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Value distribution of difference and q-difference polynomials
Advances in Difference Equations volume 2013, Article number: 98 (2013)
In this paper, we investigate the value distribution of difference polynomial and obtain the following result, which improves a recent result of K. Liu and L.Z. Yang: Let f be a transcendental meromorphic function of finite order σ, c be a nonzero constant, and be a small function of f, and let
be a polynomial with a multiple zero. If , then has infinitely many zeros. We also obtain a result concerning the value distribution of q-difference polynomial.
1 Introduction and main results
Throughout the paper, we assume that the reader is familiar with the standard symbols and fundamental results of Nevanlinna theory as found in [1–3]. A function is called the meromorphic function, if it is analytic in the complex plane except at isolated poles. For any non-constant meromorphic function f, we denote by any quantity satisfying
possibly outside of a set of finite linear measure in . A meromorphic function is called a small function of provided that . As usual, we denote by the order of a meromorphic function , and denote by () the exponent of convergence of the zeros (poles) of .
Recently, a number of papers concerning the complex difference products and the differences analogues of Nevanlinna’s theory have been published (see [4–12] for example), and many excellent results have been obtained. In 2007, Laine and Yang  investigated the value distribution of difference products of entire functions, and obtained the following result.
Theorem A Let be a transcendental entire function of finite order, and c be a non-zero complex constant. Then for , assumes every non-zero value infinitely often.
Liu and Yang  improved Theorem A, and proved the next result.
Theorem B Let be a transcendental entire function of finite order, and c be a non-zero complex constant. Then for , has infinitely many zeros, where is a polynomial in z.
The purpose of this paper is to investigate the value distribution of difference polynomial and q-difference polynomial , where with constant coefficients , and is a mall function of .
For the sake of simplicity, we denote by and the number of the simple zeros and the number of multiple zeros of a polynomial
We obtain the following result which improves Theorem A and Theorem B.
Theorem 1.1 Let f be a transcendental meromorphic function of finite order , and c be a non-zero constant, and let
be a polynomial with constant coefficients and . If , then has infinitely many zeros, where is a small function of f.
Remark 1 The result of Theorem 1.1 may be false if , for example, , it is obvious that has no zeros. The following example shows that the assumption in Theorem 1.1 cannot be deleted. In fact, let , , , and . Then and has no zeros. Also, let , , , and . Then has no zeros. This shows that the restriction in Theorem 1.1 to the multiple zero case is essential.
Considering the value distribution of q-differences polynomials, we obtain the following result.
Theorem 1.2 Let be a transcendental entire function of zero order, and . Suppose that q is a non-zero complex constant and n is an integer. If , then has infinitely many zeros.
2 Some lemmas
Lemma 2.1 
Given two distinct complex constants , , let f be a meromorphic function of finite order σ. Then, for each , we have
Lemma 2.2 
Let f be a transcendental meromorphic function of finite order σ, c be a complex number. Then, for each , we have
The following lemma is a revised form of Lemma 2.4.2 in .
Lemma 2.3 Let be a transcendental meromorphic solution of
where , are differential polynomials in f and its derivatives with meromorphic coefficients, say , n be a positive integer. If the total degree of as a polynomial in f and its derivatives is less than or equal to n, then
Lemma 2.4 
Let be a non-constant meromorphic function of finite order, . Then
outside of a possible exceptional set E with finite logarithmic measure.
Lemma 2.5 
Let f be a non-constant zero-order meromorphic function, and . Then
on a set of logarithmic density 1.
Remark 2 For the similar reason in Theorem 1.1 in , we can easily deduce that
also holds on a set of logarithmic density 1.
Using the identity
and let Poisson-Jensen formula with , we see
where and are the zeros and poles of f, respectively. Integration on the set gives us the proximity function,
Since () in , we get ().
Following the similar method in the proof of Theorem 1.1 in , we get the result. □
Lemma 2.6 Let f be a non-constant zero-order entire function, and . Then
on a set of logarithmic density 1.
Proof Since f is an entire function of zero-order, we deduce from Lemma 2.5 that
On the other hand, using Remark 2, we get
The assertion follows from (2.1) and (2.2). □
3 Proof of Theorem 1.1
Let be the canonical products of the nonzero poles of . Since and is a small function of , we know that . Suppose on contrary to the assertion that has finitely many zeros. Then we have
where is a polynomial, and is a rational function. Set . Then
Differentiating (3.2) and eliminating , we obtain
Let be the distinct zeros of . Then
Substituting this into (3.3), we have
Note that has at least one multiple zero, we may assume that without loss of generality, and we have
Now we distinguish two cases.
Case 1. . In this case, we obtain from (3.4) that
Since and , by integrating, we have
where k is a non-zero constant. From (3.2) and (3.5), we have
By Lemma 2.2, we have
Since , and is a transcendental, this is impossible.
Case 2. . In this case, we set
Since and , we know that is a differential polynomial of with meromorphic coefficients, and
By Lemma 2.3, we have
for and .
Now for any given ε (), we obtain from Lemma 2.1, Lemma 2.2 and (3.1) that
The lemma of logarithmic derivative implies that
It follows from (3.7) to (3.10) that
Since , we obtain from the definition of that
Note that, a zero of which is not a pole of and , is a pole of with the multiplicity at most 1, we know from (3.6), (3.1), Lemma 2.4 and that
for the positive ε sufficiently small. Hence (see the definition of ),
It follows from (3.15) and (3.11) that
Thus, we deduce from (3.16) and (3.13) that
This contradicts that f is of order σ. Theorem 1.1 is proved.
4 Proof of Theorem 1.2
Denote . From Lemma 2.6 and the standard Valiron-Mohon’ko theorem, we deduce
Since f is a entire function, then by the second main theorem and Lemma 2.5, we have
Since f is a transcendental entire function with , we deduce that has infinitely many zeros.
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This work was supported by the NSF of Shandong Province, P.R. China (No. ZR2010AM030) and the NNSF of China (No. 11171013 and No. 11041005).
The authors declare that they have no competing interests.
All authors drafted the manuscript, read and approved the final manuscript.