Skip to main content

Uniqueness and value distribution for difference operators of meromorphic function

Abstract

We investigate the value distribution of difference operator for meromorphic functions. In addition, we study the sharing value problems related to a meromorphic function f (z) and its shift f (z + c).

1 Introduction and main results

A meromorphic function means meromorphic in the whole complex plane. We assume that the reader is familiar with standard symbols and fundamental results of Nevanlinna Theory [1]. As usual, the abbreviation CM stands for "counting multiplicities", while IM means "ignoring multiplicities", and we denote the order of meromorphic function f by σ (f). For a non-constant meromorphic function f and a set S of complex numbers, we define the set E(S, f) = aS{z|f(z) - a = 0}, where a zero of f - a with multiplicity m counts m times in E(S, f).

We define difference operator as Δ c f = f (z + c) - f (z), where c is a non-zero constant. In particular, we denote by S (f) the family of all meromorphic functions a (z) that satisfy T(r, a) = S(r, f) = o(T(r, f)), where r → ∞ outside a possible exceptional set of finite logarithmic measure. For convenience, we set Ŝ(f) := S(f) {∞}.

The difference Nevanlinna theory and its applications to the uniqueness theory have become a subject of great interest [24], recently. With these fundamental results, Heittokangas et al. considered a meromorphic function f (z) sharing values with its shift f(z + c), we recall a key result from [5].

Theorem A [[5], Theorem 2]. Let f be a non-constant meromorphic function of finite order, let c , and let a, b, c Ŝ(f) be three distinct periodic functions with period c. If f (z) and f (z + c) share a, b CM and c IM, then f (z) = f (z + c) for all z .

Recently, Yang and Liu and one of the present authors [6] considered the case F = fn, where f is a meromorphic function, assuming value sharing with F and F (z + c):

Theorem B [[6], Theorem 1.4]. Let f be a non-constant meromorphic function of finite order, n ≥ 7 be an integer, let c , and let F = fn. If F (z) and F (z + c) share a S(f)\{0} andCM, then f (z) = ωf (z + c), for a constant ω that satisfies ωn= 1.

Next, we consider the problem that related to the Theorem B, and have the following result, where a is a periodic function with period c. However, our proof is different to the one in [6].

Theorem 1.1. Let f be a non-constant meromorphic function of finite order, let c , and let a S(f) \ {0} be a periodic function with period c. If f (z)nand f(z + c)nshare a andCM, and n ≥ 4 is an integer, then f (z) = ωf (z + c), for a constant ω that satisfies ωn= 1.

Remarks.

  1. (1)

    Theorem 1.1 is not true, if a = 0. This can be seen by considering f ( z ) = e z 2 . Then f (z)nand f (z + c)nshare 0 and ∞ CM, however, f (z) ≠ ωf (z + c), where n is a positive integer.

  2. (2)

    Theorem 1.1 does not remain valid when n = 1. For example, f (z) = ez+ 1 and f (z + c) = ez+c+ 1, where c ≠ 2πi. Clearly, f(z) and f (z + c) share 1 and ∞ CM, however, f (z) ≠ ωf (z + c) for ωn= 1. Unfortunately, we have not succeeded in reducing the condition n ≥ 4 to n ≥ 2 in Theorem 1.1, and we also cannot give a counterexample when n = 2, 3 at present.

  3. (3)

    We give an example to show that the restriction of finite order in Theorem 1.1 cannot be deleted. This can be seen by taking f ( z ) = e e z ,n e c =-1. Then f (z)nand f (z + c)nshare 1 and ∞ CM, however, f(z) ≠ ωf (z + c), where n is a positive integer.

In 1976, Gross asked the following question [[7], Question 6]:

Question. Can one find (even one set) finite sets S j (j = 1, 2) such that any two entire functions f and g satisfying E(S j , f) = E(S j , g) (j = 1, 2) must be identical?

Since then, many results have been obtained for this and related topics (see [811]). We recall the following result given by Yi [9].

Theorem C [[9], Theorem 1]. Let S1 = {ω | ωn+ n- 1+ b = 0}, where n ≥ 7 is an integer, a and b are two non-zero constants such that the algebraic equation ωn+ n- 1+ b = 0 has no multiple roots. If f and g are two entire functions satisfying E(S1, f) = E(S1, g), then f = g.

Afterwards, Fang and Lahiri [12] got the result for meromorphic functions.

