二阶线性微分方程英文翻译

Some Properties of Solutions of Periodic Second Order

Linear Differential Equations

1. Introduction and main results

In this paper, we shall assume that the reader is familiar with the fundamental results and the stardard notations of the Nevanlinna's value distribution theory of meromorphic functions [12, 14,

(f)and (f)to denote respectively the order 16]. In addition, we will use the notation (f)，

of growth, the lower order of growth and the exponent of convergence of the zeros of a meromorphic function f， e(f)（[see 8]），the e-type order of f(z), is defined to be

e(f) limlogT(r,f) r r

Similarly, e(f)，the e-type exponent of convergence of the zeros of meromorphic function f, is defined to be

log N(r,1/f) e(f) lim r r

We say thatf(z)has regular order of growth if a meromorphic functionf(z)satisfies

(f) limlogT(r,f) r logr

We consider the second order linear differential equation

f Af 0

Where A(z) B(e z)is a periodic entire function with period 2 i/ . The complex oscillation theory of (1.1) was first investigated by Bank and Laine [6]. Studies concerning (1.1) have een carried on and various oscillation theorems have been obtained [2{11, 13, 17{19]. WhenA(z)is rational in e，Bank and Laine [6] proved the following theorem

Theorem A LetA(z) B(e z)be a periodic entire function with period 2 i/ and rational in e z z.IfB( )has poles of odd order at both and 0, then for every solutionf(z)( 0)of (1.1), (f)

Bank [5] generalized this result: The above conclusion still holds if we just suppose that both and 0are poles ofB( ), and at least one is of odd order. In addition, the stronger conclusion

log N(r,1/f) o(r) (1.2)

holds. WhenA(z)is transcendental ine, Gao [10] proved the following theorem

Theorem B Let B( ) g(1/ ) z pjb ,whereg(t)is a transcendental entire function jj 1

zwith (g) 1, p is an odd positive integer andbp 0，Let A(z) B(e).Then any

non-trivia solution fof (1.1) must have (f) . In fact, the stronger conclusion (1.2) holds.

An example was given in [10] showing that Theorem B does not hold when (g)is any positive integer. If the order (g) 1 , but is not a positive integer, what can we say? Chiang and Gao [8] obtained the following theorems

zTheorem C Let A(z) B(e),whereB( ) g1(1/ ) g2( ),g1andg2are entire

functionsg2transcendental and (g2)not equal to a positive integer or infinity, andg1arbitrary. (i) (g2) 1. (a) If f is a non-trivial solution of (1.1) with e(f) (g2);

thenf(z)andf(z 2 i)are linearly dependent. (b) Iff1andf2are any two linearly Suppose

independent solutions of (1.1), then

(ii) Suppose e(f) (g2). (g2) 1 (a) If f is a non-trivial solution of (1.1)

with e(f) 1,f(z)andf(z 2 i)are linearly dependent. Iff1andf2are any two linearly independent solutions of (1.1),then e(f1f2) 1.

Theorem D Letg( )be a transcendental entire function and its order be not a positive integer or infinity. LetA(z) B(e z); whereB( ) g(1/ ) jb j 1jand p is an odd positive p

integer. Then (f) or each non-trivial solution f to (1.1). In fact, the stronger conclusion (1.2) holds.

Examples were also given in [8] showing that Theorem D is no longer valid when (g)is infinity.

The main purpose of this paper is to improve above results in the case whenB( )is transcendental. Specially, we find a condition under which Theorem D still holds in the case when (g)is a positive integer or infinity. We will prove the following results in Section 3.

Theorem 1 Let A(z) B(e),whereB( ) g1(1/ ) g2( ),g1andg2are entire functions withg2transcendental and z (g2)not equal to a positive integer or infinity, andg1arbitrary. If Some properties of solutions of periodic second order linear differential equationsf(z) and f(z 2 i)are two linearly independent solutions of (1.1), then

e(f)

Or

e(f) 1 (g2) 1 2

We remark that the conclusion of Theorem 1 remains valid if we assume (g1)

is not equal to a positive integer or infinity, andg2arbitrary and still assumeB( ) g1(1/ ) g2( ),In the case wheng1is transcendental with its lower order not equal to an integer or infinity andg2is arbitrary, we need only to consider B*( ) B(1/ ) g1( ) g2(1/ )in0 , 1/ .

