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.     

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)=∞.     

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) ∞.     

一类高阶复微分方程解的增长性

本文研究高阶线性微分方程f~((k))+A_(k-1)f~((k-1))+···+A1f′+A0f=0解的增长性,其中Aj(j=0,···,k-1)为整函数.当存在某个系数A_s是方程ω′′+P(z)ω=0的一个非零解时,我们得到上述方程具有无穷级解的判定条件,并对解的超级进行了估计.这里的P(z)为非零多项式,当P(z)为特定形式的多项式时,A_s可取为Airy函数,Weber-Hermite函数或指数函数.     

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.     

一类高阶齐次线性微分方程解的零点

本文研究一类整函数系数高阶齐次线性微分方程解的零点分布.利用Nevanlinna值分布理论,得到当系数A_(k-1)的增长性起主要支配作用时,方程f~((k))+A_(k-1)f~((k-1))+···+A_0f=0任意超越解的零点收敛指数为无穷,推广了Langley和Bank等人的结果.     

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.     

Picard type theorems concerning certain small functions

Let f(z) be a meromorphic function in the complex plane, whose zeros have multiplicity at least k + 1(k ≥ 2). If sin z is a small function with respect to f(z), then f~(k)(z)-P(z) sin z has infinitely many zeros in the complex plane, where P(z) is a nonzero polynomial of deg(P(z)) ≠ 1.     

线性微分方程整函数解的Julia集的径向分布

研究了线性微分方程f~((n))+A_(n-2)f~((n-2))+…+A_0(z)f=0整函数解的Julia集的径向分布,其中n≥2,A_j(z)(j=0,1,…,n-2)是具有有限下级的整函数,得到了这类方程线性无关解的乘积的Julia集的径向分布的下界.     

一类高阶线性微分方程的次正规解

讨论了一类高阶线性微分方程F~((k))+A_(k-1)f~((k-1))+…+A_0f=0,k≥2的次正规解的存在性和形式,并估计了所有解的增长性,推广了陈宗煊的结果     

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.     

MEROMORPHIC SOLUTIONS OF A TYPE OF SYSTEM OF COMPLEX DIFFERENTIAL-DIFFERENCE EQUATIONS

Applying Nevanlinna theory of the value distribution of meromorphic functions,we mainly study the growth and some other properties of meromorphic solutions of the type of system of complex differential and difference equations of the following form∑nj=1aj(z)f1(λj1)(z+cj) = R2(z, f2(z)),∑nj=1βj(z)f2(λj2)(z+cj)=R1(Z,F1(z)).(*)where λij(j = 1, 2, ···, n; i = 1, 2) are finite non-negative integers, and cj(j = 1, 2, ···, n)are distinct, nonzero complex numbers, αj(z), βj(z)(j = 1, 2, ···, n) are small functions relative to fi(z)(i = 1, 2) respectively, Ri(z, f(z))(i = 1, 2) are rational in fi(z)(i = 1, 2)with coefficients which are small functions of fi(z)(i = 1, 2) respectively.     

高阶齐次线性微分方程解的零点

研究了一类高阶齐次线性微分方程解的零点收敛指数,并得到当方程的系数A_0为整函数,其泰勒展式为缺项级数,并且A_0起控制作用时,方程f~((k))+A_(k-2)f~((k-2))+…+A_1f′+A_0f=0的任意两个线性无关解f_1,f_2满足max{λ(f_1),λ(f_2)}=∞,其中λ(f)表示亚纯函数.f的零点收敛指数.     

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   

一类高阶线性微分方程解的增长性

研究整函数系数高阶线性微分方程f~((k))+A_(k-1)f~((k-1))+…+A_0f=0解的增长性.利用亚纯函数的Nevanlina值分布理论,得到当系数A_s(s≠0)为满足杨不等式极端情况的整函数,A_0满足一定条件时,上述方程的每个非零解均为无穷级,并给出解的超级估计.     

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.     

C上和?内某类高阶线性微分方程解的增长性

本文分别在复平面C上和单位圆△内考虑方程f~(k)+A_((k-1))(z)f~(k-1)+…+A_1(z)f'+A_0(z)f=0的解的增长性与其系数的增长性之间的关系.当A_0(z)或某个A_j(z)(0jk)严格控制其它系数时,通过比较A_0(z)和A_j(z)的迭代下级或迭代下型,得到上述方程当系数分别为整函数和单位圆△内解析函数时解的增长性的一些估计.     

A NEW ROPER-SUFFRIDGE EXTENSION OPERATOR ON BOUNDED COMPLETE REINHARDT DOMAINS

In this paper, we construct a new Roper-Suffridge extension operator Φr n,β1,,βn(f)(z) = F(z) = ((rf(z1/r)/z1)β1z1,(rf(z1/r)/z1)β2z2,...,(rf(z1/r)/z1)βnzn)',where f is a normalized locally biholomorphic function on the unit disc D, r = sup{|z1| : z =(z1, ···, zn) ∈Ω}, β1∈ [0, 1], 0 ≤βk≤β1, k = 2, ···, n, then we prove it can preserve the property of spirallikeness of type β, almost starlikeness of order α and starlikeness of orderα on bounded complete Reinhardt domain Ω, respectively.     

ON GROWTH OF MEROMORPHIC SOLUTIONS OF NONLINEAR DIFFERENCE EQUATIONS AND TWO CONJECTURES OF C.C. YANG

In this paper, we investigate the growth of the meromorphic solutions of the following nonlinear difference equationsf(z)n+ P_(n-1)(f) = 0,where n ≥ 2 and P_(n-1)(f) is a difference polynomial of degree at most n- 1 in f with small functions as coefficients. Moreover, we give two examples to show that one conjecture proposed by Yang and Laine [2] does not hold in general if the hyper-order of f(z) is no less than 1.     

Exponential polynomials as solutions of certain nonlinear difference equations

Recently, C.-C. Yang and I. Laine have investigated finite order entire solutions f of nonlinear differential-difference equations of the form fn + L(z, f ) = h, where n ≥ 2 is an integer. In particular, it is known that the equation f(z)2 + q(z)f (z + 1) = p(z), where p(z), q(z) are polynomials, has no transcendental entire solutions of finite order. Assuming that Q(z) is also a polynomial and c ∈ C, equations of the form f(z)n + q(z)e Q(z) f(z + c) = p(z) do posses finite order entire solutions. A classification of these solutions in terms of growth and zero distribution will be given. In particular, it is shown that any exponential polynomial solution must reduce to a rather specific form. This reasoning relies on an earlier paper due to N. Steinmetz.