Theorem D [[12], Theorem 1]. Let S1 be defined as Theorem C and S2 = {∞}. Assume that f and g are two meromorphic functions satisfying E(S j , f) = E (S j , g) for j = 1,2. If f has no simple poles and n ≥ 7, then f = g.

Next, we give a difference analog of Theorem D that replacing g with f (z + c), and obtain the following result.

Theorem 1.2. Let S1 be defined as Theorem C and S2 = {∞}. Assume that f is a meromorphic function of finite order satisfying E(S j , f) = E(S j , f (z + c)) for j = 1,2. If n ≥ 6 and N ¯ ( r , f ) < n - 3 n - 1 T ( r , f ) +S ( r , f ) , then f (z) = f (z + c) for all z .

We investigate the value distribution of difference polynomials of meromorphic (entire) functions. Let f be a transcendental meromorphic function, and let n be a positive integer. Concerning to the value distribution of fnf", Hayman [[13], Corollary to Theorem 9] proved that fnf' takes every non-zero complex value infinitely often if n ≥ 3. Mues [[14], Satz 3] proved that f2f' - 1 has infinitely many zeros. Later on, Bergweiler and Eremenko [[15], Theorem 2] showed that ff' - 1 has infinitely many zeros also. As an analog result in difference, Laine and Yang [16] investigated the value distribution of difference products of entire functions, and obtained the following:

Theorem E [[16], Theorem 2]. Let f be a transcendental entire function of finite order, and let c be a non-zero complex constant. Then for n ≥ 2, f (z)nf (z + c) assumes every non-zero value a infinitely often.

In a recent article, one of the present authors considered the value distribution of f (z)n(f(z) - 1) f (z + c), the result may be stated as follows:

Theorem F [[17], Theorem 1]. Let f be a transcendental meromorphic function of finite order σ(f), let a ≠ 0 be a small function with respect to f, and let c be a non-zero complex constant. If the exponent of convergence of the poles of f satisfies λ ( 1 f ) <σ ( f ) and n ≥ 2, then f (z)n(f - 1) f (z + c) - a has infinitely many zeros.

In this article, we replace f (z + c) with Δ c f, and consider the value distribution of f (z)n(f(z) - 1)Δ c f. We get the following results:

Theorem 1.3. Let f be a transcendental meromorphic function of finite order σ(f) and Δ c f ≠ 0, let a ≠ 0 be a small function with respect to f, and let c be a non-zero complex constant. If the exponent of convergence of the poles off satisfies λ ( 1 f ) <σ ( f ) and n ≥ 3, then f (z)n(f - 1)Δ c f - a has infinitely many zeros.

Corollary 1.4. Let f be a transcendental entire function of finite order and Δ c f ≠ 0, let a ≠ 0 be a small function with respect to f, and let c be a non-zero complex constant. Then for n ≥ 3, f (z)n(f - 1)Δ c f - a has infinitely many zeros.

In particular, if a is a non-zero polynomial in Corollary 1.4, then Corollary 1.4 can be improved.

Theorem 1.5. Let f be a transcendental entire function of finite order and Δ c f ≠ 0, let a be a non-zero polynomial, and let c be a non-zero complex constant. Then for n ≥ 2, f (z)n(f - 1)Δ c f - a has infinitely many zeros.

2 Preliminary lemmas

Lemma 2.1. [[4], Theorem 2.1] Let f be a meromorphic function of finite order, and let c and δ (0, 1) . Then

m r , f ( z + c ) f ( z ) + m r , f ( z ) f ( z + c ) = o T ( r , f ) r δ = S ( r , f ) .

Chiang and Feng have obtained similar estimates for the logarithmic difference [[3], Corollary 2.5], and this study is independent from [4].

Lemma 2.2. [[4], Lemma 2.3] Let f be a meromorphic function of finite order and c . Then for any small function a S (f) with period c,

m r , Δ c f f - a = S ( r , f ) .

Lemma 2.3. [[3], Theorem 2.1] Let f be a meromorphic function of finite order σ (f), and let c be a non-zero constant. Then, for each ε > 0, we have

T ( r , f ( z + c ) ) = T ( r , f ( z ) ) + O ( r σ ( f ) - 1 + ε ) + O ( log r ) .

Lemma 2.4. [[18], Theorem 2.4.2] Let f be a transcendental meromorphic solution of

f n A ( z , f ) = B ( z , f ) ,

where A(z, f), B(z, f) are differential polynomials in f and its derivatives with small meromorphic coefficients a λ , in the sense of m(r, a λ ) = S(r, f) for all λ I. If the deg(B(z, f)) ≤ n, then m(r, A(z, f)) = S(r, f).

