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On some complex differential and difference equations concerning sharing function
Advances in Difference Equations volume 2014, Article number: 274 (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
Let f be a nonconstant meromorphic function in the whole complex plane ℂ. We shall use the following standard notations of the value distribution theory:
(see Hayman , Yang  and Yi and Yang ). We denote by any quantity satisfying , as , possibly outside of a set with finite measure. A meromorphic function is called a small function with respect to f if . In addition, we will use the notation to denote the order of meromorphic function , and to denote the type of an entire function with , which are defined to be (see )
We use to denote the hyper-order of , is defined to be (see )
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 
Let f be a nonconstant entire function. Suppose that is not a positive integer or infinite, if f and share one finite value a CM, then
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.
Thus, there are natural questions to ask:
What would happen when is replaced by in Theorem 1.3?
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.
Theorem 1.4 Let and be two nonconstant entire functions and satisfy and , and be stated as in (1) such that
If and share CM, then
for some nonzero constant c.
Theorem 1.5 Let be a nonconstant transcendental entire function with , let be not an integer, and let be stated as in (1). If f and share a nonzero constant a CM and , then
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.
Theorem 1.7 Let be a nonconstant entire function of finite order , be difference polynomial of f of the form
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 
Let be a transcendental entire function, be the central index of . Then there exists a set with finite logarithmic measure, we choose z satisfying and , we get
Lemma 2.2 
Let be an entire function of finite order , and let be the central index of f. Then, for any ε (>0), we have
Lemma 2.3 
Let f be a transcendental entire function, and let be a set having finite logarithmic measure. Then there exists such that , , , and if , then, for any given and sufficiently large ,
Lemma 2.4 
Let with be a polynomial. Then, for every , there exists such that for all the inequalities
Lemma 2.5 Let and be two entire functions with , , then there exists a set that has infinite logarithmic measure such that for all and a positive number , we have
Proof By definition, there exists an increasing sequence satisfying and
For any given β (), there exists some positive integer such that for all and for any given ε (), we have
Thus, there exists some positive integer such that for all , we have
From (2)-(4), for all and for any , we have
Set , then
From the definition of type of entire function, for any sufficiently small , we have
By (5) and (6), set , for all , we have
Thus, this completes the proof of this lemma. □
Lemma 2.6 [, Theorem 2.1]
Let be a meromorphic function of finite order σ, and let η be a fixed nonzero complex number, then, for each , we have
Lemma 2.7 [, Corollary 2.5]
Let be a meromorphic function with order , , and let η be a fixed nonzero complex number, then, for each , we have
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
Since is an entire function, and and share CM, then there is an entire function such that
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.
Suppose that is a nonconstant polynomial, let
where are constants and , . Thus, it follows from (7) and Lemma 2.4 that
Since , from Lemma 2.1, then there exists a subset with finite logarithmic measure such that for some point (), and , we have
Thus, it follows that
From Lemma 2.3, there exists such that , , , , then, for any given ε satisfying
where , and sufficiently large , we have
Since , , are polynomials, let , where , . Then, from Lemma 2.4 and (10), we have
where and M is a positive constant. Since , it follows from (11) that
Since and , from Lemma 2.5, there exists a set that has infinite logarithmic measure such that for a sequence , we have
From (8), (9), (12), (13) and Lemma 2.2, we can get that
which is impossible. Thus, is not a polynomial.
Therefore, is a constant, that is, there exists some nonzero constant c such that
Thus, this completes the proof of Theorem 1.4.
4 The proof of Theorem 1.5
Since and f share the constant a CM, then there exists an entire function such that
We will consider two cases as follows.
Case 1. If , it follows from (15) that
Since and , , are polynomials, it follows from (16) that
outside of an exceptional set with finite linear measure. Thus, there exists a constant K such that
By Lemma 2.8, there exists an , and for all , we have
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.
Case 2. If , from the derivation of (15) and eliminating , we can get
If , that is, , c is a constant. Thus, we can prove the conclusion of Theorem 1.5 easily.
If , then it follows from (18) that
We can rewrite (18) in the following form:
Since and f is transcendental, set
then we have . Thus, it follows from (20) and (21) that
Since is an entire function, from (18)-(22), then we have
It follows that
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
Since is an entire function of finite order and ξ (≠0) is a Borel exceptional value of , then can be written in the form
where is a polynomial of degree l and is an entire function satisfying . Thus, we have
From Lemma 2.7, we have and for . Since , it follows from (23) and (24) that
Set and , then we can deduce from (25) that
Let , it is easy to see that and , that is, .
