Abstract and Applied Analysis

Volume 2013 (2013), Article ID 634739, 8 pages

http://dx.doi.org/10.1155/2013/634739

Research Article

Complex Oscillation of Higher-Order Linear Differential Equations with Coefficients Being Lacunary Series of Finite Iterated Order

Jin Tu,¹ Hong-Yan Xu,² Hua-ming Liu,³ and Yong Liu4,5

¹College of Mathematics and Information Science, Jiangxi Normal University, Nanchang 330022, China

²Department of Informatics and Engineering, Jingdezhen Ceramic Institute, Jingdezhen, Jiangxi 333403, China

³School of Sciences, Jiangxi Agricultural University, Nanchang 330045, China

⁴Department of Mathematics, Shaoxing College of Arts and Sciences, Shaoxing, Zhejiang 312000, China

⁵Department of Physics and Mathematics, University of Eastern Finland, P.O. Box 111, 80101 Joensuu, Finland

Received 18 January 2013; Accepted 12 April 2013

Academic Editor: Micah Osilike

Copyright © 2013 Jin Tu 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.

The authors introduce the lacunary series of finite iterated order and use them to investigate the growth of solutions of higher-order linear differential equations with entire coefficients of finite iterated order and obtain some results which improve and extend some previous results of Belaidi, 2006, Cao and Yi, 2007, Kinnunen, 1998, Laine and Wu, 2000, Tu and Chen, 2009, Tu and Deng, 2008, Tu and Deng, 2010, Tu and Liu, 2009, and Tu and Long, 2009.

1. Definitions and Notations

In this paper, we assume that readers are familiar with the fundamental results and standard notations of the Nevanlinna theory of meromorphic functions (see [1–3]). In order to describe the growth of order of entire functions or meromorphic functions more precisely, we first introduce some notations about finite iterated order. Let us define inductively, for and . For all sufficiently large , we define and . We also denote and . Moreover, we denote the logarithmic measure of a set by , and the upper logarithmic density of is defined by

Throughout this paper, we use to denote a positive integer. In the following, we recall some definitions of entire functions or meromorphic functions of finite iterated order (see [4–10]).

Definition 1. The -iterated order of a meromorphic function is defined by

Remark 2. If is an entire function, then the -iterated order of is defined by

If , the classical growth of order of is defined by

If , the hyperorder of is defined by

Definition 3. If is an entire function with , then the -iterated type of is defined by

Definition 4. The -iterated lower order of an entire function is defined by

Definition 5. The finiteness degree of the iterated order of a meromorphic function is defined by

Definition 6. The -iterated exponent of convergence of -point of a meromorphic function is defined by

If , the -iterated exponent of convergence of zero-sequence and distinct zero-sequence of a meromorphic function are defined, respectively, by

The -iterated lower exponent of convergence of zero-sequence and distinct zero-sequence of a meromorphic function are defined, respectively, by

2. Introductions and Main Results

In the past ten years, many authors have investigated the complex oscillation properties of the higher-order linear differential equations

with being entire functions or meromorphic functions of fast growing (e.g., see [4–12]), and obtained the following results.

Theorem A (see [8]). Let be entire functions, if , then and hold for all solutions of (12).

Theorem B (see [8]). Let be entire functions and let . If or for all , then and hold for all nontrivial solutions of (12).

Theorem C (see [4, 12]). Let be entire functions and let . Assume that and . Then, every solution of (12) satisfies and .

Theorem D (see [10]). Let be entire functions of finite iterated order satisfying , , and . Then, every nontrivial solution of (12) satisfies .

Theorem E (see [10]). Let be entire functions of finite iterated order satisfying and , where is a set of of finite linear measure, then every nontrivial solution of (12) satisfies .

Theorem F(see [5]). Let be entire functions of finite iterated order such that there exists one transcendental satisfying for all , then (12) has at least one solution that satisfies and .

Remark 7. Theorems B–E are investigating the growth of solutions of (12) when the coefficients are of finite iterated order and grows faster than other coefficients in (12). What can we have if there exists one middle coefficient such that grows faster than other coefficients in (12) or (13)? Many authors have investigated this question when is of finite order and obtained many results (e.g., see [13–15]). Here, our question is that under what conditions can we obtain similar results with Theorems B-C if is of finite iterated order and grows faster than other coefficients in (12) or (13).

In 2009, Tu and Liu make use of the proposition of lacunary power series to investigate the above question in the case and obtain the following result.

Theorem G (see [15]). Let be entire functions satisfying . Suppose that is an entire function of regular growth such that the sequence of exponents satisfies Fabry gap condition

then one has

(i)if 0, then every transcendental solution of (13) satisfies ; (ii)if , then every transcendental solution of (13) satisfies .

In this paper, we continue our research in this area and obtain the following results.

