Abstract
In this paper, we study the value distribution of a meromorphic function concerning its derivative and q-shift difference , where is of finite logarithmic order. We also investigate the uniqueness of differential-q-shift-difference polynomials with more general forms of entire functions of order zero.
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1 Introduction and main results
The fundamental theorems and the standard notations of the Nevanlinna value distribution theory of meromorphic functions will be used (see e.g. Hayman [1], Yang [2] and Yi and Yang [3]). In addition, for a meromorphic function , we use to denote any quantity satisfying for all r outside a possible exceptional set E of finite logarithmic measure and also use to denote any quantity satisfying for all r on a set F of logarithmic density 1, where the logarithmic density of a set F is defined by
The order of a meromorphic function is defined by
The logarithmic order of a meromorphic function is defined by (see [4])
If , then is said to be of finite logarithmic order. It is clear that if a meromorphic function has finite logarithmic order, then has order zero.
If is a meromorphic function of finite positive logarithmic order , then has proximate logarithmic order . The logarithmic-type function of is defined as . We have for sufficiently large r. The logarithmic exponent of convergence of a-points of is equal to the logarithmic order of , which is defined as
We see by [4] that for a meromorphic function of finite positive logarithmic order , the logarithmic order of is , where is the logarithmic order of .
Moreover, we assume in the whole paper that m, n, k, , are positive integers, , , and is a non-zero small function with respect to , that is, is a non-zero meromorphic function of growth .
Many mathematicians were interested in the value distribution of different expressions of meromorphic functions and obtained lots of important theorems (see e.g. [1, 6–8]). Especially, Hayman [7] discussed Picard’s values of meromorphic functions and their derivatives, and he obtained the following famous theorem in 1959.
Theorem 1.1 [7]
Let be a transcendental entire function. Then
-
(a)
for , assumes all finite values except possibly zero infinitely often;
-
(b)
for and , assumes all finite values infinitely often.
Further, for a transcendental meromorphic function , Chen and Fang [9] obtained the following result.
Theorem 1.2 [[9], Theorem 1]
Let be a transcendental meromorphic function. If is a positive integer, has infinitely many zeros.
Recently, with the establishments of difference analogies of the Nevanlinna theory (see e.g. [10–12]), many mathematicians focused on studying difference analogies of Theorems 1.1 and 1.2. The main purpose of these results (see e.g. [5, 13–16]) is to get the sharp estimation of the value of n to make difference polynomials and admit infinitely many zeros.
Meantime, q-difference analogies of the Nevanlinna theory and their applications on the value distribution of q-difference polynomials and q-shift-difference equations are also studied (see e.g. [17–19]). Especially, for a transcendental meromorphic (resp. entire) function of order zero, Zhang and Korhonen [20] studied the value distribution of q-difference polynomials of and found that if (resp. ), then assumes every non-zero value infinitely often (see [[20], Theorem 4.1]).
Further, Xu and Zhang [21] investigated the zeros of q-shift difference polynomials of meromorphic functions of finite logarithmic order and obtained the following result in 2012.
Theorem 1.3 [[21], Theorem 2.1]
If is a transcendental meromorphic function of finite logarithmic order , with the logarithmic exponent of convergence of poles less than and q, c are non-zero complex constants, then for , assumes every value infinitely often.
One main aim of this paper is to investigate the zeros of differential-q-shift-difference polynomials about , , and , where is of finite positive logarithmic order.
Theorem 1.4 Let be a transcendental meromorphic function of finite logarithmic order , with the logarithmic exponent of convergence of poles less than . Set
If , then has infinitely many zeros.
We also deal with the value distribution of a differential-q-shift-difference polynomial with another form about , and , and we obtain the following result.
Theorem 1.5 Let be a transcendental meromorphic function of finite logarithmic order , with the logarithmic exponent of convergence of poles less than . Set
If , then has infinitely many zeros, where .
Some more general differential-q-shift-difference polynomials are investigated in the following.
Theorem 1.6 Let be a transcendental meromorphic function of finite logarithmic order , with the logarithmic exponent of convergence of poles less than . Set
If m, n satisfy or , then has infinitely many zeros.
Let
be a non-zero polynomial, where (≠0) are complex constants and is the number of the distinct zeros of . Then we also obtain the following results.
Theorem 1.7 Let be a transcendental meromorphic function of finite logarithmic order , with the logarithmic exponent of convergence of poles less than . Set
If , then has infinitely many zeros.
Theorem 1.8 Let be a transcendental meromorphic function of finite logarithmic order , with the logarithmic exponent of convergence of poles less than . Set
If , then has infinitely many zeros.
Next, we investigate the uniqueness of differential-q-shift-difference polynomials of entire functions of order zero and obtain the following results.
Theorem 1.9 Let and be transcendental entire functions of order zero and . If and share a non-zero polynomial CM, then .
Theorem 1.10 Let and be transcendental entire functions of order zero and . If and share a non-zero polynomial CM, then .
Theorem 1.11 Let and be transcendental entire functions of order zero and . If and share a non-zero polynomial CM, then .
2 Some lemmas
To prove the above theorems, we need some lemmas as follows.
