On value distribution and uniqueness of meromorphic function with finite logarithmic order concerning its derivative and q-shift difference
© Zheng and Xu; licensee Springer. 2014
Received: 12 April 2014
Accepted: 30 June 2014
Published: 19 August 2014
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.
1 Introduction and main results
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.
We see by  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  discussed Picard’s values of meromorphic functions and their derivatives, and he obtained the following famous theorem in 1959.
Theorem 1.1 
for , assumes all finite values except possibly zero infinitely often;
for and , assumes all finite values infinitely often.
Further, for a transcendental meromorphic function , Chen and Fang  obtained the following result.
Theorem 1.2 [, 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  studied the value distribution of q-difference polynomials of and found that if (resp. ), then assumes every non-zero value infinitely often (see [, Theorem 4.1]).
Further, Xu and Zhang  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 [, 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.
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.
If , then has infinitely many zeros, where .
Some more general differential-q-shift-difference polynomials are investigated in the following.
If m, n satisfy or , then has infinitely many zeros.
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.
If , then has infinitely many zeros.
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 
Lemma 2.2 
Lemma 2.3 [, Theorem 2.1]
Lemma 2.4 [, p.37]
Lemma 2.5 
Remark 2.1 From the proof of Lemma 2.5 (see [, 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 .
Thus, we get (1). □
that is, we have (3), where we assume without loss of generality. □
Thus, this completes the proof of Lemma 2.8. □
Similar to Lemma 2.8, we have the following lemma.
3 Proofs of Theorems 1.4-1.8
3.1 Proof of Theorem 1.4
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
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
where η is a non-zero constant. If , then we have , that is, .
which contradicts .
This completes the proof of Theorem 1.10.
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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