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Complex oscillation of a second-order linear differential equation with entire coefficients of order
Advances in Difference Equations volume 2014, Article number: 200 (2014)
Abstract
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.
MSC:30D35, 34A20.
1 Introduction and notations
In this paper, we shall assume that readers are familiar with the standard notations of Nevanlinna value distribution theory (see [1–3]). The theory of complex linear equations has been developed since 1960s. Many authors have investigated the second-order linear differential equation
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 ([10])
If is a meromorphic function, the -order of is defined by
Especially, if is an entire function, then the -order of is defined by (see [8, 9, 11, 12])
Remark 1.1 We use and to denote the order and the iterated order of a function .
The growth index (or the finiteness degree) of the iterated order of a meromorphic function is defined by
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 .
The exponent of convergence of the (distinct) zero-sequence of a meromorphic function is respectively defined by
Definition 1.4 ([10])
The exponent of convergence of the (distinct) pole-sequence of a meromorphic function is respectively defined by
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 ([5])
Let A be a transcendental meromorphic function of order , where , and assume that . Then, if is a meromorphic solution of (1.1), we have
Theorem B ([13])
Let be an entire function with . Let , be two linearly independent solutions of (1.1) and denote . Then and
If , then holds for all solutions of type , where .
Theorem C ([13])
Let be an entire function with , let f be any non-trivial solution of (1.1), and assume . Then .
Theorem D ([13])
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 ([14])
Let A be a meromorphic function with , and assume that . Then, if f is a nonzero meromorphic solution of (1.1), we have
In the special case where either or the poles of f are of uniformly bounded multiplicities, we can conclude that
In [16], Chyzhykov and his co-authors introduced the definition of φ-order of , where is a meromorphic function in the unit disc and used it to investigate the interaction between the analytic coefficients and solutions of
in the unit disc, where the definition of φ-order of is given as follows.
Definition 1.5 ([16])
Let be a non-decreasing unbounded function, the φ-order of a meromorphic function in the unit disc is defined by
On the basis of Definition 1.5, it is natural for us to give the order of a meromorphic function in the complex plane.
Definition 1.6 Let be a non-decreasing unbounded function, the order and lower order of a meromorphic function are respectively defined by
Similar to Definition 1.6, we can also define the exponent of convergence of the (distinct) zero-sequence of a meromorphic function .
Definition 1.7 The exponent of convergence of the (distinct) zero-sequence of a meromorphic function is respectively defined by
Proposition 1.1 If , are meromorphic functions satisfying , , then
-
(i)
, ;
-
(ii)
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.
Proposition 1.2 Let satisfy the above two conditions (i)-(ii).
-
(i)
If is an entire function, then
-
(ii)
If is a meromorphic function, then
Proof (i) By the inequality (), set (), we have
By (1.13) and , it is easy to see that conclusion (i) holds.
-
(ii)
Without loss of generality, assume that , then . Since
(1.14)
then by (1.14) and , we have
On the other hand, since , we have
By (1.16) and , we have
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
Let be a transcendental entire function, and let z be a point with at which . Then, for all outside a set of r of finite logarithmic measure, we have
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 .
Let be an entire function, be the maximum term, i.e., , and let be the central index of f.
-
(i)
If , then
(3.2) -
(ii)
For , we have
(3.3)
Lemma 3.4 Let be an entire function satisfying and , and let be the central index of f, then
Proof Let . Without loss of generality, we can assume that . From (3.2), for any , we have
By the Cauchy inequality, it is easy to see , hence
where is a constant. By Proposition 1.2, (3.4) and (), we have
On the other hand, set , by (3.3), we have
Since is a bounded sequence, by (3.7), we have
where is a constant. By Proposition 1.2, (3.8), () and , we have
By (3.5), (3.6), (3.9) and (3.10), we obtain the conclusion of Lemma 3.4. □
Lemma 3.5 Let and be entire functions of order and denote . Then
Proof Let , and be unintegrated counting functions for the number of zeros of , and . For any , it is easy to see
By Definition 1.7 and (3.11), we have
On the other hand, since the zeros of must be the zeros of or the zeros of , for any , we have
By Definition 1.7 and (3.13), we have
Therefore, by (3.12) and (3.14), we have . □
Lemma 3.6 Let be a transcendental meromorphic function satisfying , where only satisfies , and let k be any positive integer. Then, for any , there exists a set having finite linear measure such that for all , we have
Proof Set , since , for sufficiently large r and for any given , we have
By the lemma of logarithmic derivative, 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 ().
