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Growth and poles of solutions of systems of complex composite functional equations
Advances in Difference Equations volume 2013, Article number: 378 (2013)
In this paper, we investigate the growth of transcendental meromorphic solutions of some types of systems of complex functional equations and obtain the lower bounds for Nevanlinna lower order for meromorphic solutions of such equations. Our results are improvement of the previous theorems given by Gao, Zheng and Chen. Some examples are also given to illustrate our results.
1 Introduction and main results
Throughout this paper, the term ‘meromorphic’ will always mean meromorphic in the complex plane ℂ. Considering a meromorphic function f, we shall assume that readers are familiar with the fundamental results and the standard notations of the Nevanlinna value distribution theory of meromorphic functions such as , , , the first and second main theorems, lemma on the logarithmic derivatives etc. of Nevanlinna theory (see Hayman , Yang  and Yi and Yang ). We also use , , and to denote the order, the lower order, the exponent of convergence of zeros and the exponent of convergence of poles of , respectively, and to denote any quantity satisfying for all r outside a possible exceptional set of finite logarithmic measure .
Recently, there have been a number of papers focusing on the growth of solutions of difference equations, value distribution and uniqueness of differences analogues of Nevanlinna’s theory (including [4–9]). Based on these results given in [10–12], people obtained many interesting theorems in the fields of complex analysis.
In 2003, Silvennoinen  studied the growth and existence of meromorphic solutions of functional equations of the form and obtained the following result.
Theorem 1.1 
Let f be a non-constant meromorphic solution of the equation
where is an entire function, , are small meromorphic functions with respect to f. Then is a polynomial.
In 2012, Gao  studied the problem when the above equation is replaced by the following system of function equations:
where is an entire function, , are irreducible rational functions, the coefficients are small functions; and he obtained the following.
Theorem 1.2 [, Theorem 1]
Let be a non-constant meromorphic solution of system (1). Then is a polynomial.
After his works, Gao [15, 16], Xu et al.  further investigated the growth and existence of meromorphic solutions of some types of systems of complex functional equations and obtained a series of results (see [15, 16, 18, 19]). Inspired by the ideas of Refs. [14–16, 20, 21], we investigate some properties of solutions of some types of systems of complex functional equations and obtain the following results.
The first theorem is about meromorphic solutions with few zeros and poles of a type of system of complex functional equations.
Theorem 1.3 Let and suppose that , are a pair of non-rational meromorphic solutions of the system
with the coefficients , , , being small functions with respect to , and . If
then system (2) is of the form
where , are meromorphic functions, , .
Theorem 1.4 Suppose that are a pair of transcendental meromorphic solutions of the system of q-shift difference equations
where , , , and the coefficients , () are rational functions. If () are entire or have finitely many poles, then there exist constants () and such that for all ,
Theorem 1.5 Suppose that are a pair of transcendental meromorphic solutions of the system of q-shift difference equations
where , , , the coefficients , , are rational functions, and , are relatively prime polynomials in over the field of rational functions satisfying , , , . If () have infinitely many poles, then for sufficiently large r,
Remark 1.1 Since system (4) is a particular case of system (5), from the conclusions of Theorem 1.5, we can get the following result.
Under the assumptions of Theorem 1.4. If () have infinitely many poles, then there exist constants () and such that for all ,
Example 1.1 The function satisfies the system of the form
with rational coefficients, where , and . Since for all , we have () and for . This shows that the conclusion of Theorem 1.4 is sharp and the equality in the consequent result of Remark 1.1 can be arrived.
Let q, be stated as in Theorem 1.5, set
Now, we will investigate the lower order of meromorphic solutions of a type of system of complex function equations and obtain a result as follows.
Theorem 1.6 Suppose that are a pair of transcendental meromorphic solutions of the system of q-difference equations
where , are finite index sets satisfying
and , , and all coefficients of (6) are of growth , . If
then for sufficiently large r,
where are constants. Thus, the lower order of , satisfy
Example 1.2 The functions satisfy the system of function equations
with small function coefficients
where , , , and , , , are small functions of , . We have , and
This shows that Theorem 1.6 may hold.
