- Research
- Open Access
- Published:
Results on the growth of meromorphic solutions of some linear difference equations with meromorphic coefficients
Advances in Difference Equations volume 2014, Article number: 306 (2014)
Abstract
In this paper, we investigate the growth of meromorphic solutions of some linear difference equations. We obtain some new results on the growth of meromorphic solutions when most coefficients in such equations are meromorphic functions, which are supplements of previous results due to Li and Chen (Adv. Differ. Equ. 2012:203, 2012) and Liu and Mao (Adv. Differ. Equ. 2013:133, 2013).
MSC:30D35, 39A10.
1 Introduction and main results
In this article, a meromorphic function always means meromorphic in the whole complex plane ℂ, and c always means a nonzero constant. We adopt the standard notations of the Nevanlinna value distribution theory of meromorphic functions such as , and as explained in [1–3]. In addition, we will use notations to denote the order of growth of a meromorphic function , to denote the exponents of convergence of the zero sequence of a meromorphic function , to denote the exponents of convergence of the pole sequence of a meromorphic function , and we define them as follows:
Recently, meromorphic solutions of complex difference equations have become a subject of great interest from the viewpoint of Nevanlinna theory due to the apparent role of the existence of such solutions of finite order for the integrability of discrete difference equations.
About the growth of meromorphic solutions of some linear difference equations, some results can be found in [4–20]. Laine and Yang [12] considered the entire functions coefficients case and got the following.
Theorem A [12]
Let (≢0) () be entire functions of finite order such that among those coefficients having the maximal order , exactly one has its type strictly greater than the others. Then, for any meromorphic solution to
we have .
Chiang and Feng [13, 14] improved Theorem A as follows.
Let (≢0) () be entire functions such that there exists an integer l, , such that
Suppose that is a meromorphic solution to
then we have .
Recently in [15], Peng and Chen investigated the order and the hyper-order of solutions of some second-order linear differential equations and proved the following results.
Theorem C [15]
Suppose that (≢0) () are entire functions and . Let , be two distinct complex numbers such that (suppose that ). If or , then every solution of the equation
has infinite order and .
Moreover, Xu and Zhang [16] extended the above result from entire coefficients to meromorphic coefficients.
It is well known that is regarded as the difference counterpart of . Thus a natural question is: Can we change the above second-order linear differential equation to the linear difference equation? What conditions will guarantee that every meromorphic solution will have infinite order when most coefficients in such equations are meromorphic functions?
Li and Chen [17] considered the following difference equation and obtained the following theorem.
Theorem D [17]
Let k be a positive integer, p be a nonzero real number and be a nonconstant meromorphic solution of the difference equation
where , , (≢0) () are all entire functions and , then we have .
The main purpose of this paper is to investigate the growth of meromorphic solutions of certain linear difference equations with meromorphic coefficients. The remainder of the paper studies the properties of meromorphic solutions of a nonhomogeneous linear difference equation. In fact, we prove the following results, in which there are still some coefficients dominating in some angles.
Theorem 1.1 Let k be a positive integer. Suppose that , (≢0) () are all entire functions and . Let , be two distinct complex numbers such that (suppose that ). Let be a strictly negative real constant. If or , then every meromorphic solution of the equation
satisfies .
Theorem 1.2 Suppose that , (≢0) () are all meromorphic functions and . Let , be two distinct complex numbers such that (suppose that ). Let be a strictly negative real constant. If or , then every meromorphic solution of the equation
satisfies .
Theorem 1.3 Under the assumption for the coefficients of (1.1) in Theorem 1.1, if is a finite order meromorphic solution to (1.1), then . What is more, either or .
Theorem 1.4 Under the assumption for the coefficients of (1.2) in Theorem 1.2, if is a finite order meromorphic solution to (1.2), then . What is more, either or .
Liu and Mao [18] considered the meromorphic solutions of the difference equation
one of their results can be stated as follows.
Theorem E [18]
Let (), where are polynomials with degree n (≥1), (≢0) are entire functions of . If () are distinct complex numbers, then every meromorphic solution f (≢0) of Eq. (1.3) satisfies .
In this paper, we extend and improve the above result from entire coefficients to meromorphic coefficients in the case where the polynomials are of degree 1.