We assume that () holds for any positive integer k. By , for all , we have
By (3.16) and (3.17), for , we have
□
Lemma 3.7 ([19])
Let be an entire function of -order, and can be represented by the form
where and are entire functions such that
If is an entire function of order, we have a similar result as follows.
Lemma 3.8 Let be an entire function of order, and can be represented by the form
where and are entire functions of order such that
4 Proofs of Theorems 2.1-2.4
Proof of Theorem 2.1 Set . First, we prove that every solution of (1.1) satisfies . If is a polynomial solution of (1.1), it is easy to know that holds. If is a transcendental solution of (1.1), by (1.1) and Lemma 3.1, there exists a set having finite logarithmic measure such that for all z satisfying and , we have
And hence, we have
By (4.1) and Lemma 3.2, there exists some () such that for all , we have
By Lemma 3.4, (4.2) and the two conditions on , we have
On the other hand, by (1.1), we have
By (4.4), we have . Therefore, we have that holds for all non-trivial solutions of (1.1). □
Proof of Theorem 2.2 Set , by Theorem 2.1, we have . Hence, we have
By Lemma 3.5 and (4.5), we have
It remains to show that . By (1.1), we have (see [[13], pp.76-77]) that all zeros of are simple and that
where is a constant. Hence,
By Lemma 3.6, for all , we have , and . By (4.8), for all , we have
Let us assume . Since all zeros of are simple, we have
By (4.9) and (4.10), for all , we have
By Definition 1.6 and Lemma 3.2, we have , this is a contradiction. Therefore, the first assertion is proved.
If , let us assume that holds for any solution of type (). We denote and , then we have and . Since (4.9) holds for and and , we have
By , and (4.10), for some , we have
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 . □
Proof of Theorem 2.3 By Theorem 2.1 and , it is easy to know that holds. It remains to show that . Let us assume . By (1.1) and a similar proof of Theorem 5.6 in [[13], p.82], we have
By (4.13), the assumption and , for some , we have
By Definition 1.6 and (4.14), we have . By
we have , this is a contradiction. Therefore, we have that holds for all non-trivial solutions of (1.1). □
Proof of Theorem 2.4 As a similar proof of Theorem 3.1 in [6], we denote and . Let us assume
By Theorem 2.1, we have , and hence, by Lemma 3.6, for any integer and for any , we have
Furthermore, by Theorem 2.1, we have , and hence we have for some . And the order of the function implies that
By (4.9), we obtain
By Definition 1.6 and (4.15), we have . On the other hand, by
we have , hence . The same reasoning is valid for the function , we have
and . Since and , by Lemma 3.8, we may write
where P, Q, R, S are entire functions satisfying , and . Substituting (4.18) into (4.16) and (4.17), we have
where and are meromorphic functions satisfying (). Equation (4.19) subtracting (4.20), we have
where is a meromorphic function satisfying . From (4.21), we have
where and are meromorphic functions satisfying (), and . Deriving (4.22), we have
where is a meromorphic function satisfying . Eliminating by (4.22) and (4.23), we have
where is a meromorphic function satisfying . Since , therefore by (4.24), we have , thus we have , . Hence
From (4.16), (4.17) and (4.25), we have
By Lemma 3.6, we obtain
This implies
Hence, by Proposition 1.1, we have . Since , we have
From (4.13) and (4.17), we have , this is a contradiction. Therefore, we obtain the conclusion of Theorem 2.4. □
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Acknowledgements
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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XS, JT and HYX completed the main part of this article, JT and HYX corrected the main theorems. All authors read and approved the final manuscript.
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Shen, X., Tu, J. & Xu, H.Y. Complex oscillation of a second-order linear differential equation with entire coefficients of order. Adv Differ Equ 2014, 200 (2014). https://doi.org/10.1186/1687-1847-2014-200
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DOI: https://doi.org/10.1186/1687-1847-2014-200