Lemma 2.1 [3]
Let be a non-constant meromorphic function and , where are complex constants and , then
Lemma 2.2 [21]
Let be a transcendental meromorphic function of finite logarithmic order and q, η be two non-zero complex constants. Then we have
Lemma 2.3 [[22], Theorem 2.1]
Let be a non-constant zero-order meromorphic function and . Then
Lemma 2.4 [[3], p.37]
Let be a non-constant meromorphic function in the complex plane and l be a positive integer. Then
Lemma 2.5 [4]
If is a transcendental meromorphic function of finite logarithmic order , then for any two distinct small functions and with respect to , we have
where is a logarithmic-type function of . Furthermore, if has a finite lower logarithmic order
with , then
Remark 2.1 From the proof of Lemma 2.5 (see [[4], Theorem 7.1]), we can easily see that complex values a and b can be changed into and , where and are two distinct small functions with respect to .
Lemma 2.6 Let be a transcendental meromorphic function of order zero, . Then we have
Proof If is a meromorphic function of order zero, from Lemmas 2.2 and 2.4, we have
On the other hand, from Lemmas 2.2 and 2.4 again, we have
Thus, we get (1). □
Lemma 2.7 Let be a transcendental meromorphic function of zero order, . Then we have
and
Proof If is a meromorphic function of order zero, from Lemmas 2.2 and 2.4, we have
that is, we have (2). On the other hand, from Lemmas 2.2 and 2.4, we have
that is, we have (3), where we assume without loss of generality. □
Lemma 2.8 Let be a transcendental meromorphic function of order zero, . Then we have
Proof Since is a transcendental meromorphic function of order zero, by Lemmas 2.1, 2.2, and 2.4, we can easily get the second inequality. On the other hand, it follows by Lemmas 2.1, 2.2, and 2.4 that
Thus, this completes the proof of Lemma 2.8. □
Similar to Lemma 2.8, we have the following lemma.
Lemma 2.9 Let be a transcendental meromorphic function of zero order, . Then we have
3 Proofs of Theorems 1.4-1.8
3.1 Proof of Theorem 1.4
It follows by Lemma 2.6 that holds for all r on a set of logarithmic density 1. Since is transcendental and , is transcendental by Lemma 2.6 again. Since the logarithmic exponent of convergence of poles of less than , we have
Assume that has only finitely many zeros. Thus, by Lemmas 2.2, 2.4-2.6, we have
Thus, it follows that
Since , the above inequality implies
which contradicts the fact that has finite logarithmic order . Thus, has infinitely many zeros, that is, has infinitely many zeros.
This completes the proof of Theorem 1.4.
3.2 Proof of Theorem 1.5
Since is a transcendental meromorphic function of finite logarithmic order, we first claim that . In fact, if , that is, . By solving the above equation, we have , where A is a non-zero complex constant. Thus, we have , which contradicts the fact that is of order zero. Thus, set
It follows by Lemmas 2.2 and 2.4 that
that is,
On the other hand, we can easily get
And it follows from (5) and that
holds for all r on a set of logarithmic density 1. By Lemma 2.2, we have
Assume that has finitely many zeros, then
Since the logarithmic exponent of convergence of poles of is less than , we have
Then, by Lemmas 2.4, 2.5, and (6), we have
It follows by the above inequality and (4) that
Since , the above inequality implies
which contradicts that has finite logarithmic order . Thus, has infinitely many zeros.
This completes the proof of Theorem 1.5.
3.3 Proofs of Theorems 1.6, 1.7, and 1.8
Similar to the argument as in Theorem 1.4, by applying Lemmas 2.7, 2.8, and 2.9 instead, we can easily prove Theorems 1.6, 1.7, and 1.8 respectively.
4 Proofs of Theorems 1.9-1.11
Here, we only give the proof of Theorem 1.10 because the methods of the proofs of Theorems 1.9, 1.10, and 1.11 are very similar.
4.1 Proof of Theorem 1.10
Denote
Since is a transcendental entire function of order zero, by Lemmas 2.1, 2.2, and 2.4, we have
and
Then it follows from (7) and (8) that
We have by (9) that . Similarly, we have and
Since and are entire functions of order zero and share CM, we have
where η is a non-zero constant. If , then we have , that is, .
If , then we have
Since has distinct zeros, by using the second main theorem and Lemma 2.2, we have
where are the distinct zeros of . Similarly, we have
Then (9), (10), (13), and (14) result in
which contradicts .
This completes the proof of Theorem 1.10.
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Acknowledgements
This work was supported by the National Natural Science Foundation of China (11301233, 61202313, 11171119), the Natural Science Foundation of Jiangxi Province in China (20132BAB211001) and Sponsored Program for Cultivating Youths of Outstanding Ability in Jiangxi Normal University of China, and the Foundation of Education Department of Jiangxi (GJJ14271, GJJ14644) of China.
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Zheng, XM., Xu, HY. On value distribution and uniqueness of meromorphic function with finite logarithmic order concerning its derivative and q-shift difference. J Inequal Appl 2014, 295 (2014). https://doi.org/10.1186/1029-242X-2014-295
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DOI: https://doi.org/10.1186/1029-242X-2014-295