Lemma 2.5. Let f be a finite order entire function and Δ c f ≠ 0, and let c be a non-zero constant. Then

m ( r , f f Δ c f ) T ( r , f ) + S ( r , f ) .

Proof. Since f is an entire function with finite order, we deduce from Lemma 2.2 and the Lemma of logarithmic derivative that

3 T ( r , f ) = T ( r , f 3 ) = m ( r , f 3 ) + S ( r , f ) m ( r , f 3 f f Δ c f ) + m ( r , f f Δ c f ) + S ( r , f ) = m ( r , f 2 f Δ c f ) + m ( r , f f Δ c f ) + S ( r , f ) T ( r , f f ) + T ( r , Δ c f f ) + m ( r , f f Δ c f ) + S ( r , f ) 2 N ( r , 1 f ) + m ( r , f f Δ c f ) + S ( r , f ) 2 T ( r , f ) + m ( r , f f Δ c f ) + S ( r , f ) .

Hence, we get

m ( r , f f Δ c f ) T ( r , f ) + S ( r , f ) .
(1)

3 Proof of Theorem 1.1

Since f(z)nand f(z + c)nshare a and ∞ CM, we obtain that

f ( z + c ) n - a ( z + c ) f ( z ) n - a ( z ) = e Q ( z ) ,
(2)

where Q(z) is a polynomial. From Lemma 2.1, we know that T (r, eQ(z)) = m (r, eQ(z)) = S (r, f). Rewrite (2) as

f ( z + c ) n = e Q ( z ) ( f ( z ) n - a ( z ) + a ( z ) e - Q ( z ) ) .
(3)

Set

G ( z ) = f ( z ) n a ( z ) ( 1 - e - Q ( z ) ) .

If eQ(z) 1, then we apply the Valiron-Mohon'ko theorem and the second main theorem to G (z), and get

n T ( r , f ) + S ( r , f ) = T ( r , G ) N ¯ r , 1 G + N ¯ ( r , G ) + N ¯ r , 1 G - 1 + S ( r , G ) N ¯ r , 1 f + N ¯ ( r , f ) + N ¯ r , 1 f ( z ) n - a ( z ) + a ( z ) e - Q ( z ) + S ( r , f ) N ¯ r , 1 f + N ¯ ( r , f ) + N ¯ r , 1 f ( z + c ) + S ( r , f ) 2 T ( r , f ) + T ( r , f ( z + c ) ) + S ( r , f ) .
(4)

Combining (4) with Lemma 2.3, we get

n T ( r , f ) 3 T ( r , f ) + O ( r σ ( f ) - 1 + ε ) + S ( r , f ) ,

which contradicts that n ≥ 4. Therefore, eQ(z)≡ 1, that is, f (z)n= f (z + c)n, so we have f (z) = ω f (z + c), for a constant ω with ωn= 1.

4 Proof of Theorem 1.2

From the assumption of Theorem 1.2, we get that

f ( z + c ) n + a f ( z + c ) n - 1 + b f ( z ) n + a f ( z ) n - 1 + b = e Q ( z ) ,
(5)

where Q(z) is a polynomial. Applying Lemma 2.1, we obtain that T (r, eQ(z)) = m (r, eQ(z)) = S (r, f). Rewrite (5) as

f ( z + c ) n + a f ( z + c ) n - 1 = e Q ( z ) f ( z ) n + a f ( z ) n - 1 + b - b e Q ( z ) .
(6)

If eQ(z) 1, applying the second main theorem for three small functions, we get

n T ( r , f ) + S ( r , f ) = T ( r , f ( z ) n + a f ( z ) n - 1 ) N ¯ r , 1 f ( z ) n + a f ( z ) n - 1 + N ¯ ( r , f ( z ) n + a f ( z ) n - 1 ) + N ¯ r , 1 f ( z ) n + a f ( z ) n - 1 + b - b e Q ( z ) + S ( r , f ) N ¯ ( r , f ) + N ¯ r , 1 f ( z + c ) n - 1 ( f ( z + c ) + a ) + N ¯ r , 1 f ( z ) n - 1 ( f ( z ) + a ) + S ( r , f ) 3 T ( r , f ) + 2 T ( r , f ( z + c ) ) + S ( r , f ) .
(7)