Corollary 1 LetA(z) B(e z),whereB( ) g1(1/ ) g2( ),g1andg2are

entire functions with g2 transcendental and

(a)

(b) (g2)no more than 1/2, and g1 arbitrary. If f is a non-trivial solution of (1.1) with e(f) ,thenf(z) and f(z 2 i)are linearly dependent. Iff1andf2are any two linearly independent solutions of (1.1),

then e(f1f2) .

Theorem 2 Letg( )be a transcendental entire function and its lower order be no more than 1/2.

zLetA(z) B(e),whereB( ) g(1/ ) j 1bj

ppjand p is an odd positive integer, then (f) for each non-trivial solution f to (1.1). In fact, the stronger conclusion (1.2) holds. We remark that the above conclusion remains valid if

B( ) g( ) b j j

j 1

We note that Theorem 2 generalizes Theorem D when (g)is a positive integer or infinity but (g) 1/2. Combining Theorem D with Theorem 2, we have

Corollary 2 Letg( )be a transcendental entire function. LetA(z) B(ez) where B( ) g(1/ ) j 1bj jand p is an odd positive integer. Suppose that either (i) or (ii) below holds:

(i) (g) is not a positive integer or infinity;

(ii) (g) 1/2;

then (f) for each non-trivial solution f to (1.1). In fact, the stronger conclusion (1.2) holds.

2. Lemmas for the proofs of Theorems

Lemma 1 ([7]) Suppose thatk 2and thatA0,.....Ak 2are entire functions of period2 i,and that f is a non-trivial solution of p

y(k) Aj(z)y(j)(z) 0

i 0k 2

Suppose further that f satisfieslog N(r,1/f) o(r); that A0 is non-constant and rational

zine，and that ifk 3，thenA1,.....Ak 2are constants. Then there exists an integer q with1 q k such thatf(z) and f(z q2 i)are linearly dependent. The same conclusion

zholds ifA0is transcendental ine，and f satisfieslog N(r,1/f) o(r)，and if k 3，then

through a setL1r

k 2. haveT(r,Aj) o(T(r,Aj))forj 1,.....as

z

zof infinite measure, 1we and be Lemma 2 ([10]) LetA(z) B(e)be a periodic entire function with period 2 i transcendental ine, B( )is transcendental and analytic on0 .IfB( )has a pole of

odd order at or 0(including those which can be changed into this case by varying the

period ofA(z) andEq. (1.1) has a solutionf(z) 0which satisfies logN(r,1/f) o(r),

thenf(z) and f(z )are linearly independent.

3. Proofs of main results

The proof of main results are based on [8] and [15].

Proof of Theorem 1 Let us assume e(f) .Sincef(z) and f(z 2 i)are linearly independent, Lemma 1 implies that f(z)and f(z 4 i)must be linearly dependent. LetE(z) f(z)f(z 2 i),ThenE(z)satisfies the differential equation

E (z)2E (z)c2

, (2.1) 4A(z) () 2 2E(z)E(z)E(z)

Where c 0is the Wronskian off1andf2(see [12, p. 5] or [1, p. 354]), andE(z 2 i) c1E(z)or some non-zero constantc1.Clearly, E /E

and E /Eare both periodic functions with period2 i,whileA(z)is periodic by definition.

2Hence (2.1) shows thatE(z)is also periodic with period 2 i.Thus we can find an analytic

function ( )in0

yields ,so thatE(z)2 (ez)Substituting this expression into (2.1) c2 3 4B( ) 2()2 2 (2.2) 4

Since bothB( )and ( )are analytic inC* :1 ,the Valiron theory [21, p. 15] gives their representations as

n B( ) R( )b( )， ( ) n1R1( ) ( )， （2.3）

n1are some integers, R( )andR1( )are functions that are analytic and non-vanishing wheren，

on C* { }，b( )and ( ) are entire functions. Following the same arguments as used in [8], we have

T( , ) N( ,1/ ) T( ,b) S( , )， （2.4）

whereS( , ) o(T( , )).Furthermore, the following properties hold [8]

e(f) e(E) e(E2) max{ eR(E2), eL(E2)},

eR(E2) 1( ) ( ),

Where eR(E2)(resp, eL(E2)) is defined to be

log NR(r,1/E2)log NR(r,1/E2)lim(resp, lim), r r rr

Some properties of solutions of periodic second order linear differential equations

