正交多项式零点处的Gr\"{u}nwald插值算子的收敛性

设$W_{\beta}(x)=\exp(-\frac{1}{2}|x|^{\beta})~(\beta > 7/6)$ 为Freud权, Freud正交多项式定义为满足下式$\int_{- \infty}^{\infty}p_{n}(x)p_{m}(x)W_{\beta}^{2}(x)\rd x=\left \{ \begin{array}{ll} 0 & \hspace{3mm} n \neq m , \\ 1 & \hspace{3mm}n = m \end{array} \right.$的     

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   

Zygmund 空间到 Bloch 空间上径向导数算子与积分型算子的乘积

设$H(\mathbb{B})$为单位球上全纯函数类,研究了单位球上 Zygmund 空间到 Bloch 空间上径向导数算子$\Re$与积分型算子$I_\varphi^g$乘积的有界性和紧性, 这里 $$ I_\varphi^g f(z)=\int_0^1 \Re f(\varphi(tz))g(tz)\frac{{\rm d}t}{t},\quad z\in\mathbb{B}, $$ 其中$g\in H(\mathbb{B}),\ g(0)=0$, $\varphi$ 是$\mathbb{B}$上全纯自映射.     

{C}^n$中单位球上Bergman型空间的一种积分算子

设$\omega_1,\omega_2$为正规函数, $\varphi$是$B_n$ 上的全纯自映射,$ g\in H(B_n)$ 满足 $g(0)=0$. 对所有的$0相似文献   

THE SUFFICIENT AND NECESSARY CONDITIONS FOR NOETHER’S SOLVABILITY OF SINGULAR INTEGRAL EQUATIONS WITH TWO CARLEMAN’S SHIFTS

In this paper we consider the problem of solvability of singular integral equtions with two Carleman's shifts \[\begin{gathered} (\mathcal{K}\varphi )(t) \equiv {a_0}(t)\varphi (t) + {a_1}(t)\varphi [\alpha (t)] + {a_2}(t)\varphi [\beta (t)] + {a_3}(t)\varphi [\gamma (t)] \hfill \ + \frac{{{b_0}(t)}}{{\pi i}}\int_\Gamma {\frac{{\varphi (\tau )}}{{\tau - t}}} d\tau + \frac{{{b_1}(t)}}{{\pi i}}\int_\Gamma {\frac{{\varphi (\tau )}}{{\tau - \alpha (t)}}} d\tau + \frac{{{b_2}(t)}}{{\pi i}}\int_\Gamma {\frac{{\varphi (\tau )}}{{\tau - \beta (t)}}d\tau } \hfill \ + \frac{{{b_s}(t)}}{{\pi i}}\int_\Gamma {\frac{{\varphi (\tau )}}{{\tau - \gamma (t)}}} d\tau + \int_\Gamma {K(t,\tau )\varphi (\tau )d\tau = g(t){\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (1,1)} \hfill \\ \end{gathered} \] Suppose that Г is a closed simple Lyapunoff's curve and \[\alpha (t)\], \[\beta (t)\] which satisfy Carleman's. conditions and \[\alpha [\beta (t)] = \beta [\alpha (t)]\] are two different homeomorphisms of Г onto itself, and that \[{a_k}(t),{b_k}(t)\], k = 0, 1, 2, 3 belong to the,space \[{H_\mu }(\Gamma ),g(t)\] belongs to the space \[{L_p}(\Gamma ),p > 1\]), p>l and \[K(t,\tau )\] has only weak singularity. The following main results are obtained: 1. Singular integral eqution (1.1) is solvable if and only if the Noether's conditions \[det(p(t) \pm q(t)) \ne 0\] are satisfied. 2. Index of sigular integral eqution (1.1) is calculated by the formula \[Ind{\kern 1pt} {\kern 1pt} {\kern 1pt} \mathcal{K} = \frac{1}{{8\pi }}{\{ arg\frac{{\det (p(t) - q(t))}}{{\det (p(t) + q(t))}}\} _\Gamma }\] where p(t) and q(t) are matrices of coeffioents of so-called corresponding system of equtions. All these results have been generalized for systems of singular integral equtions with two Carleman's shifts and complex conjugate of unknown functions.     

