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1、英文翻译专 业 数学与应用数学 Some Properties of Solutions of Periodic Second Order Linear Differential Equations1. Introduction and main resultsIn this paper, we shall assume that the reader is familiar with the fundamental results and the stardard notations of the Nevanlinnas value distribution theory of meromo
2、rphic functions 12, 14, 16. In addition, we will use the notation，and to denote respectively the order of growth, the lower order of growth and the exponent of convergence of the zeros of a meromorphic function ，（see 8），the e-type order of f(z), is defined to be Similarly, ，the e-type exponent of co
3、nvergence of the zeros of meromorphic function , is defined to beWe say thathas regular order of growth if a meromorphic functionsatisfiesWe consider the second order linear differential equationWhere is a periodic entire function with period . The complex oscillation theory of (1.1) was first inves
4、tigated by Bank and Laine 6. Studies concerning (1.1) have een carried on and various oscillation theorems have been obtained 211, 13, 1719. Whenis rational in ，Bank and Laine 6 proved the following theoremTheorem A Letbe a periodic entire function with period and rational in .Ifhas poles of odd ord
5、er at both and , then for every solutionof (1.1), Bank 5 generalized this result: The above conclusion still holds if we just suppose that both and are poles of, and at least one is of odd order. In addition, the stronger conclusion (1.2)holds. Whenis transcendental in, Gao 10 proved the following t
6、heoremTheorem B Let ,whereis a transcendental entire function with, is an odd positive integer and，Let .Then any non-trivia solution of (1.1) must have. In fact, the stronger conclusion (1.2) holds.An example was given in 10 showing that Theorem B does not hold when is any positive integer. If the o
7、rder , but is not a positive integer, what can we say? Chiang and Gao 8 obtained the following theoremsTheorem C Let ,where,andare entire functionstranscendental andnot equal to a positive integer or infinity, andarbitrary.(i) Suppose . (a) If f is a non-trivial solution of (1.1) with; thenandare li
8、nearly dependent. (b) Ifandare any two linearly independent solutions of (1.1), then .(ii) Suppose (a) If f is a non-trivial solution of (1.1) with,andare linearly dependent. Ifandare any two linearly independent solutions of (1.1),then.Theorem D Letbe a transcendental entire function and its order
9、be not a positive integer or infinity. Let; whereand p is an odd positive integer. Thenor 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 whenis infinity.The main purpose of this paper is to im
10、prove above results in the case whenis transcendental. Specially, we find a condition under which Theorem D still holds in the case when is a positive integer or infinity. We will prove the following results in Section 3.Theorem 1 Let ,where,andare entire functions withtranscendental andnot equal to
11、 a positive integer or infinity, andarbitrary. If Some properties of solutions of periodic second order linear differential equations and are two linearly independent solutions of (1.1), thenOrWe remark that the conclusion of Theorem 1 remains valid if we assumeis not equal to a positive integer or
12、infinity, andarbitrary and still assume,In the case whenis transcendental with its lower order not equal to an integer or infinity andis arbitrary, we need only to consider in,.Corollary 1 Let,where,andareentire functions with transcendental and no more than 1/2, and arbitrary.(a) If f is a non-triv
13、ial solution of (1.1) with,then and are linearly dependent.(b) Ifandare any two linearly independent solutions of (1.1), then.Theorem 2 Letbe a transcendental entire function and its lower order be no more than 1/2. Let,whereand p is an odd positive integer, then for each non-trivial solution f to (
14、1.1). In fact, the stronger conclusion (1.2) holds. We remark that the above conclusion remains valid ifWe note that Theorem 2 generalizes Theorem D whenis a positive integer or infinity but . Combining Theorem D with Theorem 2, we haveCorollary 2 Letbe a transcendental entire function. Let where an
15、d p is an odd positive integer. Suppose that either (i) or (ii) below holds:(i) is not a positive integer or infinity;(ii) ;thenfor each non-trivial solution f to (1.1). In fact, the stronger conclusion (1.2) holds.2. Lemmas for the proofs of TheoremsLemma 1 (7) Suppose thatand thatare entire functi
16、ons of period,and that f is a non-trivial solution ofSuppose further that f satisfies; that is non-constant and rational in，and that if，thenare constants. Then there exists an integer q with such that and are linearly dependent. The same conclusion holds ifis transcendental in，and f satisfies，and if
17、 ，then asthrough a setof infinite measure, we havefor.Lemma 2 (10) Letbe a periodic entire function with periodand be transcendental in, is transcendental and analytic on.Ifhas a pole of odd order at or(including those which can be changed into this case by varying the period of and. (1.1) has a sol
18、utionwhich satisfies , then and are linearly independent.3. Proofs of main resultsThe proof of main results are based on 8 and 15.Proof of Theorem 1 Let us assume.Since and are linearly independent, Lemma 1 implies that and must be linearly dependent. Let,Thensatisfies the differential equation, (2.
