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1.
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)<∞. 相似文献
2.
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)=∞. 相似文献
3.
4.
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. 相似文献
5.
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. 相似文献
6.
Let $s_n(f,z):=\sum_{k=0}^{n}a_kz^k$ be the $n$th partial sum of
$f(z)=\sum_{k=0}^{\infty{}}a_kz^k$. We show that $\RE s_n(f/z,z)>0$ holds for all $z\in\D,\ n\in\N$, and all starlike functions $f$ of order
$\lambda$ iff $\lambda_0\leq\lambda<1$ where
$\lambda_0=0.654222...$ is the unique solution
$\lambda\in(\frac{1}{2},1)$ of the equation
$\int_{0}^{3\pi/2}t^{1-2\lambda}\cos t \,dt=0$. Here $\D$ denotes
the unit disk in the complex plane $\C$. This result is the best
possible with respect to $\lambda_0$. In particular, it
shows that for the Gegenbauer polynomials $C_{n}^{\mu}(x)$ we
have $\sum_{k=0}^n C_{k}^{\mu}(x)\cos k \theta>0$ for all
$n\in\N,\ x\in[-1,1]$, and
$0<\mu\leq\mu_0:=1-\lambda_0=0.345778...$. This result complements
an inequality of Brown, Wang, and Wilson (1993) and extends a
result of Ruscheweyh and Salinas (2000). 相似文献
7.
Wu Shengjian 《数学年刊B辑(英文版)》1994,15(4):453-462
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. 相似文献
8.
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. 相似文献
‖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. 相似文献
9.
《复变函数与椭圆型方程》2012,57(1):25-57
A function f is in the class $ V_2p $ iff $ f(z) = e^{-az^{2p+2}}g(z) $ where a S 0 and g is a constant multiple of a real entire function of genus h 2 p + 1 with only real zeros. The class $ U_2p $ is defined as follows: $ U_0 = V_0 $ , $ U_{2p} = V_{2p}-V_{2p-2} $ . Functions in the class $ U_{2p}^{*} $ are represented as $ g(z) = c(z)f(z) $ where $ f\in U_{2p} $ and c is a real polynomial with no real zeros. Every real entire function g , of finite order with at most finitely many non-real zeros satisfies $ g\in U_{2p}^{*} $ for a unique p . We show the exact number of non-real zeros of f" , for $ f\in U_{2p} $ , in terms of the number of non-real zeros of f' and a geometrical condition on the components of Im Q ( z ) > 0, where $ \displaystyle Q(z) = z-({f(z)}/{f'(z)}) $ . Further, for a subclass of $ f\in U_{2p} $ , we show necessary and sufficient conditions for f" to have exactly 2 p non-real zeros. For a subclass of $ U_{2p}^{*} $ we show that if f' has only real zeros, then f" has exactly 2 p non-real zeros. For $ f\in U_{2p}^{*} $ we show that 2 p is a lower bound for the number of non-real zeros of $ f^{(k)} $ for k S 2. 相似文献
10.
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. 相似文献
11.
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
12.
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. 相似文献
13.
关于超越亚纯系数微分方程亚纯解的零点 总被引:6,自引:0,他引:6
本文研究了非齐次线性微分方程的复振荡问题,其中,D0,D1,…,D(k-1),是超越亚纯函数.当存在某个Ds(1≤s≤k-1)比其它Dj(j≠s)有较快增长的意义下起支配作用时,得到了微分方程(I)亚纯解的零点收敛指数的精确估计式. 相似文献
14.
本文研究了线性微分方程$f^{(k)}+A(z)f=0$ 和 $f^{(k)}+A(z)f=F(z)$解的性质,推广了原有的一些结果. 相似文献
15.
The paper proves on the basis of [1] the following theorem: Let $\[f(z)\]$ be an entire function of lower order $\[\mu < \infty \]$, and $\[{a_i}(z)(l = 1,2, \cdots ,k)\]$ be meromorphic functions which satisfy $\[T(r,{a_i}(z)) = o\{ T(r,f)\} \]$. If
$$\[\sum\limits_{i = 1}^k {\delta ({a_i}(z),f) = 1\begin{array}{*{20}{c}}
{({a_i}(z) \ne \infty )}&{(1)}
\end{array}} \]$$
then the deficiencies $\[\delta ({a_i}(z),f)\]$ are equal to $\[\frac{{{n_1}}}{\mu }\]$, where $\[{n_i}\]$ is an integer,$\[l = 1,2, \cdots ,k\]$. 相似文献
16.
Zhu Yaochen 《数学年刊B辑(英文版)》1984,5(1):109-118
Letf_v(z)=∑a_(v,,k)z~(λ_(v,k))(v=1,…,s)be s power series with algebraic coefficients a_(v,k),convergence radii R_v>0 and sufficientlyrapidly increasing integers λ_(v,k).It is shown that under certain conditions depending only ona_(v,k) and λ_(v,k),(i)f_1(θ_1),…,f_s(θ_s)are algebraically independent for arbitrary algebraicnumbers θ_1,…,θ_s with θ<丨θ_v丨相似文献
17.
Hu Ke 《数学年刊B辑(英文版)》1983,4(2):187-190
AIn this paper, the author obtains the following results:(1) If Taylor coeffiients of a function satisfy the conditions:(i),(ii),(iii)A_k=O(1/k) the for any h>0 the function φ(z)=exp{w(z)} satisfies the asymptotic equality the case h>1/2 was proved by Milin.(2) If f(z)=z α_2z~2 …∈S~* and,then for λ>1/2 相似文献
18.
吴昭君 《数学年刊A辑(中文版)》2015,36(1):81-90
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. 相似文献
19.
The present paper investigates the convergence of Hermite interpolation operators on the real line. The main result is: Given 0 〈 δo 〈 1/2, 0 〈 εo 〈 1. Let f ∈ C(-∞,∞) satisfy |y|= O(e^(1/2-δo)xk^2,) and |f(x)|t= O(e^(1-εo )x2^). Then for any given point x ∈ R, we have limn→Hn,(f, x) = f(x). 相似文献
20.
Anja Baesch 《Results in Mathematics》1996,29(1-2):43-55
We consider the class of differential equations $ y^{(k)}+\Sigma_{k- 2}^{\nu=1}A_{\nu}y^{(\nu)}+A_0(z)y=0\ {\rm where}\ A_{1},\dots,A_{k- 2}$ are constants, k ≥ 3 and where A0(z) is a non-constant periodic entire function, which is a rational function of e z. In this paper we develop a method that enables us to decide if this equation can have solutions with few zeros, and we also present the construction of these solutions. 相似文献