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Meromorphic functions sharing small functions with their linear difference polynomials
Advances in Difference Equations volume 2013, Article number: 58 (2013)
In this paper, we prove some results on the uniqueness of meromorphic functions sharing small functions CM with their linear difference polynomials. Examples are provided to show the existence of meromorphic functions satisfying the conditions of our results.
MSC:30D35, 39A10, 39B32.
1 Introduction and main results
In the following, we use the standard notations of Nevanlinna theory of meromorphic functions (see [1–3]). For any given nonconstant meromorphic function , we recall the hyper order of defined as follows (see ):
Denote by any quantity satisfying as , possibly outside of a set of r with a finite logarithmic measure. A meromorphic function is said to be a small function of if . In what follows, we use to denote the set of all small functions of .
For two meromorphic functions and , and , we say that and share a CM when and have the same zeros counting multiplicity.
For a nonzero complex constant , is called a shift of . And a difference monomial of type is called a difference product of , where and .
A difference polynomial of is a finite sum of difference products of , with all coefficients being small functions of . In the following, we mainly consider a linear difference polynomial of of the form
where , .
It is well known that the difference operators of are defined as follows:
In particular, for the case . We point out that a difference operator is just a special linear difference polynomial of such that the sum of its coefficients equals 0.
The subject on the uniqueness of the entire function sharing values with its derivative was initiated by Rubel and Yang . For a nonconstant entire function , they proved that provided that and share two distinct finite values CM.
Recently, a number of papers have focused on the Nevanlinna theory with respect to difference operators; see, e.g., the papers [5, 6] by Chiang and Feng and [7, 8] by Halburd and Korhonen. Then, many authors started to investigate the uniqueness of meromorphic functions sharing values or small functions with their shifts (see, e.g., [9–14]) or difference operators (see, e.g., [9, 12]). The following Theorem A is indeed a corollary of Theorem 2.1 in  and Theorem 2 in .
Let be a meromorphic function of finite order, let , and let be two distinct periodic functions with period c. If and share , , ∞ CM, then for all .
Theorem B ()
Let be a transcendental meromorphic function such that its order of growth is not an integer or infinite, and let be a constant such that . If and share three distinct values a, b, ∞ CM, then .
Theorem C ()
Let be a nonconstant entire function of finite order, , and n be a positive integer. Suppose that and share two distinct finite values a, b CM and one of the following cases is satisfied:
Remark 1 The methods in  and  are quite different. Due to a result of Ozawa  (he proved that for any given , there exists a periodic entire function of order ρ), Chen and Yi  and Li and Gao  gave some examples to show the existence of functions satisfying the conditions of Theorem B and Theorem C respectively.
Theorem 1.1 Let be a meromorphic function of hyper order , let be a difference polynomial of , and let be two distinct meromorphic functions. Suppose that and share a, b, ∞ CM and one of the following cases holds:
or , and for some ;
Example 1 We give two examples for Theorem 1.1.
For the cases (i) and (ii): Let and . Then , and hence for and any given , and share a, b, ∞ CM.
For the case (iii): Let and , where is a periodic entire function with period 1 such that . Then , and hence for any given , and share a, b, ∞ CM.
For one CM shared value case, Li and Gao  proved the following results.
Theorem D ()
Let be a nonconstant entire function of finite order , . If and share one finite value a CM, and for a finite value , and have distinct common zeros of multiplicity ≥2, then .
Theorem E ()
Let be a nonconstant entire function of finite order , , and n be a positive integer. If and share one finite value a CM, and for a finite value , and have distinct common zeros of multiplicity ≥2, then .
To generalize Theorems D and E, we prove Theorem 1.2 below.
Theorem 1.2 Let be a meromorphic function of finite order , let be a difference polynomial of , and let be two distinct meromorphic functions. Suppose that and share a, ∞ CM and and have distinct common zeros of multiplicity ≥2, denoted by , such that . Then .
Example 2 Let and , where is a periodic entire function with period 1 such that . Then , and hence for any given and , and share a, ∞ CM.
Remark 2 Chen and Yi  (resp. Li and Gao ) conjectured that the condition on the order of growth of in Theorem B (resp. Theorem C) could be omitted. The same conjecture should be made for Theorems 1.1 and 1.2.
Lemma 2.1 ()
Let be a meromorphic function of hyper order , , and . Then
possibly outside of a set of r with a finite logarithmic measure.
The following lemma is a Clunie-type lemma  for the difference-differential polynomials of a meromorphic function f, which is a finite sum of products of f, derivatives of f, and of their shifts, with all the coefficients being small functions of f. It can be proved by applying Lemma 2.1 with a similar reasoning as in  and stated as follows.