)(resp. NL(r,1/E2)denotes a counting function that only counts the zeros

2of E(z)in the right-half plane (resp. in the left-half plane), 1( )is the exponent of convergence of the zeros of inC*, which is defined to be

log N( ,1/ ) 1( ) lim log

Recall the condition e(f) ,we obtain ( ) . whereNR(r,1/E

Now substituting (2.3) into (2.2) yields 2

n1R1 32n1R1 2c2

4 R( )b( ) n1 ( ) ( ) R1 4 R1 R1( ) ( )

R1 n1R1n1 R1 R1 2n1(n1 1) 2 2 2 ) （2.5） (2 R1 R1 R1 n

Proof of Corollary 1 We can easily deduce Corollary 1 (a) from Theorem 1 .

Proof of Corollary 1 (b). Supposef1andf2are linearly independent and e(f1f2) ,then e(f1) ,and

Corollary 1 (a) that

Letfj(z)and e(f2) .We deduce from the conclusion of fj(z 2 i)are linearly dependent, j = 1; 2. E(z) f1(z)f2(z).Then we can find a non-zero constant c2such thatE(z 2 i) c2E(z).Repeating the same arguments as used in Theorem 1 by using the fact that E(z)2is also periodic, we obtain

e(E) 1 (g2) 1 2,a contradiction since (g2) 1/2.Hence e(f1f2) .

Proof of Theorem 2 Suppose there exists a non-trivial solution f of (1.1) that satisfies log N(r,1/f) o(r). We deduce e(f) 0, so f(z)andf(z 2 i) are linearly dependent by Corollary 1 (a). However, Lemma 2 implies that f(z)andf(z 2 i)are linearly

independent. This is a contradiction. Hence logN(r,1/f) o(r)holds for each non-trivial

solution f of (1.1). This completes the proof of Theorem 2.

Acknowledgments The authors would like to thank the referees for helpful suggestions to improve this paper.

References

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[2] BAESCH A. On the explicit determination of certain solutions of periodic differential

equations of higher order [J]. Results Math., 1996, 29(1-2): 42{55.

[3] BAESCH A, STEINMETZ N. Exceptional solutions of nth order periodic linear differential

equations [J].Complex Variables Theory Appl., 1997, 34(1-2): 7{17.

[4] BANK S B. On the explicit determination of certain solutions of periodic differential equations

[J]. Complex Variables Theory Appl., 1993, 23(1-2): 101{121.

[5] BANK S B. Three results in the value-distribution theory of solutions of linear differential

equations [J].Kodai Math. J., 1986, 9(2): 225{240.

[6] BANK S B, LAINE I. Representations of solutions of periodic second order linear differential

equations [J]. J. Reine Angew. Math., 1983, 344: 1{21.

[7] BANK S B, LANGLEY J K. Oscillation theorems for higher order linear differential equations

with entire periodic coe±cients [J]. Comment. Math. Univ. St. Paul., 1992, 41(1): 65{85.

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一些周期性的二阶线性微分方程解的方法

1． 简介和主要成果

在本文中，我们假设读者熟悉的函数的数值分布理论[12，14，16]的基本成果和数学符号。此外，我们将使用的符号 (f)， (f)and (f)，表示的顺序分别增长，低增长的一个纯函数的零点收敛指数，f， e(f)（[8]），E型的f(z)，被定义为

e(f) limlogT(r,f) r r

同样， e(f)，E型的亚纯函数f的零点收敛指数，被定义为

log N(r,1/f) e(f) lim r r

我们说，如果一个亚纯函数f(z)满足增长的正常秩序

logT(r,f) (f) lim r logr

我们考虑的二阶线性微分方程

f Af 0

在A(z) B(e z)是一个整函数在 2 i/ 。在（1.1）的反复波动理论的第一次探讨中由银行和莱恩[6]。已经进行了研究在（1.1）中，并已取得各种波动定理在[2{11，13，17{19]。在e函数中A(z)正确的，银行和莱恩[6]证明了如下定理

定理A 设A(z) B(e z)这函数是一个周期性函数，周期为 2 i/ 在整个函数e存 z z在。如果B( )有奇数阶极点在 和 0，然后对于任何一个结果答案f(z)( 0)在(1.1)中 (f)

