Abstract
In this paper, we consider the value distribution of meromorphic solutions for linear difference equations with meromorphic coefficients.
MSC:30D35, 39A10.
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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 ([2])
Let be polynomials such that there exists an integer l, , such that
holds. Suppose is a meromorphic solution of the difference equation
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 [1] weakened the condition (1.1) of Theorem A and proved the following results.
Theorem B ([1])
Let be polynomials such that and
Then every finite order meromorphic solution (≢0) of equation (1.2) satisfies , and assumes every nonzero value infinitely often and .
Theorem C ([1])
Let , be polynomials such that and (1.3). Then every finite order transcendental meromorphic solution of the equation
satisfies and .
Theorem D ([1])
Let , be polynomials such that . Suppose that is a meromorphic solution with infinitely many poles of (1.2) (or (1.4)). Then .
For the linear difference equation with transcendental coefficients
Chiang and Feng proved the following result.
Theorem E ([2])
Let be entire functions such that there exists an integer l, , such that
If is a meromorphic solution of (1.5), then we have .
Laine and Yang proved the following theorem.
Theorem F ([5])
Let be entire functions of finite order so that among those having the maximal order , exactly one has its type strictly greater than the others. Then for any meromorphic solution of
we have .
Remark 1.1 If are meromorphic functions satisfying (1.6), then Theorem E does not hold. For example, the equation
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.
If satisfies any one of the following three conditions:
-
(i)
and ;
-
(ii)
;
-
(iii)
where b is a nonzero constant, (≢0) is a meromorphic function with ,
then every meromorphic solution f (≢0) of the difference equation
satisfies .
Further, if (≢0) is a meromorphic function with
then
Corollary Under conditions of Theorem 1.1, every finite order solution (≢0) of (1.8) has infinitely many fixed points, satisfies , and for any nonzero constant c,
Example 1.1
The equation
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.
Theorem 1.2 Let , , , a, satisfy conditions of Theorem 1.1, and let (≢0) be a meromorphic function with . Then all meromorphic solutions with finite order of the equation
satisfy
with at most one possible exceptional solution with .
Remark 1.2 Under conditions of Theorem 1.1, equation (1.8) has no rational solution. But equation (1.9) in Theorem 1.2 may have a rational solution. For example, the equation
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.
Given two distinct complex constants , , let f be a meromorphic function of finite order σ. Then, for each , we have
Lemma 2.2 (see [11])
Suppose that (α, β are real numbers, ) is a polynomial with degree , that (≢0) is an entire function with . Set , , . Then, for any given , there exists a set that has the linear measure zero such that for any , there is such that for , we have that
-
(i)
if , then
(2.1) -
(ii)
if , then
(2.2)
where is a finite set.
Lemma 2.3 Let , (), a be nonzero constants, (), be nonzero meromorphic functions. Suppose that is a finite order meromorphic solution of the equation
If , then .
Proof Suppose that , . Then . Equation (2.3) can be rewritten as the form
Thus, by (2.4), we deduce that
For any given ε (), and for sufficiently large r, we have that
By Lemma 2.1, we obtain
where M (>0) is some constant.
By , there exists a sequence satisfying , such that
Thus, for sufficiently large , we have that
Substituting (2.6)-(2.9) into (2.5), we obtain for sufficiently large
Since and ε is arbitrary, by (2.10), we obtain
Hence, . □
Proof of Theorem 1.1 Suppose that (≢0) is a meromorphic solution of equation (1.8) with .
-
(1)
Suppose that satisfies the condition (i): and . Thus, for sufficiently large r,
(2.11)
Clearly, by (1.8). By Lemma 2.1, we see that for any given ε (),
and
By (1.8), we have that
Substituting (2.11)-(2.13) into (2.14), we deduce that
By , there is a sequence ( , ) satisfying
Thus, by (2.15) and (2.16), we obtain
where M (>0) is some constant. Combining (2.17) and , it follows that
So that, it follows that .
-
(2)
Suppose that satisfies the condition (ii): . Using the same method as in (1), we can obtain .
-
(3)
Suppose that satisfies the condition (iii): , where b is a nonzero constant, (≢0) is a meromorphic function with .
Now we need to prove . Contrary to the assertion, suppose that . We will deduce a contradiction. Set . Then
In what follows, we divide this proof into three subcases: (a) ; (b) and ; (c) .
Subcase (a). Since and (2.18), it is easy to see that there exists a ray such that
By (1.8) and (2.19), we see that cannot be a rational function. By Lemma 2.1, (2.12) holds. By Lemma 2.2 and (2.19), it is easy to see that for any given () and for sufficiently large r,
and
Thus, by (1.8), (2.12), (2.20) and (2.21), we deduce that
By , and , it is easy to see that (2.22) is a contradiction. Hence, .
Subcase (b). By and , we see that cannot be a rational function. By Lemma 2.1, (2.12) holds. By and (2.18), we take , then and
Now suppose that . By Lemma 2.2, for any given (),
and
Thus, by (1.8), (2.12), (2.24) and (2.25), we deduce that
Since , we have that . Combining this and (2.26), we obtain
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.
By Lemma 2.1, (2.12) holds. By , equation (1.8) can be rewritten as
Using the same method as in the proof of (1), we can obtain .
-
(4)
Suppose that (≢0) is a meromorphic function with . Set . Substituting into (1.8), we obtain
(2.29)
If , then is a nonzero meromorphic solution of (1.8). Thus, by the proof above, we have that . This contradicts our condition that . Hence, , and
Applying this and Lemma 2.3 to (2.29), we deduce that
Thus, Theorem 1.1 is proved. □
3 Proof of Theorem 1.2
Suppose that is a meromorphic solution of (1.9) with
If () is another meromorphic solution of (1.9) satisfying , then
But is a solution of the corresponding homogeneous equation (1.8) of (1.9). By Theorem 1.1, we have , a contradiction. Hence equation (1.9) possesses at most one exceptional solution with .
Now suppose that f is a meromorphic solution of (1.9) with
Since , applying Lemma 2.3 to (1.9), we obtain
Thus, Theorem 1.2 is proved.
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Acknowledgements
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).
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Liu, Y. On growth of meromorphic solutions for linear difference equations with meromorphic coefficients. Adv Differ Equ 2013, 60 (2013). https://doi.org/10.1186/1687-1847-2013-60
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DOI: https://doi.org/10.1186/1687-1847-2013-60