Combining (4.3) with Lemma 2.3, we get

n T ( r , f ) 5 T ( r , f ) + O ( r σ ( f ) - 1 + ε ) + S ( r , f ) ,

which contradicts n ≥ 6. Hence, eQ(z)≡ 1, we conclude by (5) that

f ( z + c ) n + a f ( z + c ) n - 1 = f ( z ) n + a f ( z ) n - 1 .
(8)

Set G ( z ) = f ( z ) f ( z + c ) . If G (z) is non-constant, then we have from (8)

f ( z ) = - a G ( G n - 1 - 1 ) G n - 1 = - a G n - 1 + + G G n - 1 + + 1 .
(9)

Making use of the standard Valiron-Mohon'ko lemma, we get from (9) that

T ( r , f ) = ( n - 1 ) T ( r , G ) + S ( r , f ) .
(10)

Noting that n ≥ 6, we deduce that 1 is not a Picard value of Gn. Suppose that a j { \ 1} (j = 1, 2,..., n - 1) are the distinct roots of equation hn- 1 = 0. Applying the second main theorem to G, we conclude by (9) that

( n - 3 ) T ( r , G ) j = 1 n - 1 N ¯ r , 1 G - a j + S ( r , G ) = N ¯ ( r , f ) .
(11)

From (10) and (11), we get N ¯ ( r , f ) n - 3 n - 1 T ( r , f ) +S ( r , f ) , which contradicts the assumption.

So G(z) is a constant, and we get f (z) = tf (z + c), where t is a non-zero constant. From (8), we know t = 1, therefore, f = g.

5 Proof of Theorem 1.3

The main idea of this proof is from [[17], Theorem 1], while the details are somewhat different. For the convenience of the reader, we give a complete proof.

Set F(z) = fn(z) (f(z) - 1)Δ c f. Since f is a transcendental meromorphic function with finite order σ(f), we conclude by Lemma 2.3 that

T ( r , F ) T ( r , f n ( z ) ( f ( z ) - 1 ) ) + T ( r , Δ c f ) + S ( r , f ) ( n + 2 ) T ( r , f ) + T ( r , f ( z + c ) ) + S ( r , f ) ( n + 3 ) T ( r , f ) + O ( r σ ( f ) - 1 + ε ) + S ( r , f ) .

Thus, we get S(r, F) = o(T(r, f)) = S(r, f). Moreover, we get

T ( r , Δ c f ) m ( r , Δ c f ) + O r λ ( 1 f ) + ε + S ( r , f ) m r , Δ c f f + m ( r , f ) + O r λ ( 1 f ) + ε + S ( r , f ) T ( r , f ) + O r λ ( 1 f ) + ε + S ( r , f ) .
(12)

On the other hand, we deduce by Lemma 2.2 that

( n + 2 ) T ( r , f ) = T ( r , f n + 1 ( f - 1 ) ) + S ( r , f ) = m ( r , f n + 1 ( f - 1 ) ) + O r λ ( 1 f ) + ε + S ( r , f ) m r , f n + 1 ( f - 1 ) F + m ( r , F ) + O r λ ( 1 f ) + ε + S ( r , f ) T r , Δ c f f + m ( r , F ) + O r λ ( 1 f ) + ε + S ( r , f ) m r , Δ c f f + N r , 1 f + m ( r , F ) + O r λ ( 1 f ) + ε + S ( r , f ) T ( r , f ) + T ( r , F ) + O r λ ( 1 f ) + ε + S ( r , f ) .

Hence

( n + 1 ) T ( r , f ) T ( r , F ) + O r λ ( 1 f ) + ε + S ( r , f ) .
(13)

The second main theorem yields

T ( r , F ) N ¯ ( r , F ) + N ¯ r , 1 F + N ¯ ( r , 1 F - a ) + S ( r , F ) N ¯ ( r , 1 F - a ) + N ¯ r , 1 f + N ¯ r , 1 f - 1 + N ¯ ( r , 1 Δ c f ) + O r λ ( 1 f ) + ε + S ( r , f ) N ¯ r , 1 F - a + 2 T ( r , f ) + T ( r , Δ c f ) + O r λ ( 1 f ) + ε + S ( r , f ) .

From (12) and above inequality, we get that

T ( r , F ) N ¯ r , 1 F - a + 3 T ( r , f ) + O r λ ( 1 f ) + ε + S ( r , f ) .
(14)

Combining (13) and (14), we have

( n - 2 ) T ( r , f ) N ¯ r , 1 F - a + O r λ ( 1 f ) + ε + S ( r , f ) ,

which is a contradiction to the fact that f is of order σ (f) if F - a has finitely many zeros. The conclusion follows.