Suppose that . Since , it follows from (26) that
By the second fundamental theorem concerning small functions, for any , we have
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.
Hayman WK: Meromorphic Functions. Clarendon, Oxford; 1964.
Yang L: Value Distribution Theory. Springer, Berlin; 1993.
Yi HX, Yang CC: Uniqueness Theory of Meromorphic Functions. Kluwer Academic, Dordrecht; 2003. Chinese original: Science Press, Beijing (1995)
Rubel L, Yang CC: Values shared by an entire function and its derivative. Lecture Notes in Mathematics 599. In Complex Analysis. Springer, Berlin; 1977:101-103. (Proc. Conf., Univ. Kentucky, Lexington, Ky., 1976)
Brück R: On entire functions which share one value CM with their first derivative. Results Math. 1996, 30: 21-24. 10.1007/BF03322176
Gundersen GG, Yang LZ: Entire functions that share one value with one or two of their derivatives. J. Math. Anal. Appl. 1998, 223: 85-95.
Mues E, Steinmetz N: Meromorphe funktionen, die mit ihrer ersten und zweiten Ableitung einen endlichen Wert teilen. Complex Var. Theory Appl. 1986, 6: 51-71. 10.1080/17476938608814158
Zhang JL, Yang LZ: A power of a meromorphic function sharing a small function with its derivative. Ann. Acad. Sci. Fenn., Math. 2009, 34: 249-260.
Zhang QC: Meromorphic function that shares one small function with its derivative. J. Inequal. Pure Appl. Math. 2005., 6: Article ID 116
Ai LJ, Yi CF: The growth for solutions of a class of higher order linear differential equations with meromorphic coefficients. J. Jiangxi Norm. Univ., Nat. Sci. 2014, 38(3):250-253.
Tu J, Huang HX, Xu HY, Chen CF: The order and type of meromorphic functions and analytic functions in the unit disc. J. Jiangxi Norm. Univ., Nat. Sci. 2013, 37(5):449-452.
He J, Zheng XM: The iterated order of meromorphic solutions of some classes of higher order linear differential equations. J. Jiangxi Norm. Univ., Nat. Sci. 2012, 36(6):584-588.
Chang JM, Zhu YZ: Entire functions that share a small function with their derivatives. J. Math. Anal. Appl. 2009, 351: 491-496. 10.1016/j.jmaa.2008.07.080
Heittokangas J, Korhonen R, Laine I, Rieppo J, Zhang JL: Value sharing results for shifts of meromorphic functions, and sufficient conditions for periodicity. J. Math. Anal. Appl. 2009, 355: 352-363. 10.1016/j.jmaa.2009.01.053
Li XM, Yi HX: Entire functions sharing an entire function of smaller order with their shifts. Proc. Jpn. Acad., Ser. A, Math. Sci. 2013, 89: 34-39.
Laine I: Nevanlinna Theory and Complex Differential Equations. de Gruyter, Berlin; 1993.
He YZ, Xiao XZ: Algebroid Functions and Ordinary Differential Equations. Science Press, Beijing; 1988.
Mao ZQ: Uniqueness theorems on entire functions and their linear differential polynomials. Results Math. 2009, 55: 447-456. 10.1007/s00025-009-0419-4
Chiang YM, Feng SJ:On the Nevanlinna characteristic of and difference equations in the complex plane. Ramanujan J. 2008, 16: 105-129. 10.1007/s11139-007-9101-1
Barnett DC, Halburd RG, Korhonen RJ, Morgan W: Nevanlinna theory for the q -difference operator and meromorphic solutions of q -difference equations. Proc. R. Soc. Edinb., Sect. A, Math. 2007, 137: 457-474.
Li XM, Yi HX: An entire function and its derivatives sharing a polynomial. J. Math. Anal. Appl. 2007, 330: 66-79. 10.1016/j.jmaa.2006.07.038
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).
The authors declare that they have no competing interests.
HW, L-ZY and H-YX completed the main part of this article, HW, H-YX corrected the main theorems. All authors read and approved the final manuscript.
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Wang, H., Yang, LZ. & Xu, HY. On some complex differential and difference equations concerning sharing function. Adv Differ Equ 2014, 274 (2014). https://doi.org/10.1186/1687-1847-2014-274
- entire function
- Brück’s conjecture
- difference equation