Meromorphic functions sharing small functions with their linear difference polynomials
© Li and Chen; licensee Springer 2013
Received: 30 November 2012
Accepted: 24 February 2013
Published: 18 March 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
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 .
where , .
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 ()
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.
or , and for some ;
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 ()
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 ()
holds and if the total degree of in f and its derivatives and their shifts is ≤n, then .
3 Proof of Theorem 1.1
where , are entire functions such that .
Since , we get and hence finish our proof from (3.1).
Next, we assume that and complete our proof in three steps.
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.
where , , , , .
and hence , which contradicts the condition for some .
4 Proof of Theorem 1.2
where p is a polynomial such that .
holds for some nonconstant c. For the point such that , we get from (4.5) that and hence prove that .
This work was supported by the National Natural Science Foundation of China (No. 11226091, No. 11171013).
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