6 Proof of Theorem 1.5

Suppose that fn(f - 1)Δ c f - a admits finitely many zeros only. Then, there are two non-zero polynomials P(z), Q(z) such that

f n ( f - 1 ) Δ c f - a = P ( z ) e Q ( z ) .
(15)

Differentiating (15) and eliminating eQ(z), we obtain

( f n - f n - 1 ) F ( z , f ) = a P ( z ) - a P * ( z ) - P ( z ) f ( z ) n - 1 f ( z ) Δ c f ,
(16)

where

F ( z , f ) = ( n + 1 ) P ( z ) f ( z ) Δ c f + P ( z ) f ( z ) ( Δ c f ) - P * ( z ) f ( z ) Δ c f

and P*(z) = P'(z) + P(z)Q'(z).

First, we conclude that a'P (z) - aP*(z) 0. Otherwise, if a'P(z) - aP*(z) = 0, by integrating, then we have

a P ( z ) = A e Q ( z ) ,

where A is a non-zero constant. Hence, we get eQ(z)is a constant and

f n ( z ) ( f ( z ) - 1 ) Δ c f = B P ( z ) + a ,
(17)

where B is a non-zero constant. Then, from Lemma 2.3 and (17), we obtain that

( n + 1 ) T ( r , f ) 2 T ( r , f ) + O r σ ( f ) - 1 + ε + S ( r , f ) ,

which is a contradiction when n ≥ 2.

If F(z, f) vanish identically, then

a P * ( z ) + P ( z ) f ( z ) n - 1 f ( z ) Δ c f - a P ( z ) 0 .
(18)

Rewrite (18), we get

f n 2 f f ( z ) Δ c f = a P ( z ) a P * ( z ) P ( z ) ,

hence

f n - 2 f 2 f ( z ) Δ c f f = a P ( z ) - a P * ( z ) P ( z ) .
(19)

Then, combining Lemmas 2.2, 2.4 and Equation (19), we conclude that

m ( r , f f ( z ) Δ c f ) = S ( r , f ) ,

which contradicts (1).

It remains to consider the case that F (z, f) 0. We rewrite (16) in the form that

( f ( z ) n + 2 - f ( z ) n + 1 ) F ( z , f ) f ( z ) 2 = a P ( z ) - a P * ( z ) - P ( z ) f ( z ) n - 1 f ( z ) Δ c f
(20)

and

f ( z ) n + 1 ( f ( z ) - 1 ) F ( z , f ) f ( z ) 2 = a P ( z ) - a P * ( z ) - P ( z ) f ( z ) n - 1 f ( z ) Δ c f f ( z ) 2 .

By Lemmas 2.2 and 2.4, we know that

m r , F ( z , f ) f ( z ) 2 = S ( r , f )

and

m r , ( f ( z ) - 1 ) F ( z , f ) f ( z ) 2 = S ( r , f ) .

As f (z) is entire, we get that the poles of F ( z , f ) f ( z ) 2 may be located only at the zeros of f (z). If F ( z , f ) f ( z ) 2 has infinitely many poles, then from that a zero of f (z) with multiplicity t should be a pole of t + 1 of F ( z , f ) f ( z ) 2 . Since n ≥ 2, we know that the left side of (20) must have infinitely many zeros, which is a contradiction that a'P(z) - aP*(z) is a non-zero polynomial. We get

N r , F ( z , f ) f ( z ) 2 = O ( log r ) and N r , ( f ( z ) - 1 ) F ( z , f ) f ( z ) 2 = O ( log r ) .

Hence

T r , F ( z , f ) f ( z ) 2 = S ( r , f )

and

T r , ( f ( z ) - 1 ) F ( z , f ) f ( z ) 2 = S ( r , f )

as well. Combining these two estimates, we obtain

T ( r , f ) = S ( r , f )

contradiction. This completes the proof of Theorem 1.5.