广义这样的结果：上述结论仍然认为，如果我们只是假设， 既 0和B( )的极点，并且至少有一个是奇数阶。此外，较强的结论

log

zN(r,1/f) o(r) （1.2） 认为。当A(z)是超越在e，高[10]证明了如下定理

定理B

设B( ) g(1/ ) pjb ，其中g(t)是一个超越整函数与 (g) 1，p是奇正整jj 1

并且bp 0，设A(z) B(e)，那么任何微分解在（1.1）的函数f必须有 (f) 。事实上，在（1.2）已经有证明的结论。

是在[10] 一个例子表明当定理B不成立时， (g)是任意正整数。如果在另一方面z (g) 1，但如果没有一个正整数，我们可以说些什么呢？蒋和高[8]得到以下定理 定理C

设A(z) B(e)，其中B( ) g1(1/ ) g2( )，函数g1和函数g2是整函数g2先验和 (g2)不等于一个正整数或无穷大，并函数g1任意。

（一） 假设 (g2) 1（a）如果函数f是一个非平凡解 e(f) (g2)在（1.1），那么f(z)

和f(z 2 i)是线性相关。

（b）如果函数f1和函数f2在（1.1）是两个线性无关函数，则存在这样一个条件 z e(f) (g2)。

（二） 假设 (g2) 1（a）如果函数f有一个非平凡解在（1.1）且 e(f) 1，f(z)和

f(z 2 i)是线性相关的。

如果函数f1和函数f2在（1.1）在（1.1）是两个线性无关函数，则存在这样一个条件 e(f1f2) 1。

定理 D

让g( )是一个超越整函数和它的秩序是正整数或无穷大。设A(z) B(e z)，B( ) g(1/ ) j 1bj j和p是一个奇正整数。然后 (f) 或F得到每一个非平凡解在（1.1）。事实上，在（1.2）中已经有证明的结论。

例子表明在高[8]定理D不再成立，当 (g)是无穷的。

本文的主要目的是改善上述结果的情况下，当B( )是超越。特别地，我们找到的条件下定理D仍然成立的情况下，当 (g)是一个正整数或无穷大。

我们将证明在第3节的结果如下：

定理1

设A(z) B(e)，其中B( ) g1(1/ ) g2( )，g1和g2g2先验和 (g2)不等于一个正整数或无穷，g1任意整函数。如果定期二阶线性微分方程f(z)和f(z 2 i)的解不是一些属性是两个线性无关的解在（1.1），然后 zp

e(f)

或者

e(f) 1 (g2) 1 2

我们的说法，定理1的结论仍然有效，如果我们假设函数 (g1)不等于一个正整数或无穷大，任意和承担的情况下B( ) g1(1/ ) g2( )，当其低阶不等于一个整数或无穷超然是任意的，我们只需要考虑B*( ) B(1/ ) g1( ) g2(1/ )在0 ， 1/ 。 推论1

设A(z) B(e)，其中B( ) g1(1/ ) g2( )，函数g1和函数g2是整个g2先验和 z

(g2)不超过1 / 2，并且g1任意的。

（一） 如果函数f是一个非平凡解 e(f) 在（1.1）中，那么f(z)和f(z 2 i)是线

性相关。

（二） 如果f1和f2是两个线性无关解在（1.1）中，那么 e(f1f2) 。

定理2

设g( )是一个超越整函数及其低阶不超过1 / 2。设A(z) B(e)，其中zB( ) g(1/ ) j 1bj j和p是一个奇正整数，则 (f) 为每个非平凡解F到在（1.1）中。事实上，在（1.2）中证明正确的结论。

我们注意到，上述结论仍然有效的假设 p

B( ) g( ) b j j

j 1p

我们注意到，我们得出定理2推广定理D，当是一个正整数或无穷，但 (g) 1/2结合定理2定理的研究。

推论2

z设g( )是一个超越整函数。设A(z) B(e)，其中B( ) g(1/ ) pjb 和 pjj 1

是一个奇正整数。假设要么（一）或（二）中认为：

（一） (g)不是正整数或无穷;