Lemma 2.2 ()
Let be a meromorphic function of hyper order and , be two difference-differential polynomials of f. If
holds and if the total degree of in f and its derivatives and their shifts is ≤n, then .
3 Proof of Theorem 1.1
Since and share the value a, b, ∞ CM, we have
where , are entire functions such that .
It follows from (3.1) and (3.2) that
If , then from (3.3) we obtain
Since , we get and hence finish our proof from (3.1).
Next, we assume that and complete our proof in three steps.
Step 1. We prove the case (i): . From (3.1), we get from Lemma 2.1 that
Similarly, we have . Now, we can deduce a contradiction from (3.3) that
Step 2. We prove the case (iii): . In this case, , are polynomials, and we have
From (3.3), we obtain
which gives , a contradiction to (3.5).
Step 3. We prove the case (ii): or , and for some . Without loss of generality, we assume that and hence (3.4) still holds.
Differentiating (3.1) and (3.2), we get
Combining two equations above, we get
where , , , , .
Notice that the right-hand side of (3.6) is a difference-differential polynomial of f with degree in f, its derivatives and their shifts being ≤1. Then from Lemma 2.2 and its remark, we have . Considering this, with (3.1) and (3.4), we obtain
and hence , which contradicts the condition for some .
4 Proof of Theorem 1.2
Since and share a CM, we have
where p is a polynomial such that .
It follows from (4.1) that
For each point , , satisfying the assumption in Theorem 1.2, we get
from (4.1) and (4.3), we see that . Then we can obtain from (4.2) and (4.4) that . By assumption, has at least m zeros. This means that . Therefore,
holds for some nonconstant c. For the point such that , we get from (4.5) that and hence prove that .
Hayman WK: Meromorphic Functions. Clarendon Press, Oxford; 1964.
Laine I: Nevanlinna Theory and Complex Differential Equations. de Gruyter, Berlin; 1993.
Yang CC, Yi HX: Uniqueness Theory of Meromorphic Functions. Kluwer Academic, Dordrecht; 2003.
Rubel LA, Yang CC Lecture Notes in Math. 599. In Values Shared by an Entire Function and Its Derivative. Springer, Berlin; 1977:101-103.
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: 3767-3791. 10.1090/S0002-9947-09-04663-7
Halburd RG, Korhonen RJ: 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.2005.04.010
Halburd RG, Korhonen RJ: Nevanlinna theory for the difference operator. Ann. Acad. Sci. Fenn., Math. 2006, 31: 463-478.
Chen ZX, Yi HX: On sharing values of meromorphic functions and their differences. Results Math. 2013, 63: 557-565. 10.1007/s00025-011-0217-7
Heittokangas J, Korhonen R, Laine I, Rieppo J: Uniqueness of meromorphic functions sharing values with their shifts. Complex Var. Elliptic Equ. 2011, 56: 81-92. 10.1080/17476930903394770
Heittokangas J, Korhonen R, Laine I, Rieppo J, Zhang J: Value sharing results for shifts of meromorphic functions, and sufficient conditions for periodicity. J. Math. Anal. Appl. 2009, 355: 352-363. 10.1016/j.jmaa.2009.01.053
Li S, Gao ZS: Entire functions sharing one or two finite values CM with their shifts or difference operators. Arch. Math. 2011, 97: 475-483. 10.1007/s00013-011-0324-4
Qi XG: Value distribution and uniqueness of difference polynomials and entire solutions of difference equations. Ann. Pol. Math. 2011, 102: 129-142. 10.4064/ap102-2-3
Qi XG, Liu K: Uniqueness and value distribution of differences of entire functions. J. Math. Anal. Appl. 2011, 379: 180-187. 10.1016/j.jmaa.2010.12.031
Ozawa M: On the existence of prime periodic entire functions. Kodai Math. Semin. Rep. 1978, 29: 308-321. 10.2996/kmj/1138833654
Halburd, RG, Korhonen, RJ, Tohge, K: Holomorphic curves with shift-invariant hyperplane preimages. arXiv:0903.3236
Clunie J: On integral and meromorphic functions. J. Lond. Math. Soc. 1962, 37: 17-27. 10.1112/jlms/s1-37.1.17
Yang CC, Laine I: On analogies between nonlinear difference and differential equations. Proc. Jpn. Acad., Ser. A, Math. Sci. 2010, 86: 10-14. 10.3792/pjaa.86.10
This work was supported by the National Natural Science Foundation of China (No. 11226091, No. 11171013).
The authors declare that they have no competing interests.
Both authors drafted the manuscript, read and approved the final manuscript.