The Scientific World Journal

Volume 2013 (2013), Article ID 243873, 8 pages

http://dx.doi.org/10.1155/2013/243873

Research Article

The Oscillation on Solutions of Some Classes of Linear Differential Equations with Meromorphic Coefficients of Finite -Order

Hong-Yan Xu,¹ Jin Tu,² and Zu-Xing Xuan³

¹Department of Informatics and Engineering, Jingdezhen Ceramic Institute, Jingdezhen, Jiangxi 333403, China

²Institute of Mathematics and Informatics, Jiangxi Normal University, Nanchang, Jiangxi 330022, China

³Beijing Key Laboratory of Information Service Engineering, Department of General Education, Beijing Union University, No. 97 Bei Si Huan Dong Road, Chaoyang District, Beijing 100101, China

Received 21 October 2013; Accepted 17 November 2013

Academic Editors: F. J. Garcia-Pacheco, L. Kong, and J. Sun

Copyright © 2013 Hong-Yan Xu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

This paper considers the oscillation on meromorphic solutions of the second-order linear differential equations with the form where is a meromorphic function with -order. We obtain some theorems which are the improvement and generalization of the results given by Bank and Laine, Cao and Li, Kinnunen, and others.

1. Introduction and Main Results

The purpose of this paper is to study the oscillation on solutions of linear differential equations in the complex plane. It is well known that Nevanlinna theory has appeared to be a powerful tool in the field of complex differential equations. We assume that readers are familiar with the standard notations and the fundamental results of the Nevanlinna’s value distribution theory of meromorphic functions (see [1, 3, 4]). Throughout the paper, a meromorphic function means meromorphic in the complex plane . In addition, we use and to denote the order and the exponent of convergence of zero sequence of meromorphic function , respectively. For sufficiently large , we define and and , .

For the second-order linear differential equation,

where is an entire function or meromorphic function of finite order. In 1982, Bank and Laine [5] mainly studied the distribution of zeros of solutions of (1) when is an entire function of finite order.

Obviously, all solutions of (1) are entire when is entire. However, there are some immediate difficulties when is meromorphic; for example, the solutions of (1) may not be entire, and it is possible that no solution of (1) except the zero solution is single-valued on the plane. In 1983, Bank and Laine [6] investigated the exponent of convergence of zero sequence of nontrivial solutions of (1), when is meromorphic function and obtained the results as follows.

Theorem 1 (see [6, Theorem  5]). Let be a transcendental meromorphic function of order , where . Assume that . Then, if is a meromorphic solution of (1), one has

Theorem 2 (see [6, Theorem  6]). Let be a transcendental meromorphic function. Assume that (1) possesses two linearly independent meromorphic solutions and satisfying , . Then, any solution of (1) which is not a constant multiple of either or satisfies

unless all solutions of (1) are of finite order. In the special case where , one can conclude that unless all solutions of (1) are of finite order.

After their work, many authors have investigated the growth and the exponent of convergence of zero sequence of non-trivial solutions of (1) and obtained many classical results (see [5, 7, 8]).

In 1998, Kinnunen [9] further investigated the oscillation results of entire solutions of (1) when is an entire function with finite iterated order and obtained some theorems which improved some theorems given by Bank and Laine [5]. Later, Chen [10] and Wang and Lü [11] studied the oscillation of solutions of (1) when is a meromorphic function with finite order by using the Wiman-Valiron theory; they obtained some results which extend some theorems of Kinnunen [9]. In 2007, Liang and Liu [12] considered the complex oscillation on (1) when is a meromorphic function with many finite poles. They extended the above oscillation results by using the method of the Wiman-Valiron theory. Although the Wiman-Valiron theory is a powerful tool to investigate entire solutions, it is only useful for the meromorphic function with the exponent of the convergence sequence of poles which is less than the order of if we consider (1). In 2010, Cao and Li [13] made use of a result due to Chiang and Hayman [14] instead of the Wiman-Valiron theory and obtained four oscillation theorems and three corollaries which extended the above results due to Bank-Laine, Kinnunen, and Liang and Liu.