19、1)Where is the Wronskian ofand(see 12, p. 5 or 1, p. 354), andor some non-zero constant.Clearly, and are both periodic functions with period,whileis periodic by definition. Hence (2.1) shows thatis also periodic with period .Thus we can find an analytic functionin,so thatSubstituting this expression
20、 into (2.1) yields (2.2)Since bothand are analytic in,the Valiron theory 21, p. 15 gives their representations as ， （2.3）where，are some integers, andare functions that are analytic and non-vanishing on ，and are entire functions. Following the same arguments as used in 8, we have， （2.4）where.Furtherm
21、ore, the following properties hold 8,Where (resp, ) is defined to be(resp, ),Some properties of solutions of periodic second order linear differential equationswhere(resp. denotes a counting function that only counts the zeros of in the right-half plane (resp. in the left-half plane), is the exponen
22、t of convergence of the zeros of in, which is defined to beRecall the condition ,we obtain.Now substituting (2.3) into (2.2) yields （2.5）Proof of Corollary 1 We can easily deduce Corollary 1 (a) from Theorem 1 .Proof of Corollary 1 (b). Supposeandare linearly independent and,then,and .We deduce from
23、 the conclusion of Corollary 1 (a) thatand are linearly dependent, j = 1; 2. Let.Then we can find a non-zero constant such that.Repeating the same arguments as used in Theorem 1 by using the fact that is also periodic, we obtain,a contradiction since .Hence .Proof of Theorem 2 Suppose there exists a
24、 non-trivial solution f of (1.1) that satisfies . We deduce , so and are linearly dependent by Corollary 1 (a). However, Lemma 2 implies that andare linearly independent. This is a contradiction. Hence holds for each non-trivial solution f of (1.1). This completes the proof of Theorem 2.中文翻译一些周期性的二阶
25、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，and ，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，（8），E型的f(z)，被定义为同样，E型的亚纯函数的零点收敛指数，被定义为我们说，如果一个亚纯函数满足增长的正常秩序我们考虑的二阶线性微分方程在是一个整函数在。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211，13，1719。在函数中正确的，银行和莱恩6证明了如下定理定理A 设这函数是一个周期性函数，周期为在整个函数存在
26、。如果有奇数阶极点在和，然后对于任何一个结果答案在(1.1)中广义这样的结果：上述结论仍然认为，如果我们只是假设，既和的极点，并且至少有一个是奇数阶。此外，较强的结论 （1.2）认为。当是超越在，高10证明了如下定理定理B设，其中是一个超越整函数与，是奇正整并且，设，那么任何微分解在（1.1）的函数必须有。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，是任意正整数。如果在另一方面，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，其中，函数和函数是整函数先验和不等于一个正整数或无穷大，并函数任意。（一） 假设（a）如果函数f是一个非平凡解
27、在（1.1），那么和是线性相关。（b）如果函数和函数在（1.1）是两个线性无关函数，则存在这样一个条件。（二） 假设（a）如果函数f有一个非平凡解在（1.1）且，和是线性相关的。 如果函数和函数在（1.1）在（1.1）是两个线性无关函数，则存在这样一个条件。定理 D 待添加的隐藏文字内容1让是一个超越整函数和它的秩序是正整数或无穷大。设，和p是一个奇正整数。然后或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，当是无穷的。本文的主要目的是改善上述结果的情况下，当是超越。特别地，我们找到的条件下定理D仍然成立的情况下，当是一个正整数或无穷大
28、。我们将证明在第3节的结果如下：定理1设，其中，和先验和不等于一个正整数或无穷，任意整函数。如果定期二阶线性微分方程和的解不是一些属性是两个线性无关的解在（1.1），然后或者我们的说法，定理1的结论仍然有效，如果我们假设函数不等于一个正整数或无穷大，任意和承担的情况下，当其低阶不等于一个整数或无穷超然是任意的，我们只需要考虑在，。推论1设，其中，函数和函数是整个先验和不超过1 / 2，并且任意的。（一） 如果函数f是一个非平凡解在（1.1）中，那么和是线性相关。（二） 如果和是两个线性无关解在（1.1）中，那么。定理2设是一个超越整函数及其低阶不超过1 / 2。设，其中和p是一个奇正整数，则为
29、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，但结合定理2定理的研究。推论2设是一个超越整函数。设，其中和 p是一个奇正整数。假设要么（一）或（二）中认为：（一）不是正整数或无穷;（二）然后为每一个非平凡解在（1.1）中函数f对于。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），和的假设是整个周期，并且函数f是有一个非平凡解进一步假设函数f满足;，是在非恒定和理性的，而且，如果，且是常数。则存在一个整数q与 ，和是线性相关。相同的结论认为，如果是超越，
30、和f满足，如果，然后通过一个无限措施的集合为，且引理2（10） 设是一个周期为在（包括那些可以改变这种情况下极奇数阶设是定期与整函数周期在的先验。在（1.1）中由不同的时期，有一个满足，那么和是线性无关的解。3主要结果的证明主要结果的证明的基础上8和15。定理1的证明让我们假设。正弦和是线性无关的，引理1意味着和必须是线性相关的。设，则满足微分方程, (2.1)其中是和(见12, p. 5 or 1, p. 354)，且或某些非零的常数。显然，和是两个周期，而是定义函数。在（2.1），也定期与周期。因此，我们可以找到一个解析函数在，使代入（2.1）得这种表达 (2.2)由于和在，理论21，p.
31、15给出了他们的结论， （2.3）其中，是一些整数，和函数分析和上非零，和是整函数。按照相同的 8中，我们得出， （2.4）其中，此外，下列结论由8得,其中是定义为(resp, ),定期二阶线性微分方程解的一些性质其中，(resp. 表示一个计数功能，只计算在右半平面的零点（在左半平面），是在 的零点收敛指数，它的定义为由条件，我们得到。现在（2.3）代入（2.2）中 （2.5）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）。假设和与线性无关，那么，我们证明推论1的结论（一），与线性相关，J =1;2。假设，然后我们可以找到的一个非零的常数，重复同样的论点定理1中使用的事实，也是能找到，我们得到与自矛盾，因此。定理2的证明假设存在一个非平凡解的f在（1.1）中，满足。我们推断，和的线性依赖推论1（a）。然而，引理2意味着和是线性无关的。这是一对矛盾。因此，认为都有非平凡解的F在（1.1）中，这就完成了定理2的证明。
