Properties of q-shift difference-differential polynomials of meromorphic functions
© Wang et al.; licensee Springer 2014
Received: 28 May 2014
Accepted: 9 September 2014
Published: 24 September 2014
In this paper, we deal with the zeros of the q-shift difference-differential polynomials and , where is a nonzero polynomial of degree n, () are constants, and is a small function of f. The results of this paper are an extension of the previous theorems given by Chen and Chen and Qi. We also investigate the value sharing for q-shift difference polynomials of entire functions and obtain some results which extend the recent theorem given by Liu, Liu and Cao.
1 Introduction and main results
In addition, for some , if the zeros of and (if , zeros of and are the poles of and , respectively) coincide in locations and multiplicities we say that and share the value a CM (counting multiplicities) and if they coincide in locations only we say that and share a IM (ignoring multiplicities).
Let l be a nonnegative integer or infinity. For , we denote by the set of all a-points of f where an a-point of multiplicity k is counted k times if and times if . If , we say that f, g share the value a with weight l.
Definition 1.2 (see )
When f and g share 1 IM, we denote by the counting function of the 1-points of f whose multiplicities are greater than 1-points of g, where each zero is counted only once; similarly, we have . Let be a zero of of multiplicity p and a zero of of multiplicity q, we also denote by the counting function of those 1-points of f where .
In recent years, there has been an increasing interest in studying difference equations, difference products and q-differences in the complex plane ℂ, and a number of papers (including [6–13]) have focused on the value distribution and uniqueness of differences and differences operator analogs of Nevanlinna theory.
For a transcendental meromorphic function f of finite order, herein and hereinafter, c is a nonzero complex constant and is small function with respect to f, Liu et al. , Chen et al. , and Luo and Lin  studied the zeros distributions of difference polynomials of meromorphic functions and obtained: if , then has infinitely many zeros [, Theorem 1.2]; if , then has infinitely many zeros (see [, Theorem 1]), where is a nonzero polynomial, where (≠0) are complex constants, and m is the number of the distinct zeros of .
For transcendental meromorphic (resp. entire) function f of zero order and nonzero complex constant q, Zhang and Korhonen  studied the value distribution of q-difference polynomials of meromorphic functions and obtained the result that if (resp. ), then assumes every nonzero value infinitely often (see [, Theorem 4.1]).
Recently, Liu and Qi  firstly investigated the value distributions for q-shift of meromorphic function and obtained the following result.
Theorem A (see [, Theorem 3.6])
Let f be a zero-order transcendental meromorphic function, , , , and let be a rational function. Then the q-shift difference polynomial has infinitely many zeros.
where is a nonzero polynomial of degree n, m is the number of the distinct zeros of , and we obtain the following results.
Theorem 1.1 Let f be a transcendental meromorphic (resp. entire) function of zero order and be stated as in (1). If and (resp. ). Then has infinitely many zeros, where , if .
Theorem 1.2 Let f be a transcendental meromorphic (resp. entire) function of zero order and be stated as in (2). Assume and (resp. ). Then has infinitely many zeros, provided that , .
Recently, there were obtained some results on the existence and growth of solutions of difference-differential equations (see [19, 20]). Here, from Theorem 1.1 and Theorem 1.2, we get the following result on some nonlinear q-shift difference-differential equations.
has no transcendental meromorphic solution of zero order, provided that .
has no transcendental meromorphic solution of zero order, provided that , , and .
Theorem B (see [, Theorem 2])
- (i)for a constant t such that , where and
f and g satisfy the algebraic equation , where ;
, , where and are two polynomials, b is a constant satisfying and .
In this paper, we will investigated the uniqueness problem of q-shifts of entire functions and obtain the following results.
- (i)for a constant t such that where and
- (ii)f and g satisfy the algebraic equation , where
Then the conclusions of Theorem 1.3 hold, where .
2 Some lemmas
For and k is a positive integer, we denote by the counting function of those a-points of f whose multiplicities are not less than k in counting the a-points of f we ignore the multiplicities (see ) and .
Lemma 2.1 (see )
Lemma 2.2 (see )
, where a (≠0), b are two constants.
Lemma 2.3 (see )
, where a (≠0), b are two constants.
Lemma 2.5 (see [, Theorem 2.1])
on a set of logarithmic density 1.
Thus, we get (6).
Thus, we get (5). □
Using the same method as in Lemma 2.6, we get the following lemma easily.
Since f, g are transcendental entire functions of zero order, from (11), Lemma 2.4 and , we get a contradiction.
This completes the proof of Lemma 2.9. □
3 Proofs of Theorems 1.1 and 1.2
Proof of Theorem 1.1 From (1), by the Valiron-Mohon’ko lemma and Lemma 2.6, we find that is not constant and . Next, we will consider the following two cases when .
From the definitions of , and , we get a contradiction to (12). Then has infinitely many zeros.
which is a contradiction with .
For , similar to the proofs of Case 1 and Case 2, and by the Second Fundamental Theorem and Lemma 2.6, we get the conclusions of Theorem 1.1.
Thus, we complete the proof of Theorem 1.1. □
Proof of Theorem 1.2 Similar to the proof of Theorem 1.1, and using Lemma 2.8, we can prove Theorem 1.2 easily. □
4 Proofs of Corollaries 1.1 and 1.2
The proofs of Corollaries 1.1 and 1.2 are similar. Here, we just give the proof of Corollary 1.2.
which is contradiction with .
This completes the proof of Corollary 1.2. □
5 Proofs of Theorems 1.3 and 1.4
Proof of Theorem 1.3 From the assumptions of Theorem 1.3, we see that , share 1 CM. Then the following three cases will be considered.
Since and f, g are transcendental functions, we get a contradiction.
where are constants.
If and , then from (19) and g is transcendental function, we get .
If and there exists (). Suppose that , from (19), we have which is contradiction with transcendental function g. Then . Similar to this discussion, we can see that when for some .
Thus, from the definition of l, we can see that where t is a constant such that , .
Case 3. If . From Lemma 2.9, we get that for a constant μ such that .
Thus, this completes the proof of Theorem 1.3. □
- (I). Since
Since and f, g are transcendental, a contradiction is obtained.
where a (≠0), b are two constants.
We now will consider three subcases as follows.
which is a contradiction with f, g are transcendental.
Then, from Lemma 2.6 and , we know , a contradiction.
Subcase 2.2. . Then (20) becomes .
If , then is a Picard exceptional value of G. Similar to the discussion as in Subcase 2.1, we can deduce a contradiction again.
Since , by Lemma 2.9, we see that for a constant μ such that .
Subcase 2.3. . Then (20) becomes .
If , then . Similar to the discussion as in Subcase 2.1, we can deduce a contradiction again.
- (II). Since(21)
Since and f, g are transcendental functions, we get a contradiction.
- (III). Since(24)
Since , from (27) and f, g are transcendental, we get a contradiction.
- (IV), that is, , share 1 IM. From the definitions of , , we have(28)
Since , we get a contradiction.
Case 2. Suppose that , satisfy Lemma 2.3(ii). Similar to the proof of Case 2 in (I), we get the conclusions of Theorem 1.4 easily.
Thus, the proof of Theorem 1.4 is completed. □
The authors thank the referee for his/her valuable suggestions to improve the present article. This work was supported by the NSFC (11301233, 61202313), the Natural Science Foundation of Jiang-Xi Province in China (20132BAB211001), the Foundation of Education Department of Jiangxi (GJJ14644) of China, and the Humanities and Social Sciences of the Chinese Education Ministry (13YJA760064).
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