2 The proof of Theorem 1.3
Denote , . By applying Valiron-Mohon’ko theorem  to (2), we have
From (3), we can take constants , such that
then we have
From (8) and the definitions of (), similar to the above argument, we have
where . From (3), we know that zeros and poles are Borel exceptions of (), and from [, Satz 13.4], we have that () is of regular growth. Hence, there exists such that for . So, we can get that
Now, we rewrite system (2) as
without loss of generality, assume that , are monic polynomials in with coefficients of growth , . Set , , . From (9), we have . And because
it follows that
Substituting , , to the above equalities and comparing the leading coefficients, we can get
Solving the above equations, we get
From (9) and (10), it follows that
Thus, we complete the proof of Theorem 1.3.
3 Proofs of Theorems 1.4 and 1.5
3.1 The proof of Theorem 1.4
Because the coefficients , () are rational functions, we can rewrite (4) as follows:
where the coefficients , () are polynomials. We will consider two cases as follows.
Case 1. Since are a pair of solutions of system (4) or (11) and , , are transcendental entire, set (), (), , and . Taking , and from and , we have that
when r is sufficiently large. Since (; ) are polynomials and () are transcendental entire functions, we have and . Then, for sufficiently large r, it follows that
From (12) and (13), for sufficiently large r it follows that
where , , for some constants . From (14), for sufficiently large r, we get
Iterating (15), we have
Observe that , then for sufficiently large r, we have
And since , it follows that the series is convergent. Thus, for sufficiently large r, we have
where () are some constants. Since is a transcendental entire function, for sufficiently large r, we have
where . Hence, from (16)-(17), there exists such that for , we have
Thus, for each sufficiently large R, there exists such that
From (19) and (20), we have
Similar to the above argument, we can get that there exist constants and such that for all ,
Case 2. Suppose that are a pair of solutions of system (4) and () are meromorphic with finitely many poles. Then there exist polynomials such that () are entire functions. Substituting into (11) and again multiplying away the denominators, we can get a system similar to (11). By using the same argument as in the above, we can get that for sufficiently large r,
where (>0) () are some constants.
From Case 1 and Case 2, this completes the proof of Theorem 1.4.
3.2 The proof of Theorem 1.5
Since the coefficients of , are rational functions, we can choose a sufficiently large constant R (>0) such that the coefficients of , () have no zeros or poles in . Assume that is a solution of system (5) and () are transcendental, since () have infinitely many poles. Thus, without loss of generality, we choose a pole of of multiplicity satisfying . Since , then the right-hand side of the second equation in (5) has a pole of multiplicity at . Therefore, there exists at least one index such that is a pole of of multiplicity . If , this process will be terminated and we have to choose another pole of in the way we did above. If , since , then the right-hand side of the first equation in (5) has a pole of multiplicity . Therefore, there exists at least one index such that is a pole of of multiplicity .
We proceed to follow the step above, we can get a sequence
where is a pole of with multiplicity , and . From the above discussion, we can get . Obviously, we have as . Then there exists a positive integer such that for sufficiently large k (),
where . Thus, for each sufficiently large r, there exists such that , where , it follows that
From (23) and (24), we have
And there exists and for all , we have
Similar to the above discussion, we can get that there exists and for all , we have
From these inequalities, we can get easily.
Thus, the proof of Theorem 1.5 is completed.
4 Proof of Theorem 1.6
Lemma 4.1 [, Lemma 2]
Let be meromorphic functions. Then
where is an index set consisting of s elements, and .
outside of a possible exceptional set of finite linear measure. Then from (25) there exists such that
holds for all . Iterating (26), for any and , we have
By employing the same argument as in the proof of Theorem 1.5, for sufficiently large ϱ, from the above inequalities, we can get
Letting , from (27) we have
where , are constants satisfying
Thus, from (28) the lower order of , satisfy
Hence, we complete the proof of Theorem 1.6. □
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The authors thank the referees for their valuable suggestions to improve the present article. The first author was supported by the NNSF of China (61202313), the Natural Science Foundation of Jiang-Xi Province in China (2010GQS0119 and 20132BAB211001).
The authors declare that they have no competing interests.
HW, HYX completed the main part of this article, HW, YH, HYX corrected the main theorems. All authors read and approved the final manuscript.
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Wang, H., Huang, Y. & Xu, H. Growth and poles of solutions of systems of complex composite functional equations. Adv Differ Equ 2013, 378 (2013). https://doi.org/10.1186/1687-1847-2013-378
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