Complex oscillation of a second-order linear differential equation with entire coefficients of order
© Shen et al.; licensee Springer. 2014
Received: 22 February 2014
Accepted: 22 May 2014
Published: 23 July 2014
In this paper, the authors investigate the interaction between the growth, zeros of solutions with the coefficients of second-order linear differential equations in terms of order and obtain some results in general form.
1 Introduction and notations
where is an entire function or a meromorphic function of finite order or finite iterated order, and have obtained many results about the interaction between the solutions and the coefficient of (1.1) (see [4–7]). What about the case when is an entire function of -order or more general growth? In the following, we will introduce some notations about -order, where p and q are two positive integers and satisfy throughout this paper (see [8–11]). Firstly, for , we define and , , and for all sufficiently large r, we define and , . Especially, we have and . Secondly, we denote the linear measure and the logarithmic measure of a set by and .
Definition 1.1 ()
Remark 1.1 We use and to denote the order and the iterated order of a function .
Remark 1.2 By Definition 1.2, we can similarly give the definition of the growth index of the iterated exponent of convergence of the zero-sequence of a meromorphic function by .
Definition 1.4 ()
Remark 1.3 We use , and , to denote the (iterated) exponent of convergence of the zero-sequence and pole-sequence of a meromorphic function .
Recently, some authors have investigated the exponent of convergence of the zero-sequence and pole-sequence of the solutions of second-order linear differential equations (see [13–15]) and have obtained the following results.
Theorem A ()
Theorem B ()
If , then holds for all solutions of type , where .
Theorem C ()
Let be an entire function with , let f be any non-trivial solution of (1.1), and assume . Then .
Theorem D ()
Let be an entire function with and . Let and be two linearly independent solutions of (1.1) such that . Let be any entire function for which either or and . Then any two linearly independent solutions and of the differential equation satisfy .
Theorem E ()
in the unit disc, where the definition of φ-order of is given as follows.
Definition 1.5 ()
On the basis of Definition 1.5, it is natural for us to give the order of a meromorphic function in the complex plane.
Similar to Definition 1.6, we can also define the exponent of convergence of the (distinct) zero-sequence of a meromorphic function .
If , , .
In this paper, we add two conditions on as follows: is a non-decreasing unbounded function and satisfies (i) , (ii) for some . Throughout this paper, we assume that always satisfies the above two conditions without special instruction.
- (i)If is an entire function, then
- (ii)If is a meromorphic function, then
- (ii)Without loss of generality, assume that , then . Since(1.14)
By (1.15) and (1.17), it is easy to see that . By the same proof above, we can obtain the conclusion . □
Remark 1.4 If , Definitions 1.1 and 1.3 are special cases of Definitions 1.6 and 1.7.
2 Main results
In this paper, our aim is to make use of the concept of order of entire functions to investigate the growth, zeros of the solutions of equation (1.1).
Theorem 2.1 Let be an entire function satisfying . Then holds for all non-trivial solutions of (1.1).
Theorem 2.2 Let be an entire function satisfying , let , be two linearly independent solutions of (1.1) and denote . Then . If , then holds for all solutions of type , where .
Theorem 2.3 Let be an entire function satisfying . Then holds for all non-trivial solutions of (1.1).
Theorem 2.4 Let be an entire function satisfying , let and be two linearly independent solutions of (1.1) such that . Let be any entire function satisfying . Then any two linearly independent solutions and of the differential equation satisfy .
3 Some lemmas
where is the central index of .
Let and be monotone non-decreasing functions such that outside of an exceptional set of finite linear measure or finite logarithmic measure. Then, for any , there exists such that for all .
- (i)If , then(3.2)
- (ii)For , we have(3.3)
By (3.5), (3.6), (3.9) and (3.10), we obtain the conclusion of Lemma 3.4. □
Therefore, by (3.12) and (3.14), we have . □
where is a set of finite linear measure, not necessarily the same at each occurrence. By (3.15), (3.16) and , we have ().
Lemma 3.7 ()
If is an entire function of order, we have a similar result as follows.
4 Proofs of Theorems 2.1-2.4
By (4.4), we have . Therefore, we have that holds for all non-trivial solutions of (1.1). □
By Definition 1.6 and Lemma 3.2, we have , this is a contradiction. Therefore, the first assertion is proved.
By Definition 1.6 and (4.12), we have , this is a contradiction with Theorem 2.1. Therefore, we have that holds for all solutions of type , where . □
we have , this is a contradiction. Therefore, we have that holds for all non-trivial solutions of (1.1). □
From (4.13) and (4.17), we have , this is a contradiction. Therefore, we obtain the conclusion of Theorem 2.4. □
The authors thank the referee for his/her valuable suggestions to improve the present article. This work was supported by the NSFC (11301233), the Natural Science Foundation of Jiang-Xi Province in China (20132BAB211001, 20132BAB211002), and the Foundation of Education Department of Jiangxi (GJJ14272, GJJ14644) of China.
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