Value distribution of q-difference differential polynomials of entire functions
© Xu et al.; licensee Springer. 2014
Received: 26 September 2013
Accepted: 18 February 2014
Published: 7 March 2014
For a complex value , and a transcendental entire function with order, , we study the value distribution of q-difference differential polynomials and .
1 Introduction and main results
A Borel exceptional value of is any value satisfying , where means .
Recently, the difference variant of the Nevanlinna theory has been established independently in [3–6]. Using these theories, value distributions of difference polynomials have been studied by many papers. For example, Laine and Yang  proved that if is a transcendental entire function of finite order, c is a nonzero complex constant and , then takes every nonzero value infinitely often.
Chen  considered the value distribution of and obtained the following theorem.
Theorem A [, Corollary 1.3]
Let be a transcendental entire function of finite order, and let c be a nonzero complex constant. If has the Borel exceptional value 0, then takes every nonzero value infinitely often.
Chen  considered zeros of difference product and gave some conditions that guarantee has finitely many zeros or infinitely many zeros.
Theorem B [, Theorem 1]
If satisfies , or has infinitely many zeros, then has infinitely many zeros.
If has only finitely many zeros and , then has only finitely many zeros.
The zero distribution of differential polynomials is a classical topic in the theory of meromorphic functions. Hayman [, Theorem 10] firstly considered the value distribution of , where f is a transcendental function.
Recently, Liu, Liu and Cao  investigated the zeros of and , where is a nonzero small function with respect to .
Theorem C [, Theorems 1.1 and 1.3]
Let be a transcendental entire function of finite order and be a nonzero small function with respect to . If , then has infinitely many zeros. If is not a periodic function with period c and , then has infinitely many zeros.
The main purpose of this paper is to consider a transcendental entire function with positive and finite order and obtain some results on the value distributions of the q-difference differential polynomials and . The first theorem will consider what conditions guarantee that has infinitely many zeros.
Theorem 1.1 Let be a transcendental entire function of finite and positive order , be a constant such that and . Set , is an integer. If has finitely many zeros, then has infinitely many zeros, where is a nonzero small entire function with respect to .
In the following, we will study the value distribution of .
If , then 0 is also the Borel exceptional value of . So that has no nonzero finite Borel exceptional value.
If , then has no finite Borel exceptional value.
takes every nonzero value infinitely often and satisfies .
Using the similar method of the proof of Theorem 1.2(1), we get the following result immediately.
Corollary 1.1 Let be a transcendental entire function of finite and positive order , be a constant such that . If 0 is a Borel exceptional value of , then 0 is also the Borel exceptional value of .
2 Some lemmas
The following are the well-known Weierstrass factorization and Hadamard factorization theorems.
Lemma 2.1 
satisfying . Further, if is of finite order, then in the above form is a polynomial of degree less or equal to the order of .
Lemma 2.2 
are not constants for ;
For , , (, ).
3 The proofs
3.1 Proof of Theorem 1.1
This is a contradiction.
This is a contradiction.
If , where c is a constant, then using the same method as above, we also obtain a contradiction.
Hence has infinitely many zeros.
3.2 Proof of Theorem 1.2
Here , , are differential polynomials of , and , .
Since and , this implies that 0 is the Borel exceptional value of .
Since , we have .
This is a contradiction.
Which is a contradiction.
If or , then using the same method as above, we also obtain a contradiction.
Case 3. From Case 1 and Case 2, we get that if has a finite Borel exceptional value, then any nonzero finite value c must not be the Borel exceptional value of , so takes every nonzero value infinitely often, since , then .
The proof of Theorem 1.2 is completed.
This research was partly supported by the NSFC (no. 11101201, 11301260), Foundation of Post Ph.D. of Jiangxi, the NSF of Jiangxi (no. 20122BAB211001, 20132BAB211003) and NSF of education department of Jiangxi (no. GJJ13077, GJJ13078) of China.
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