Theorem 3 (see [13, Theorem  1.6]). Let be a meromorphic function with , and assume that . Then, if is a nonzero meromorphic solution of (1), one has . In the special case where either or the poles of are of uniformly bounded multiplicities, one can conclude that .

Theorem 4 (see [13, Theorem  1.7]). Let be a meromorphic function with . Assume that (1) possesses two linearly independent meromorphic solutions and . Denote . If , then any nonzero solution of (1) which is not a constant multiple of either or satisfies , unless all solutions of (1) are of finite iterated -order. In the special case where   , or (e.g., is an entire function), one can conclude that .

Theorem 5 (see [13, Theorem  1.9]). Let be a meromorphic function with . Assume that and are two linearly independent meromorphic solutions of (1) such that . Let be any meromorphic function for which . Let and be two linearly independent solutions of the differential equation . Denote and . If

then .

In 1976, Juneja and his coauthors [15, 16] firstly introduced the concept of -order of entire functions. Recently, Belaïdi [17] and Liu et al. [18] investigated the growth of solutions of complex differential equations

where , are entire or meromorphic functions with finite -order, by using the idea of -order, and obtained some interest results which improved and extended some previous theorems given by [5–7, 9, 19].

Thus, it is interesting to consider the complex oscillation on the meromorphic solutions of (1) for the case when is entire or meromorphic functions in the terms of the idea of -order.

In this paper, we further investigated the complex oscillation of meromorphic solutions of (1) when is meromorphic by using the idea of -order. To state our theorems, we first introduce the concepts of entire functions of -order (see [15, 16, 18]). Throughout this paper, we always assume that , are positive integers satisfying .

Definition 6. If is a transcendental entire function, the -order of is defined by

Remark 7. If is a polynomial, then for any . By Definition 6, we have that , .

Remark 8. If is an entire function satisfying , then

(i), , for .(ii)if is any pair of integers satisfying and , then if and if ;(iii) for and for .

Definition 9. A transcendental meromorphic function is said to have index-pair if and is not a nonzero finite number.

Definition 10. Let be two entire functions such that , and . Then the following results about their comparative growth can be easily deduced.

(i)If , then the growth of is slower than the growth of ;(ii)If , then grows faster than .(iii)If , then the growth of is slower than the growth of if while the growth of is faster than the growth of if .(iv)Let ; then , are of the same index-pair . If , then grows faster than , and if , then grows slower than . If , Definition 6 does not give any precise estimate about the relative growth of and .

Definition 11 (see [15, 16, 18]). The exponent of convergence of the zero sequence and the exponent of convergence of the distinct zero sequence of are defined respectively, by

Remark 12. It is easy to know that

Now, we will show our main results on the complex oscillation on meromorphic solutions of (1) when is meromorphic with finite -order as follows.

Theorem 13. Let be a transcendental meromorphic function with . Assume that . Then, if is a nonzero meromorphic solution of (1), one has

In the special case where either or the poles of are of uniformly bounded multiplicities, one can get that

Theorem 14. Let be a transcendental meromorphic function with . Assume that (1) possesses two linearly independent meromorphic solutions and . Denote . If , then any nonzero solution of (1) which is not a constant multiple of either or satisfies , unless all solutions of (1) are of finite -order. In the special case where or , one can conclude that .

Theorem 15. Let be a transcendental meromorphic function with . Assume that (1) possesses two linearly independent meromorphic solutions and . Denote . If or , and if either or the poles of are of uniformly bounded multiplicities, then one has

Theorem 16. Let be a meromorphic function with . Assume that and are two linearly independent meromorphic solutions of (1) such that

Let be any meromorphic function for which . Let and be two linearly independent solutions of the differential equation

Denote and . If

then .