Theorem 8. Let be entire functions of finite iterated order and satisfying and . Suppose that is an entire function such that the sequence of exponents satisfies

for some , then one has

(i)if or and , then every transcendental solution of (13) satisfies ; furthermore if , then every transcendental solution of (13) satisfies ;(ii)if and , then all solutions of (13) satisfy and ; (iii)if , then all solutions of (13) satisfy , and holds for all solutions of (13) with at most one exceptional solution satisfying .

Remark 9. If is an entire function of finite order and the sequence of exponents satisfies (14), then (18) in Lemma 15 holds for , but for entire functions of infinite order, (14) certainly does not imply (18) in Lemma 15 (see [8]); therefore, we need more stringent gap condition (15) which is sufficient and unnecessary for Theorem 8.

Theorem 10. Let be entire functions of finite iterated order satisfying ,. Suppose that is an entire function such that the sequence of exponents satisfies gap condition (15), then every transcendental solution of (13) satisfies . Furthermore if , then every transcendental solution of (13) satisfies .

Theorem 11. Let be entire functions satisfying ,. Suppose that as outside a set of of finite logarithmic measure, then one has

(i)if , then every transcendental solution of (13) satisfies ; furthermore, if , then every transcendental solution of (13) satisfies ;(ii)if and , then all solutions of (13) satisfy and ; (iii)if , then all solutions of (13) satisfy , and holds for all solutions of (13) with at most one exceptional solution satisfying ; (iv)if and and , then every transcendental solution of (13) satisfies .

Remark 12. Theorem 10 implies that all the solutions of (13) are of regular growth if is of regular growth under some conditions; Theorem 11 is an improvement of the Theorem in [14, page 2694] and Theorems in [16, page 624]. In fact, by Lemma 15, the gap condition (15) in Theorem 8 implies that as outside a set of of finite logarithmic measure; therefore, Theorem 11 is a generalization of Theorem 8 in a sense, but the condition on in Theorem 8 is more stringent than that in Theorem 11. In addition, Theorems 8–11 may have polynomial solutions of degree if .

3. Preliminary Lemmas

Lemma 13 (see [17]). Let be a transcendental meromorphic function, and let be a given constant, for any given constant and for any given ,

(i)there exist a constant and a set having finite logarithmic measure such that for all satisfying , one has

(ii)There exists a set that has linear measure zero a constant that depends only on , for any , there exists a constant such that for all satisfying and , one has

Remark 14. Throughout this paper, we use to denote a set having finite logarithmic measure or finite linear measure, not always the same at each occurrence.

Lemma 15 (see [18]). Let be an entire function and the sequence of exponents satisfies the gap condition (15). Then for any given ,

holds outside a set of finite logarithmic measure, where , .

Lemma 16 (see [4]). Let be an entire function of finite iterated order satisfying and , then for any given , there exists a set having infinite logarithmic measure such that for all , one has

Lemma 17. Let be an entire function of finite iterated order satisfying and such that the sequence of exponents satisfies the gap condition (15). Then, for any given , there exists a set having infinite logarithmic measure such that for all , one has

Proof. By Lemma 15, for any given , we have

For any given , we can choose and sufficiently small such that and ; by Lemma 16, there exists a set having infinite logarithmic measure such that for all , we have

where is a set having infinite logarithmic measure.

Lemma 18 (see [13]). Let be a transcendental entire function. Then, there is a set having finite logarithmic measure such that for all satisfying and , one has

Lemma 19 (see [7, 9, 10]). Let be meromorphic functions, and let be a meromorphic solution of (13) satisfying one of the following conditions:

(i);(ii),

then .

Lemma 20 (see [8]). Let be entire functions of finite iterated order, if . Then holds for all solutions of (13).

Lemma 21 (see [2]). Let be monotone increasing functions such that

(i) outside of an exceptional set of finite linear measure. Then, for any , there exists such that for all .(ii) outside of an exceptional set of finite logarithmic measure. Then, for any , there exists such that for all .

Lemma 22 (see [19]). Let be an entire function of finite iterated order satisfying . Then, for any given , there exists a set having infinite logarithmic measure such that for all , one has

Lemma 23 (see [2, 20]). Let be a transcendental entire function, let and a point such that and that holds. Then, there exists a set of finite logarithmic measure such that

holds for all and all , where is the central index of .

Lemma 24 (see [7, 9]). Let be an entire function of finite iterated order satisfying . Then, one has

Lemma 25. Let be entire functions of finite iterated order satisfying . Then, every solution of (13) satisfies .

Proof. By (13), we have

By Lemma 23, there exists a set having finite logarithmic measure such that for all and , we have

By Lemma 22, for any given , there exists a set having infinite logarithmic measure such that for all and , we have

Hence from (27)–(29), for any given and for all satisfying and , we have

By (30) and Lemma 24, we have . Therefore, we complete the proof of Lemma 25.

Lemma 26. Let be meromorphic functions of finite iterated order; if is a meromorphic solution of the (13) and satisfies , then .

