On some complex differential and difference equations concerning sharing function
© Wang et al.; licensee Springer. 2014
Received: 24 June 2014
Accepted: 6 October 2014
Published: 27 October 2014
By using the theory of complex differential equations, the purpose of this paper is to investigate a conjecture of Brück concerning an entire function f and its differential polynomial sharing a function and a constant β. We also study the problem on entire function and its difference polynomials sharing a function.
1 Introduction and main results
In 1976, Rubel and Yang  proved the following result.
Theorem 1.1 
Let f be a nonconstant entire function. If f and share two finite distinct values CM, then .
In 1996, Brück  gave the following conjecture.
Conjecture 1.1 
for some nonzero constant c.
In 1998, Gundersen and Yang  proved that Brück’s conjecture holds for entire functions of finite order and obtained the following result.
Theorem 1.2 [, Theorem 1]
Let f be a nonconstant entire function of finite order. If f and share one finite value a CM, then for some nonzero constant c.
The shared value problems relative to a meromorphic function f and its derivative have been a more widely studied subtopic of the uniqueness theory of entire and meromorphic functions in the field of complex analysis (see [7–12]).
In 2009, Chang and Zhu  further investigated the problem related to Brück’s conjecture and proved that Theorem 1.2 remains valid if the value a is replaced by a function.
Theorem 1.3 [, Theorem 1]
Let f be an entire function of finite order and be a function such that . If f and share CM, then for some nonzero constant c.
What would happen when is replaced by in Theorem 1.3?
- (ii)For Theorems 1.1-1.3, what would happen when is replaced by differential polynomial(1)
where are polynomials?
The main purpose of this article is to study the above questions and obtain the following theorems.
for some nonzero constant c.
for some nonzero constant c.
Theorem 1.6 [, Theorem 1]
Let f be a nonconstant meromorphic function of finite order , and let η be a nonzero complex number. If and share a finite complex value a CM, then for all , where c is some nonzero complex number.
In this paper, we further investigate Brück’s conjecture related to entire function and its difference polynomial and obtain the following result.
where are nonzero complex numbers. If and ξ (≠0) is a Borel exceptional value of , then .
2 Some lemmas
To prove our theorems, we will require some lemmas as follows.
Lemma 2.1 
Lemma 2.2 
Lemma 2.3 
Lemma 2.4 
Thus, this completes the proof of this lemma. □
Lemma 2.6 [, Theorem 2.1]
Lemma 2.7 [, Corollary 2.5]
Let , be monotone increasing functions such that outside of an exceptional set E with finite linear measure, or , , where is a set of finite logarithmic measure. Then, for any , there exists such that for all .
3 The proof of Theorem 1.4
Next, we will claim that is a constant.
Suppose that is transcendental. It follows that . However, since , it follows from the left-hand side of (7) that , a contradiction. Thus, is not transcendental.
which is impossible. Thus, is not a polynomial.
Thus, this completes the proof of Theorem 1.4.
4 The proof of Theorem 1.5
We will consider two cases as follows.
Thus, we can deduce from (17) that , that is, is a polynomial.
By using the same argument as in [, Theorem 1.1], we can get that , which is a contradiction to is not a positive integer. Thus, is only a constant, it follows from (15) that , where c is a nonzero constant.
If , that is, , c is a constant. Thus, we can prove the conclusion of Theorem 1.5 easily.
which is a contradiction to the assumption of Theorem 1.5.
Thus, from Case 1 and Case 2, we complete the proof of Theorem 1.5.
5 The proof of Theorem 1.7
Let , it is easy to see that and , that is, .
Since ε is arbitrary, we can get a contradiction from the above inequality. Thus, we can get that .
Therefore, we prove that , that is, the conclusion of Theorem 1.7 holds.
The authors thank the referee for his/her valuable suggestions to improve the present article. The first author was supported by the NSF of China (11301233, 61202313), the Natural Science Foundation of Jiang-Xi Province in China (20132BAB211001), and the Foundation of Education Department of Jiangxi (GJJ14644) of China. The second author was supported by the NSF of China (11371225, 11171013).
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