Remark 17 (Following Hayman [20]). we will use the abbreviation “n.e.” (nearly everywhere) to mean “everywhere in except in a set of finite measure” in the proofs of our main results of this paper.

Remark 18. Obviously, Theorems 13–16 are the improvement of Theorems 3–5 given by Cao and Li [13].

2. Some Lemmas

For the proof of our results we need the following lemmas.

Lemma 19 (see [20, Theorem  4]). Let be a transcendental meromorphic function not of the form . Then

Using the same proof of Remark  1.3 in [9], one can easily prove the following lemma.

Lemma 20. Let be a meromorphic function of -order. Then

Lemma 21. Let be a meromorphic function with -order and , and let be an integer. Then for any ,

holds outside of an exceptional set of finite linear measure.

Proof. Let . Since , we have for all sufficiently large

By the lemma of the logarithmic derivative, we have

where is a set of finite linear measure, not necessarily the same at each occurrence. Hence we have

Next, assume that we have

for some . Since , it holds that

By (20), we again obtain

and hence, for sufficiently large ,

Using the same proof of Lemma  3.6 in [19], we can easily prove the following lemma.

Lemma 22. Let be a continuous and positive increasing function, defined for on , with -order . Then for any subset of that has finite linear measure, there exists a sequence , such that

Lemma 23. Let and be two entire functions of -order, and denote . Then

Proof. Let denote the number of the zeros of in and so on for and . Since, for any given , we have and , thus by Definition 11, we have

On the other hand, since the zero of must be the zero of or , then for any given , we have

Therefore, by Definition 11, we have

Thus we complete the proof of Lemma 22.

Lemma 24. A meromorphic function with index can be represented by the form

where , , and are entire functions such that

Proof. By using the essential part of the factorization theorem for meromorphic function of finite -order and similar to the proof of Lemma  1.8 in [9], we can get the conclusions of this lemma easily.

Lemma 25 (see [14, Theorem  6.2]). Let be a meromorphic solution of

where are meromorphic functions in the plane . Assume that not all coefficients are constants. Given a real constant , and denoting , one has

outside of an exceptional set with .

Remark 26. From the above lemma, we can see that corresponds to Euclidean measure and to logarithmic measure.

Using the above lemma, we can get the following lemma.

Lemma 27. Let be meromorphic functions with index and . If is a meromorphic solution of (20) whose poles are of uniformly bounded multiplicities or , then .

Proof. Firstly, suppose that ; then we have . Since , then we have . Second, suppose that . From (32), we know that the poles of can only occur at the poles of . Since the multiplicities of poles of are uniformly bounded, we have

where , are some suitable positive constants. Thus, we can get

Set ; for sufficiently large , we have

From Lemma 24 and (35) or (36), we obtain

or

outside of an exceptional set with finite logarithmic measure.

Using a standard method to deal with the finite logarithmic measure set, one immediately gets from the previous inequalities that .

Thus, we complete the proof of this lemma.

Lemma 28. Let be a meromorphic function with index, and let be a nonzero meromorphic solution of (1). Then

(i)if either or the poles of are of uniformly bounded multiplicities, then ;(ii)if or , then .

Proof. Suppose that is a nonzero meromorphic solution of (1). It is easy to see that (i) holds since (i) is just a special case of Lemma 25.

Next, we assume that satisfies or . By (1), we have

By (39) and Lemma 21, we can get that

holds for all sufficiently large , where has finite linear measure. Hence

holds for all sufficiently large .

If , since is a meromorphic function with index, by Definition 6 and Lemma 22, there exists a sequence such that, for all ,

holds for any sufficiently large constant . If , by Lemma 22 there exists a sequence such that, for all ,

holds for any given .

We will consider two cases as follows.

Case  1. Suppose that . Then for sufficiently large , we have

If and has index, from (42) and (44), we have . If , then from (43) and (44), we can get that .

Case  2. Suppose that . Then the inequality

holds for any given . Then from (41), (43), and (45), we can get that .

Thus, we complete the proof of Lemma 28.