Proof. By (13), we have

By (31), we get

By the lemma of logarithmic derivative and (31), we have

By (32) and (33), we have

Since ,  , for sufficiently large , we have

By (34)-(35), we have

By Lemma 21   and by (36), we have .

Lemma 27. Let be a transcendental entire function, for each sufficiently large , and let be a point satisfying . Then, there exists a constant such that for all satisfying and , one has

Proof. If is a point satisfying , since is continuous in , then there exists a constant such that for all satisfying (large enough) and , we have

By Lemma 23, we have

holds for all satisfying and .

Lemma 28. Let be a transcendental entire function, for each sufficiently large , and let be a point satisfying . Then, there exists a constant such that for all satisfying and , one has

Proof. If is a point satisfying , then by Lemma 27 there exists a constant such that for all satisfying and , we have

Since is transcendental, we have . Hence by (41), for all satisfying and , we have

Therefore, we complete the proof of Lemma 28.

Lemma 29. Let be an entire function of order . Then for any given , there exists a set with positive upper logarithmic density such that for all , one has

Proof. Since , then for any given , there exists an increasing sequence tending to such that for , we have

Since , we can choose to satisfy . Then for all , we have

Setting , we have

Thus, Lemma 29 is proved.

Lemma 30. Let be a transcendental entire function satisfying and as outside a set of finite logarithmic measure. Then for any given , there exists a set with positive upper logarithmic density and a set with linear measure zero such that for all satisfying and , one has

Proof. Since as , then by the definition of , there exists a set having linear measure zero such that for all satisfying , we have

By Lemma 29, for any given , there exists a set with positive upper logarithmic density, we have

By (48) and (49), for any given and for all satisfying and , we have

Therefore, we complete the proof of Lemma 30.

4. Proofs of Theorems 8–11

Proof of Theorem 8. Assume that is a transcendental solution of (13). By (13), we have

By Lemma 13  , there exists a set having finite logarithmic measure and a constant such that

holds for all and for sufficiently large . Since and , we choose to satisfy ; by Lemma 17, there exists a set having infinite logarithmic measure such that for all satisfying and for sufficiently large , we have

By Lemma 18, there exists a set having finite logarithmic measure such that for all satisfying and , we have

Hence from (51)–(54), for all satisfying and , we have

By (55), we have

On the other hand, by Lemma 20, we have . Therefore, every transcendental solution of (13) satisfies . Furthermore if , then by Lemma 19, we have that every transcendental solution of (13) satisfies .

We assume that is a solution of (13). By the elementary theory of differential equations, all the solutions of (13) are entire functions and have the form

where are complex constants, is a solution base of (12), and is a solution of (13) and has the form

where are certain entire functions satisfying

where are differential polynomials in and their derivative with constant coefficients, and is the Wronskian of . By Theorem A, we have ; then by (57)–(59), we get

Since , it is easy to see that by (13).

Suppose that is a solution of (13), it is easy to see that by (13). On the other hand, since and by (57)–(59), we have

Therefore, all solutions of (13) satisfy .

By the same proof in Theorem 4.2 in [8, page 401], we can obtain that all solutions of (13) satisfying with at most one exceptional solution satisfying .

Proof of Theorem 10. Suppose that is a transcendental solution of (13), by the same proof in Theorem 8, we have . Thus, it remains to show that . We choose to satisfy

Since the sequence of exponents of satisfies (15) and , then by Lemma 15, there exists a set having finite logarithmic measure such that for all sufficiently large , we have

Hence from (51), (52), (54), and (63), for all satisfying and , we have

Since is arbitrarily close to , by (64) and Lemma 21  , we have

On the other hand, by Lemma 26, we have ; therefore, every transcendental solution of (13) satisfies .

Proof of Theorem 11. By Lemma 20, we know that every solution of (13) satisfies . In the following, we show that every transcendental solution of (13) satisfies . Suppose that is a transcendental solution of (13). For each sufficiently large circle , we take a point satisfying . By Lemma 28, there exist a constant and a set such that for all satisfying and , we have

By Lemma 13  , there exist a set having linear measure zero and a constant such that for sufficiently large and for all satisfying , we have

Setting , for all satisfying and and for any given , we have

Since as , by Lemma 30, for any given , there exists a set with and a set with linear measure zero such that for all satisfying and , we have

Substituting (66)–(69) into (51), for all satisfying and , we have

From (70), we have . Therefore, every transcendental solution of (13) satisfies . Furthermore, if , then every transcendental solution of (13) satisfies .

By the same proof in Theorems 8 and 10, we can obtain the conclusions .

Acknowledgments

This project was supported by the National Natural Science Foundation of China (Grant nos. 11171119, 11261024, and 61202313), the Natural Science Foundation of Jiangxi Province in China (20122BAB211005, 20114BAB211003, 2010GQS0119, and 20122BAB201016), and the Foundation of Education Bureau of Jiangxi Province in China (Grant no. GJJ12206).

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