从$A^{p}_{\alpha}$到$A^{\infty}(\varphi)$的加权复合算子

设$u \in H(D), \ \phi$为$D$上的解析自映射,定义$H(D)$上的加权复合算子为$u C_{\phi}(f)=$$uf\circ\phi$, \ $f\in H(D)$.本文得到了从$A^{p}_{\alpha}$到$A^{\infty}(\varphi)\ (A_{0}^{\infty}(\varphi))$的加权复合算子$u C_{\phi}$的有界性和紧性的充要条件.     

涉及分担超平面的正规定则

在本文中, 作者继续讨论涉及分担超平面的全纯曲线的正规性, 得到了如下结果:设$\mathcal F$是一族从区域$D\subset\mathbb C$到$\mathbb P^N(\mathbb C)$上的全纯曲线,$H_j=\{x\in\mathbb P^N(\mathbb C):\langle\bm{x},\alpha_j\rangle=0\}$是$\mathbb P^N(\mathbb C)$中处于一般位置的超平面, 这里$\alpha_j=(a_{j0},\cdots,a_{jN})^{\rm T}$且$a_{j0}\ne0$, $j=1,2,\cdots,2N+1$.若对于任意的$f\in\mathcal F$, 满足下列两个条件:(i) 如果$f(z)\in H_j$, 那么$\nabla f\in H_j$, 这里$j=1,2,\cdots,2N+1$;(ii) 如果$f(z)\in\bigcup\limits_{j=1}^{2N+1} H_j$, 那么$\frac{|\langle f(z),H_0\rangle|}{\|f\|\|H_0\|}\ge \delta$, 这里$0<\delta<1$是一个常数,而$H_0=\{w_0=0\}$,\noindent 则$\mathcal F$在$D$上正规.     

图映射的非游荡点和深度

设 $G$ 是一个图, $f:G\rightarrow G$ 是连续映射. 用$R(f)$和$\Omega (f)$分别表示$f$的回归点集和非游荡集. 设$\Omega_0 (f)=G$, $\Omega_n (f)=\Omega (f|_{\Omega_{n-1} (f)})$(对任$n\in {\N}$). 满足$\Omega_{m} (f)=\Omega_{m+1} (f)$的最小的$m\in {\N}\cup \{\infty\}$称为$f$的深度. 证明了$\Omega_2(f)=\overline{R(f)}$且 $f$的深度不超过2. 进一步, 还得到$f$的非游荡点的若干性质.     

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.     

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).     

阶极小渐近基

Let N denote the set of all nonnegative integers and A be a subset of N.Let W be a nonempty subset of N.Denote by F~*(W) the set of all finite,nonempty subsets of W.Fix integer g≥2,let A_g(W) be the set of all numbers of the form sum f∈Fa_fg~f where F∈F~*(W)and 1≤a_f≤g-1.For i=0,1,2,3,let W_i = {n∈N|n≡ i(mod 4)}.In this paper,we show that the set A = U_i~3=0 A_g(W_i) is a minimal asymptotic basis of order four.     

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.     

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.     

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)=∞.     

Inequalities and Separation for a Biharmonic Laplace-Beltrami Differential Operator in a Hilbert Space Associated with the Existence and Uniqueness Theorem

In this paper, we have studied the separation for the biharmonic Laplace-Beltrami differential operator\begin{equation*}Au(x)=-\Delta \Delta u(x)+V(x)u(x),\end{equation*}for all $x\in R^{n}$, in the Hilbert space $H=L_{2}(R^{n},H_{1})$ with the operator potential $V(x)\in C^{1}(R^{n},L(H_{1}))$, where $L(H_{1})$ is the space of all bounded linear operators on the Hilbert space $H_{1}$, while $\Delta \Delta u$\ is the biharmonic differential operator and\begin{equation*}\Delta u{=-}\sum_{i,j=1}^{n}\frac{1}{\sqrt{\det g}}\frac{\partial }{{\partial x_{i}}}\left[ \sqrt{\det g}g^{-1}(x)\frac{\partial u}{{\partial x}_{j}}\right]\end{equation*}is the Laplace-Beltrami differential operator in $R^{n}$. Here $g(x)=(g_{ij}(x))$ is the Riemannian matrix, while $g^{-1}(x)$ is the inverse of the matrix $g(x)$. Moreover, we have studied the existence and uniqueness Theorem for the solution of the non-homogeneous biharmonic Laplace-Beltrami differential equation $Au=-\Delta \Delta u+V(x)u(x)=f(x)$ in the Hilbert space $H$ where $f(x)\in H$ as an application of the separation approach.     

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.