3. Proof of Theorem 13

Since and is a solution of (1), it is easy to see that can not be rational nor be of the form for constants and . Thus, by Lemma 19 we have

From (1), we have

Suppose that (9) fails to hold that is

by the assumption and from (46)–(48), we can get .

Set ; by the first main theorem, we can get that

From (1) and , we can get . Thus, from (49) and Lemma 20, we have , a contradiction. Thus, (9) is true.

In the special case where either or the poles of are of uniformly bounded multiplicities, by Lemma 28 we have

Combining the above discussions, we can get (10).

Thus, this completes the proof of Theorem 13.

4. Proof of Theorem 14

Suppose that (1) possesses two linearly independent meromorphic solutions and such that , where . Let , where and are nonzero constants, and set . From (1), we can see that any pole of is a pole of . Since , and by the assumptions of Theorem 14, we have . If , from the above discussion, we can get that and . By Lemma D(e) in [6], there exists a constant such that n.e. as ,

for . Since , from (51) we can get that n.e. as ,

From and Lemma 21, we can get

And from (1), we can see that any pole of is at most double and is either a zero or pole of . Then we get

By assumptions , and (54), we can get that as for some . Together with (52), (53), and (54), we can get that n.e. as . Thus, it follows that is of finite -order.

By the identity of Abel, we have

where is equal to the Wronskian of and . Hence, by Lemma 20 and (55), we get

Reversing the roles of and , we can get that . Thus, we can get that all solutions of (1) are of finite -order if .

In special case where or , by Lemma 28, we can get that all meromorphic solutions of (1) satisfy . Hence, we can obtain that holds for any solution of (1) which is not a constant multiple of either or .

5. Proof of Theorem 15

From [6, Page 664], we can get easily. Suppose that or and that either or the poles of are of uniformly bounded multiplicities. Then by Lemma 28 we obtain

By Lemma D(e) in [6], there a constant such that n.e. as ,

By Lemmas 21 and 28, we have

for some outside of a possible exceptional set with finite linear measure. If , then for sufficiently large , we have

If , there exists a constant such that

From the above equality and (60), we have

where .

Therefore, together with (58) and either (60) or (62), we obtain

Suppose that ; then from Definitions 6 and 11 we have for some . From (63), , , and then by standard reasoning, we obtain . Thus, we get a contradiction. Therefore, we have . And since , then we have .

By Lemma D(a) in [6], and have no common zeros. Let , where and have no common zeros. This implies that and have no common zeros, that for , and that . Then by Lemma 23, we have .

Thus, we can get the following conclusion

Therefore, we complete the proof of Theorem 15.

6. Proof of Theorem 16

From and , by using a similar argument as in [9, Lemma  1.7], we can get . Suppose that from (12), we have

for some . Since and from Definition 6, for any , we have

By Lemma D(e) in [6], we have . Thus, we can get that

Hence, from the above equality, we have . On the other hand, by Lemma B(iv) in [6], we have

which implies that . Since , using the same argument as in the above for the function , we have .

From the assumptions of Theorem 16 and Lemma 24, we can write

where and . And since , we have

Substituting (69) into (68), and using the same argument as in the proof of Theorem  3.1 in [7], we can get

where . From (68), we have

For the function , similar to (68), we have

From (68), (72), and (73), we have

Since , by Lemma 22, for any , we can get

From (75), we can get easily. Thus, we can get . Since , we have

Then from (68) and (73), we have , a contradiction.

Therefore, we complete the proof of Theorem 16.

Acknowledgments

This project is supported by the NSF of China (11301233 and 61202313) and the Natural Science Foundation of Jiangxi Province in China (20132BAB211001 and 20132BAB211002). Zu-Xing Xuan is supported by the Beijing Natural Science Foundation (no. 1132013) and The Project of Construction of Innovative Teams and Teacher Career Development for Universities and Colleges under Beijing Municipality (CIT and TCD20130513).

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