References

  1. Yang CC, Yi HX: Uniqueness Theory of Meromorphic Functions. Kluwer Academic Publishers, Dordrecht; 2003.

    Chapter  Google Scholar 

  2. Bergweiler W, Langley JK: Zeros of difference of meromorphic functions. Math Proc Cambridge Philos Soc 2007, 142: 133–147. 10.1017/S0305004106009777

    MATH  MathSciNet  Article  Google Scholar 

  3. Chiang YM, Feng SJ: On the Nevanlinna characteristic of f ( z + η ) and difference equations in the complex plane. Ramanujan J 2008, 16: 105–129. 10.1007/s11139-007-9101-1

    MATH  MathSciNet  Article  Google Scholar 

  4. Halburd RG, Korhonen RJ: Nevanlinna theory for the difference operator. Ann Acad Sci Fenn 2006, 31: 463–478.

    MATH  MathSciNet  Google Scholar 

  5. Heittokangas J, Korhonen R, Laine I, Rieppo J, Zhang JL: Value sharing results for shifts of meromorphic function, and sufficient conditions for periodicity. J Math Anal Appl 2009, 355: 352–363. 10.1016/j.jmaa.2009.01.053

    MATH  MathSciNet  Article  Google Scholar 

  6. Qi XG, Yang LZ, Liu K: Uniqueness and periodicity of meromorphic functions concerning difference operator. Comput Math Appl 2010, 60: 1739–1746. 10.1016/j.camwa.2010.07.004

    MATH  MathSciNet  Article  Google Scholar 

  7. Gross F: Factorization of meromorphic functions and some open problems, Complex analysis (Proc. Conf., Univ. Kentucky, Lexington, Ky., 1976). In Lecture Notes in Math. Volume 599. Springer, Berlin; 1977:51–67. 10.1007/BFb0096825

    Google Scholar 

  8. Frank G, Reinders M: A unique range set for meromorphic functions with 11 elements. Complex var Theory Appl 1998, 37: 185–193. 10.1080/17476939808815132

    MATH  MathSciNet  Article  Google Scholar 

  9. Yi HX: A question of Gross and the uniqueness of entire functions. Nagoya Math J 1995, 138: 169–177.

    MATH  MathSciNet  Google Scholar 

  10. Yi HX: Unicity theorems for meromorphic or entire functions II. Bull Austral Math Soc 1995, 52: 215–224. 10.1017/S0004972700014635

    MATH  MathSciNet  Article  Google Scholar 

  11. Yi HX: Unicity theorems for meromorphic or entire functions III. Bull Austral Math Soc 1996, 53: 71–82. 10.1017/S0004972700016737

    MATH  MathSciNet  Article  Google Scholar 

  12. Fang ML, Lahiri I: Unique range set for certain meromorphic functions. Indian J Math 2003, 45: 141–150.

    MATH  MathSciNet  Google Scholar 

  13. Hayman WK: Picard values of meromorphic functions and their derivatives. Ann Math 1959, 70(2):9–42.

    MATH  MathSciNet  Article  Google Scholar 

  14. Mues E: Über ein Problem von Hayman. Math Z 1979, 164: 239–259. 10.1007/BF01182271

    MATH  MathSciNet  Article  Google Scholar 

  15. Bergweiler W, Eremenko A: On the singularities of the inverse to a meromorphic function of finite order. Rev Mat Iberoamericana 1995, 11: 355–373.

    MATH  MathSciNet  Article  Google Scholar 

  16. Laine I, Yang CC: Value Distribution of Difference Polynomials. Proc Japan Acad Ser A Math Sci 2007, 83: 148–151. 10.3792/pjaa.83.148

    MATH  MathSciNet  Article  Google Scholar 

  17. Qi XG: Value distribution and uniqueness of difference polynomials and entire solutions of difference equations. Ann Polon Math 2011, 102(2):129–142. 10.4064/ap102-2-3

    MATH  MathSciNet  Article  Google Scholar 

  18. Laine I: Nevanlinna Theory and Complex Differential Equations. In Walter de Gruyter. Berlin-New York; 1993.

    Google Scholar 

Download references

Acknowledgements

The authors thank the referee for his/her valuable suggestions to improve the present article. This study was supported by the NSF of Shandong Province, China (No. ZR2010AM030).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Xiaoguang Qi.

Additional information

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

XQ completed the main part of this article, JD and LY corrected the main theorems. 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 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and Permissions

About this article

Cite this article

Qi, X., Dou, J. & Yang, L. Uniqueness and value distribution for difference operators of meromorphic function. Adv Differ Equ 2012, 32 (2012). https://doi.org/10.1186/1687-1847-2012-32

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/1687-1847-2012-32

Keywords

  • Entire Function
  • Meromorphic Function
  • Finite Order
  • Meromorphic Solution
  • Small Function