- Open Access
On growth of meromorphic solutions for linear difference equations with meromorphic coefficients
© Liu; licensee Springer 2013
Received: 2 December 2012
Accepted: 26 February 2013
Published: 19 March 2013
In this paper, we consider the value distribution of meromorphic solutions for linear difference equations with meromorphic coefficients.
1 Introduction and preliminaries
Recently, several papers (including [1–7]) have been published regarding value distribution of meromorphic solutions of linear difference equations. We recall the following results. Chiang and Feng proved the following theorem.
Theorem A ()
Then we have .
In this paper, we use the basic notions of Nevanlinna’s theory (see [8, 9]). In addition, we use the notation to denote the order of growth of the meromorphic function , and to denote the exponent of convergence of zeros of .
Chen  weakened the condition (1.1) of Theorem A and proved the following results.
Theorem B ()
Then every finite order meromorphic solution (≢0) of equation (1.2) satisfies , and assumes every nonzero value infinitely often and .
Theorem C ()
satisfies and .
Theorem D ()
Let , be polynomials such that . Suppose that is a meromorphic solution with infinitely many poles of (1.2) (or (1.4)). Then .
Chiang and Feng proved the following result.
Theorem E ()
If is a meromorphic solution of (1.5), then we have .
Laine and Yang proved the following theorem.
Theorem F ()
we have .
has a solution , which .
This example shows that for the linear difference equation with meromorphic coefficients, the condition (1.6) cannot guarantee that every transcendental meromorphic solution of (1.7) satisfies .
Thus, a natural question to ask is what conditions will guarantee every transcendental meromorphic solution of (1.7) with meromorphic coefficients satisfies .
In this note, we consider this question and prove the following results.
Theorem 1.1 Let , (), a be nonzero constants, be a nonzero meromorphic function with , be a nonzero meromorphic function.
where b is a nonzero constant, (≢0) is a meromorphic function with ,
satisfies conditions of Theorem 1.1 and has a solution satisfying and . This example shows that under conditions of Theorem 1.1, a meromorphic solution of (1.8) may have no zero.
with at most one possible exceptional solution with .
satisfies conditions of Theorem 1.2 and has a solution . This shows that in Theorem 1.2, there exists one possible exceptional solution with .
2 Proof of Theorem 1.1
We need the following lemmas to prove Theorem 1.1.
Lemma 2.2 (see )
- (i)if , then(2.1)
- (ii)if , then(2.2)
where is a finite set.
If , then .
where M (>0) is some constant.
Hence, . □
- (1)Suppose that satisfies the condition (i): and . Thus, for sufficiently large r,(2.11)
Suppose that satisfies the condition (ii): . Using the same method as in (1), we can obtain .
Suppose that satisfies the condition (iii): , where b is a nonzero constant, (≢0) is a meromorphic function with .
In what follows, we divide this proof into three subcases: (a) ; (b) and ; (c) .
By , and , it is easy to see that (2.22) is a contradiction. Hence, .
By , we see that (2.27) is a contradiction.
Now suppose that . Using the same method as above, we can also deduce a contradiction.
Hence, in Subcase (b).
Subcase (c). We first affirm that cannot be a nonzero rational function. In fact, if is a rational function, then is a rational function. So that , that is, , a contradiction.
- (4)Suppose that (≢0) is a meromorphic function with . Set . Substituting into (1.8), we obtain(2.29)
Thus, Theorem 1.1 is proved. □
3 Proof of Theorem 1.2
Thus, Theorem 1.2 is proved.
The author is grateful to the referees for a number of helpful suggestions to improve the paper. This research was partly supported by the National Natural Science Foundation of China (grant no. 11171119).
- 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.2010.06.049MATHMathSciNetView ArticleGoogle Scholar
- Chiang YM, Feng SJ:On the Nevanlinna characteristic of and difference equations in the complex plane. Ramanujan J. 2008, 16: 105-129. 10.1007/s11139-007-9101-1MATHMathSciNetView ArticleGoogle Scholar
- Ishizaki K: On difference Riccati equations and second order linear difference equations. Aequ. Math. 2011, 81: 185-198. 10.1007/s00010-010-0060-zMATHMathSciNetView ArticleGoogle Scholar
- Ishizaki K, Yanagihara N: Wiman-Valiron method for difference equations. Nagoya Math. J. 2004, 175: 75-102.MATHMathSciNetGoogle Scholar
- Laine I: Nevanlinna Theory and Complex Differential Equations. de Gruyter, Berlin; 1993.View ArticleGoogle Scholar
- Li S, Gao ZS: Finite order meromorphic solutions of linear difference equations. Proc. Jpn Acad., Ser. A 2011, 87(5):73-76. 10.3792/pjaa.87.73MATHView ArticleGoogle Scholar
- Zheng XM, Tu J: Growth of meromorphic solutions of linear difference equations. J. Math. Anal. Appl. 2011, 384: 349-356. 10.1016/j.jmaa.2011.05.069MATHMathSciNetView ArticleGoogle Scholar
- Hayman WK: Meromorphic Functions. Clarendon, Oxford; 1964.MATHGoogle Scholar
- Laine I, Yang CC: Clunie theorems for difference and q -difference polynomials. J. Lond. Math. Soc. 2007, 76(3):556-566. 10.1112/jlms/jdm073MATHMathSciNetView ArticleGoogle Scholar
- Halburd RG, Korhonen R: Difference analogue of the lemma on the logarithmic derivative with applications to difference equations. J. Math. Anal. Appl. 2006, 314: 477-487. 10.1016/j.jmaa.2005.04.010MATHMathSciNetView ArticleGoogle Scholar
- Chen ZX:The growth of solutions of where the order .Sci. China Math. 2002, 45(3):290-300.MATHGoogle Scholar
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