Theorem 1.5 Let (), are distinct complex constants, suppose that (≢0) are meromorphic functions and , then every meromorphic solution f (≢0) of the equation
satisfies .
Theorem 1.6 Let (), are distinct complex constants, suppose that (≢0), (≢0) are meromorphic functions and , , then every meromorphic solution f (≢0) of Eq. (1.4) satisfies .
Next we consider the properties of meromorphic solutions of the nonhomogeneous linear difference equation corresponding to (1.4)
where (≢0) is a meromorphic function.
Theorem 1.7 Let () satisfy the hypothesis of Theorem 1.5 or Theorem 1.6, and let be a meromorphic function of , then at most one meromorphic solution of Eq. (1.5) satisfies and , the other solutions f satisfy .
2 Some lemmas
In this section, we present some lemmas which will be needed in the sequel.
Lemma 2.1 [13]
Let be a meromorphic function of finite order ρ, ϵ be a positive constant, and be two distinct nonzero complex constants. Then
and there exists a subset of finite logarithmic measure such that, for all z satisfying , and as ,
Lemma 2.2 [19]
Let be a meromorphic function of finite order ρ, then, for any given , there exists a set of finite logarithmic measure such that, for all z satisfying , and as ,
Lemma 2.3 [20]
Suppose that (α, β are real numbers, ) is a polynomial with degree , (≢0) is an entire function with . Set , , . Then, for any given , there exists a set that has linear measure zero such that for any , there is such that for , we have
-
(i)
if , then
-
(ii)
if , then
where is a finite set.
Lemma 2.3 applies in Theorem 1.1 where (≢0) is an entire function.
Lemma 2.4 [15]
Suppose that is a positive integer. Let () be nonconstant polynomials, where () are distinct complex numbers and . Set , , , , then there is a set that has linear measure zero. If , then there exists a ray , such that
or
where is a finite set, which has linear measure zero.
In Lemma 2.4, if is replaced by , then we have the same result.
Lemma 2.5 [19]
Consider , where (≢0) is a meromorphic function with , a is a complex constant, (). Set , then is a finite set. Then, for any given ϵ (), there exists a set that has linear measure zero, if , , then we have (when r is sufficiently large):
-
(i)
if , then
-
(ii)
if , then
where .
Lemma 2.5 applies in Theorem 1.2 where (≢0) is a meromorphic function.
Lemma 2.6 [3]
Let (, ) be meromorphic functions and (, ) be entire functions such that
-
(i)
,
-
(ii)
are not constant functions for ,
-
(iii)
(, ), where E is an exceptional set of finite linear measure, and .
Then ().
Lemma 2.7 [18]
Let , where are polynomials with degree n (≥1), (≢0) are meromorphic functions of order . If () are distinct complex numbers, then .
3 Proofs of the results
3.1 The proof of Theorem 1.1
Suppose that (1.1) admits a nontrivial meromorphic solution such that , then by Lemma 2.1, for any given ϵ such that , we have
for all r outside of a possible exceptional set with finite logarithmic measure.
Applying Lemma 2.2, we have
for all r outside of a possible exceptional set with finite linear measure.
Therefore from (1.1) we can get that
i.e.,
Setting , ().
Case 1. , which is .
Subcase 1.1. Assume that . By Lemma 2.4, for the above ϵ, there is a ray such that (where and are defined as in Lemma 2.4, is of linear measure zero) satisfying , or , for a sufficiently large r.
Since , (≢0) () are entire functions and , then when , for a sufficiently large r, by Lemma 2.3, we have
By (3.5) and (3.6), we have
Since , we know that , then . Therefore, by (3.2) we obtain
Substituting (3.1), (3.2), (3.3), (3.7) and (3.8) into (3.4), we obtain
By , and , we know that (3.9) is a contradiction.
When , , using a proof similar to the above, we can get a contradiction.
Subcase 1.2. Assume that . By Lemma 2.4, for the above ϵ, there is a ray such that (where and are defined as in Lemma 2.4, is of linear measure zero) satisfying .
Since , , and , then , thus . For a sufficiently large r, by Lemma 2.3, we get
By (3.10) and (3.11), we get
where .
Since , we see that , then , .