（二） (g) 1/2

然后为每一个非平凡解在（1.1）中函数f对于 (f) 。事实上，在（1.2）中已经有证明的结论。

2． 引理为定理的证明

引理1

（[7]），k 2和的假设A0,.....Ak 2是整个周期2 i，并且函数f是有一个非平凡解

y(k) Aj(z)y(j)(z) 0

i 0k 2

进一步假设函数f满足log N(r,1/f) o(r);，A0是在e非恒定和理性的，而且，如果z

k 3，且A1,.....Ak 2是常数。则存在一个整数q与1 q k ，f(z)和f(z q2 i)是线

z性相关。相同的结论认为，如果A0是超越e，和f满足log N(r,1/f) o(r)，如果k 3，

然后通过一个无限措施的集合L1为r ，T(r,Aj) o(T(r,Aj))且j 1,.....k 2 引理2

（[10]） 设A(z) B(e z)是一个周期为 2 i

极奇数阶设B( )是定期与整函数周期 2 i

无关的解。

3．主要结果的证明

主要结果的证明的基础上[8]和[15]。 1 1在e z（包括那些可以改变这种情况下 的先验。在（1.1）中由不同的时期f(z) 0，log N(r,1/f) o(r)有一个满足，那么f(z)和f(z )是线性在0

定理1的证明

让我们假设 e(f) 。正弦f(z)和f(z 2 i)是线性无关的，引理1意味着f(z)和f(z 4 i)必须是线性相关的。设E(z) f(z)f(z 2 i)，则E(z)满足微分方程

E (z)2E (z)c2

, (2.1) 4A(z) () 2 2E(z)E(z)E(z)

其中c 0是f1和f2(见[12, p. 5] or [1, p. 354])，且E(z 2 i) c1E(z)或某些非零的常数c1。显然，E /E和E /E是两个周期2 i，而A(z)是定义函数。在（2.1），E(z)2也定期与周期2 i。因此，我们可以找到一个解析函数 ( )在0 ，使E(z)2 (ez)代入（2.1）得这种表达

c2 3 22 4B( ) 2() (2.2) 4

由于B( )和 ( )在C* :1 ，理论[21，p.15]给出了他们的结论

B( ) nR( )b( )， ( ) n1R1( ) ( )， （2.3）

其中n，n1是一些整数，R( )和R1( )函数分析和C* { }上非零，b( )和 ( )是整函数。按照相同的 [8]中，我们得出

T( , ) N( ,1/ ) T( ,b) S( , )， （2.4）

其中S( , ) o(T( , ))，此外，下列结论由[8]得

e(f) e(E) e(E2) max{ eR(E2), eL(E2)},

eR(E2) 1( ) ( ),

其中 eR(E2)是定义为

log NR(r,1/E2)log NR(r,1/E2)lim(resp, lim), r r rr

定期二阶线性微分方程解的一些性质

其中，NR(r,1/E2)(resp. NL(r,1/E2)表示一个计数功能，只计算在右半平面的E(z)2零点（在左半平面）， 1( )是在 的C*零点收敛指数，它的定义为

log N( ,1/ ) 1( ) lim log

由条件 e(f) ，我们得到 ( ) 。

现在（2.3）代入（2.2）中

2n1R1 32n1R1 2cn 4 R( )b( ) n1 ( ) ( ) R1 4 R1 R1( ) ( )

R1 n1R1n1 R1 R1 2n1(n1 1) 2 2 2 ) （2.5） ( 2 R1 R1 R1

推论1的证明

我们可以很容易地推导出定理1的推论1（一）推论1的证明（B）。假设f1和f2与 e(f1f2) 线性无关，那么 e(f1) ，我们证明推论1的结论（一），fj(z)与fj(z 2 i)线性相关，J =1;2。假设E(z) f1(z)f2(z)，然后我们可以找到

2E(z 2 i) c2E(z)的一个非零的常数c2，重复同样的论点定理1中使用的事实，E(z)

也是能找到，我们得到 e(E) 1 (g2) 1 2与 (g2) 1/2自矛盾，因此 e(f1f2) 。

定理2的证明

N(r,1/f) o(r)。我们推断 e(f) 0，

f(z)和f(z 2 i)的线性依赖推论1（a）。然而，引理2意味着f(z)和f(z 2 i)是线

性无关的。这是一对矛盾。因此logN(r,1/f) o(r)，认为都有非平凡解的F在（1.1）假设存在一个非平凡解的f在（1.1）中，满足log

中，这就完成了定理2的证明。

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