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1、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 meromorphic functions
2、12, 14, 16. In addition, we will use the notation ， and to denote respectively the )(ff)(forder 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 f)(fe rfTr),(logimSimilarly, ，the e-type e
3、xponent of convergence of the zeros of meromorphic function , )(fe fis defined to be rfNfre )/1,(logi)(We say that has regular order of growth if a meromorphic function satisfies)(zf )(zfrfTfrlog),(im)(We consider the second order linear differential equation0AfWhere is a periodic entire function wi
4、th period . The complex )()zeBA /2ioscillation 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 211, 13, 1719. Whenis rational in ，Bank and Laine 6 proved the following theorem)(zzeTheorem A L
5、et be a periodic entire function with period and rational )()zB /2iin .If has poles of odd order at both and , then for every solutionze(0of (1.1), )0f )(fBank 5 generalized this result: The above conclusion still holds if we just suppose that bothand are poles of , and at least one is of odd order.
6、 In addition, the stronger )(Bconclusion(1.2)(/1,(logrofrNholds. When is transcendental in , Gao 10 proved the following theorem)(zAzeTheorem B Let ,where is a transcendental entire function pjjb1)/()(tgwith , is an odd positive integer and ，Let .Then any non-1)(gp0p )(zeBAtrivia solution of (1.1) m
7、ust have . In fact, the stronger conclusion (1.2) holds.f )(fAn example was given in 10 showing that Theorem B does not hold when is any )(gpositive integer. If the order , but is not a positive integer, what can we say? Chiang 1)(gand Gao 8 obtained the following theoremsTheorem 1 Let ,where , and
8、are entire functions )()zeBA)(/()21g12gwith transcendental and not equal to a positive integer or infinity, and arbitrary. If 2g2g 1Some properties of solutions of periodic second order linear differential equations and )(zfare two linearly independent solutions of (1.1), then)(izf)(feOr 2)()(11gfeW
9、e remark that the conclusion of Theorem 1 remains valid if we assume )(1gis not equal to a positive integer or infinity, and arbitrary and still assume2,In the case when is transcendental with its lower order not equal )(/1()2gB1gto an integer or infinity and is arbitrary, we need only to consider 2
10、in , .)/()/()*210/Corollary 1 Let ,where , and arezeBA)(/1(2g12gentire functions with transcendental and no more than 1/2, and arbitrary.2g)2(a) If f is a non-trivial solution of (1.1) with ,then and )(fe)(zfare linearly dependent.)2(izf(b) If and are any two linearly independent solutions of (1.1),
11、 then1.)(2feTheorem 2 Let be a transcendental entire function and its lower order be no more than 1/2. gLet ,where and p is an odd positive integer, then)(zeBApjjbg1)/()for each non-trivial solution f to (1.1). In fact, the stronger conclusion (1.2) holds.)(fWe remark that the above conclusion remai
12、ns valid ifpjjbgB1)(We note that Theorem 2 generalizes Theorem D when is a positive integer or infinity but )(g. Combining Theorem D with Theorem 2, we have/1)(gCorollary 2 Let be a transcendental entire function. Let where )(g )(zeBAand p is an odd positive integer. Suppose that either (i) or (ii)
13、pjjbB1/()below holds:(i) is not a positive integer or infinity;)(g(ii) ;2/1then for each non-trivial solution f to (1.1). In fact, the stronger conclusion (1.2) holds.)(f2. Lemmas for the proofs of TheoremsLemma 1 (7) Suppose that and that are entire functions of period ,and that 2k20,.kAi2f is a no
14、n-trivial solution of 0)()(20kijjzyySuppose further that f satisfies ; that is non-constant and rational in/1,logrofrN0A，and that if ，then are constants. Then there exists an integer q withze3k21,.kAsuch that and are linearly dependent. The same conclusion holds ifq1)(zf)(iqfis transcendental in ，an
15、d f satisfies ，and if ，then as0Aze )(/1,(logrofrN 3kthrough a set of infinite measure, we have for .r1L,jjAT2,.1Lemma 2 (10) Let be a periodic entire function with period and be )()zeBA itranscendental in , is transcendental and analytic on .If has a pole of ze 0)(Bodd order at or (including those which can be changed into this case by varying the 0period of and . (1.1) has a solution which satisfies , )(zAEq)(zf )(/1,(logrofrNthen and are linearly independent.f)f3. Proofs of main resultsThe proof of main results are based on 8 and 15.Proof of Theorem 1 Let