Since , we know that , then . Therefore, by (3.2) we obtain
Substituting (3.1), (3.2), (3.3), (3.12) and (3.13) into (3.3), we obtain
By , and , we know that (3.14) is a contradiction.
Case 2. , which is .
Subcase 2.1. Assume that , then . By Lemma 2.4, for the above ϵ, there is a ray such that (where and are defined as in Lemma 2.4, is of linear measure zero) satisfying . Since , we have . For a sufficiently large r, by Lemma 2.3, we get
By (3.15) and (3.16), we get
Using the same reasoning as in Subcase 1.1, we can get a contradiction.
Subcase 2.2. Assume that , then . By Lemma 2.4, for the above ϵ, there is a ray such that (where and are defined as in Lemma 2.4, is of linear measure zero), then , , .
Since , , and , then , thus . For a sufficiently large r, we get (3.10), (3.11) and (3.12) hold.
Using the same reasoning as in Subcase 1.2, we can get a contradiction. Thus we have .
3.2 The proof of Theorem 1.2
Since , (≢0) () are meromorphic functions and , then when , for a sufficiently large r, by Lemma 2.5, we have
Then, using a similar argument to that of Theorem 1.1, we obtain a contradiction.
3.3 The proof of Theorems 1.3 and 1.4
Suppose that is a nonconstant meromorphic solution of (1.1) such that . We first prove that . Submitting into (1.1), we get
where
Since , we have .
Now, for any given , applying Lemma 2.1 and Theorem 1.1, we can deduce that
This implies that
then follows.
Next, we assert that either or . If the assertion does not hold, we have .
Assume that is a zero (or a pole) of of order m. Applying the Hadamard factorization of a meromorphic function, we write as follows:
where , are entire functions such that , and is a polynomial such that deg .
Now, we obtain from (1.1) that
where
and
Notice that deg for and . Thus, Lemma 2.6 is valid for (3.20), hence we get that for , a contradiction to our assumption. This completes our proof.
The proof of Theorem 1.4 is similar to that of Theorem 1.3.
3.4 The proof of Theorem 1.5
Let f (≢0) be a meromorphic solution of (1.4). Suppose that , then by Lemma 2.1, for any given , there exists a subset of finite logarithmic measure such that for all z satisfying , and as r sufficiently large, we have
Set , and (). Then is a set of linear measure zero.
Consider that , (≢0) are meromorphic functions and , by Lemma 2.5, for the above , there exists a set of linear measure zero such that for any satisfying , and as , we have
-
(i)
if , then
(3.22) -
(ii)
if , then
(3.23)
Set , then is a set of linear measure zero.
Since are distinct complex constants, then there exists only one such that for any . Now we take a ray such that .
Let , , then . We discuss the following two cases.
Case 1. . We rewrite (1.4) in the form
By (3.21), (3.22) and (3.24), we get for and sufficiently large ,
When , by (3.25), we get
This is impossible.
Case 2. . By (3.21), (3.23) and (3.24), we get for and sufficiently large ,
This is a contradiction. Hence we get .
3.5 The proof of Theorem 1.6
Considering (), (≢0), (≢0) are meromorphic functions and , , by Lemmas 2.2 and 2.5, we know that for any given , there exists a subset of finite logarithmic measure such that, for all z satisfying , and as , we have
-
(i)
if , then
(3.26) -
(ii)
if , then
(3.27)
Then, using a similar argument to that of Theorem 1.5 and Theorem 1.1 and only replacing (3.22) (or (3.23)) by (3.26) (or (3.27)), we can prove Theorem 1.6.
3.6 The proof of Theorem 1.7
Let be a meromorphic solution of (1.5). Suppose that , then by Lemma 2.7, we obtain . This contradicts , therefore we have .
Suppose that there exist two distinct meromorphic solutions , of Eq. (1.5) such that , then is a meromorphic solution of the homogeneous linear difference equation corresponding to (1.5) and . By Theorem 1.5 or Theorem 1.6, we get a contradiction. So Eq. (1.5) has at most one meromorphic solution satisfying .
Next we prove in the case . Suppose that , then by the Weierstrass factorization, we obtain
where , are entire functions such that , and is a polynomial of degree 1.
