Value Distributions and Uniqueness of Difference Polynomials
© Kai Liu et al. 2011
Received: 21 January 2011
Accepted: 7 March 2011
Published: 15 March 2011
We investigate the zeros distributions of difference polynomials of meromorphic functions, which can be viewed as the Hayman conjecture as introduced by (Hayman 1967) for difference. And we also study the uniqueness of difference polynomials of meromorphic functions sharing a common value, and obtain uniqueness theorems for difference.
A meromorphic function means meromorphic in the whole complex plane. Given a meromorphic function , recall that , is a small function with respect to , if , where is used to denote any quantity satisfying , as outside a possible exceptional set of finite logarithmic measure. We use notations , to denote the order of growth of and the exponent of convergence of the poles of , respectively. We say that meromorphic functions and share a finite value IM (ignoring multiplicities) when and have the same zeros. If and have the same zeros with the same multiplicities, then we say that and share the value CM (counting multiplicities). We assume that the reader is familiar with standard notations and fundamental results of Nevanlinna Theory [1–3].
As we all know that a finite value is called the Picard exception value of , if has no zeros. The Picard theorem shows that a transcendental entire function has at most one Picard exception value, a transcendental meromorphic function has at most two Picard exception values. The Hayman conjecture , is that if is a transcendental meromorphic function and , then takes every finite nonzero value infinitely often. This conjecture has been solved by Hayman  for , by Mues  for , by Bergweiler and Eremenko  for . From above, it is showed that the Picard exception value of may only be zero. Recently, for an analog of Hayman conjecture for difference, Laine and Yang [8, Theorem 2] proved the following.
Theorem A implies that the Picard exception value of cannot be nonzero constant. However, Theorem A does not remain valid for meromorphic functions. For example, , , . Thus, we get that never takes the value −1, and never takes the value 1.
As the improvement of Theorem A to the case of meromorphic functions, we first obtain the following theorem. In the following, we assume that and are small functions with respect of , unless otherwise specified.
In the following, we will consider the zeros of other difference polynomials. Using the similar method of the proof of Theorem 1.2 below, we also can obtain the following results.
Some results about the zeros distributions of difference polynomials of entire functions or meromorphic functions with the condition can be found in [9–12]. Theorem 1.7 is a partial improvement of [11, Theorem 1.1] for is an entire function and is also an improvement of [13, Theorem 1.1] for the case of .
The uniqueness problem of differential polynomials of meromorphic functions has been considered by many authors, such as Fang and Hua , Qiu and Fang , Xu and Yi , Yang and Hua , and Lahiri and Rupa . The uniqueness results for difference polynomials of entire functions was considered in a recent paper , which can be stated as follows.
Theorem B (see [19, Theorem 1.1]).
Theorem C (see [19, Theorem 1.2]).
In this paper, we improve Theorems B and C to meromorphic functions and obtain the following results.
From the proof of Theorem 1.11 and (2.7) below, we obtain easily the next result.
2. Some Lemmas
The difference logarithmic derivative lemma of functions with finite order, given by Chiang and Feng [20, Corollary 2.5], Halburd and Korhonen [21, Theorem 2.1], plays an important part in considering the difference Nevanlinna theory. Here, we state the following version.
Lemma 2.1 (see [22, Theorem 5.6]).
Lemma 2.2 (see [20, Theorem 2.1]).
For the proof of Theorem 1.4, we need the following lemma.
thus, we get the (2.3).
In order to prove Theorem 1.5 and Corollary 1.13, we also need the next result.
Lemma 2.5 (see [17, Lemma 3]).
For the proof of Theorem 1.11, we need the following lemma.
Lemma 2.6 (see [16, Lemma 2.3]).
3. Proof of the Theorems
Proof of Theorem 1.2.
Proof of Theorem 1.7.
Proof of Theorem 1.10.
Proof of Theorem 1.11.
which is also a contradiction.
The authors thank the referee for his/her valuable suggestions to improve the present paper. This work was partially supported by the NNSF (no. 11026110), the NSF of Jiangxi (nos. 2010GQS0144 and 2010GQS0139) and the YFED of Jiangxi (nos. GJJ11043 and GJJ10050) of China.
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