In the case . Substituting (3.28) into (1.5), we get
Since are distinct complex numbers, by Lemma 2.7, we obtain that the order of the left-hand side of (3.29) is 1. This contradicts .
For , by using an argument similar to the above, we also obtain a contradiction.
It is obvious that provided that . Therefore we have .
References
Hayman WK: Meromorphic Functions. Claredon Press, Oxford; 1964.
Laine l: Nevanlinna Theory and Complex Differential Equations. de Gruyter, Berlin; 1993.
Yang CC, Yi HX: Uniqueness Theory of Meromorphic Functions. Kluwer Academic, Dordrecht; 2003.
Zhang J: Some results on zeros and the uniqueness of one certain type of high difference polynomials. Adv. Differ. Equ. 2012., 2012: Article ID 160
Chen ZX: Growth and zeros of meromorphic solution of some linear difference equations. J. Math. Anal. Appl. 2011, 373: 235-241. 10.1016/j.jmaa.2010.06.049
Ishizaki K, Yanagihara N: Wiman-Valiron method for difference equations. Nagoya Math. J. 2004, 175: 75-102.
Laine I: Nevanlinna Theory and Complex Differential Equations. de Gruyter, Berlin; 1993.
Li S, Gao ZS: Finite order meromorphic solutions of linear difference equations. Proc. Jpn. Acad., Ser. A, Math. Sci. 2011, 87(5):73-76. 10.3792/pjaa.87.73
Zheng XM, Tu J: Growth of meromorphic solutions of linear difference equations. J. Math. Anal. Appl. 2011, 384: 349-356. 10.1016/j.jmaa.2011.05.069
Chen ZX: The growth of solutions of a class of second-order differential equations with entire coefficients. Chin. Ann. Math., Ser. B 1999, 20(1):7-14. (in Chinese) 10.1142/S0252959999000035
Liu Y: On growth of meromorphic solutions for linear difference equations with meromorphic coefficients. Adv. Differ. Equ. 2013., 2013: Article ID 60
Laine I, Yang CC: Clunie theorems for difference and q -difference polynomials. J. Lond. Math. Soc. 2007, 76(2):556-566.
Chiang YM, Feng SJ:On the Nevanlinna characteristic and difference equations in the complex plane. Ramanujan J. 2008, 16: 105-129. 10.1007/s11139-007-9101-1
Chiang YM, Feng SJ: On the growth of logarithmic differences, difference quotients and logarithmic derivatives of meromorphic functions. Trans. Am. Math. Soc. 2009, 361(7):3767-3791. 10.1090/S0002-9947-09-04663-7
Peng F, Chen ZX: On the growth of solutions of some second-order linear differential equations. J. Inequal. Appl. 2011., 2011: Article ID 635604
Xu J, Zhang X: Some results of meromorphic solutions of second-order linear differential equations. J. Inequal. Appl. 2013., 2013: Article ID 304
Li S, Chen B: Results on meromorphic solutions of linear difference equations. Adv. Differ. Equ. 2012., 2012: Article ID 203
Liu H, Mao Z: On the meromorphic solutions of some linear difference equations. Adv. Differ. Equ. 2013., 2013: Article ID 133
Chen ZX, Shon KH: On the growth and fixed points of solutions of second order differential equation with meromorphic coefficients. Acta Math. Sin. Engl. Ser. 2005, 21(4):753-764. 10.1007/s10114-004-0434-z
Halburd RG, Korhonen R: Finite-order meromorphic solutions and the discrete Painlevé equations. Proc. Lond. Math. Soc. 2007, 94: 443-474.
Acknowledgements
The author thanks the referee for his/her valuable suggestions to improve the present article. The research was supported by the Beijing Natural Science Foundation (No. 1132013) and the Foundation of Beijing University of Technology (No. 006000514313002).
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
ZLY performed and drafted manuscript. All authors read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Yuan, Z.L., Ling, Q. Results on the growth of meromorphic solutions of some linear difference equations with meromorphic coefficients. Adv Differ Equ 2014, 306 (2014). https://doi.org/10.1186/1687-1847-2014-306
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1687-1847-2014-306
Keywords
- complex difference equation
- meromorphic coefficients
- growth