 Research
 Open access
 Published:
Growth of the solutions of some qdifference differential equations
Advances in Difference Equations volumeÂ 2015, ArticleÂ number:Â 172 (2015)
Abstract
In view of Nevanlinna theory, we study the growth and poles of solutions of some complex qdifference differential equations. We obtain the estimates on the Nevalinna order, the lower order, and the counting function of poles for meromorphic solutions of such equations.
1 Introduction and main results
In this paper, the fundamental theorems and the standard notations of the Nevanlinna value distribution theory of meromorphic functions will be used (see Hayman [1], Yang [2] and Yi and Yang [3]). For a meromorphic function \(f(z)\), we also use \(S(r,f)\) to denote any quantity satisfying \(S(r,f)=o(T(r,f))\) for all r outside a possible exceptional set E of finite logarithmic measure \(\lim_{r\rightarrow\infty}\int_{[1,r)\cap E}\frac{dt}{t}<\infty\), and a meromorphic function \(a(z)\) is called a small function with respect to f, if \(T(r,a)=S(r,f)=o(T(r,f))\).
In 1925, Ritt [4] gave the form of solutions of the SchrÃ¶dinger equation
where \(c\in\mathbb{C}\), \(c\neq0,1\), and \(R(f)\) is a rational function in f. In 1983, Rubel [5] posed the following question:
What can be said about the more general equation
where \(R(z,f)\) is rational in both variables?
Later, Ishizaki [6] and Wittich [7] investigated the existence of meromorphic solutions of the equation of the following form:
where \(a(z)\) and \(b(z)\) are meromorphic functions.
In 2002, Gundersen et al. [8] studied the growth of meromorphic solutions of qdifference equations and obtained results as follows.
Theorem 1.1
([8], TheoremÂ 3.2)
Suppose that f is a transcendental meromorphic solution of an equation of the form
with meromorphic coefficients \(a_{j}(z)\), \(b_{j}(z)\) are of growth \(S(r,f)\), and a constant c (\(c>1\)), assuming that \(d:= \max\{p,q\}\geq1\), \(a_{p}(z)\neq0\), \(b_{q}(z)\neq0\), and that \(R(z, f(z))\) is irreducible in f. Then \(\rho(f)=\frac{\log d}{\logc}\), where \(\rho(f)= \limsup_{r\rightarrow+\infty}\frac{\log\log T(r,f)}{\log r}\).
Theorem 1.2
([8], TheoremÂ 3.4)
Let c be a complex constant satisfying \(c>1\), and suppose that f is a nonconstant meromorphic solution of a functional equation of the form
where \(A(z,y)\) and \(B(z,y)\) are rational functions with meromorphic coefficients of growth \(S(r,f)\) such that \(A(z,y)\) and \(B(z,y)\) are irreducible in y. If \(0< a:=\deg_{f}A\leq\deg_{f}B=:b\), then \(\rho(f)=\frac{\log b\log a}{\logc}\).
In 2012, Beardon [9] studied entire solutions of the generalized function equation
where q is a nonzero complex number. To state the results of Beardon [9], we first introduce some notations as follows.
Let the formal series \(\mathcal{O}\) and \(\mathcal{I}\) be defined by
and the sets \(\mathcal{K}_{p}=\{z: z^{p}=p+2\} \) (\(p=1,2,\ldots\)), and \(\mathcal{K}=\mathcal {K}_{1}\cup\mathcal{K}_{2}\cup\cdots\)â€‰. Thus, we see that \(\mathcal {K}_{p}\) contains p elements and \(z=r_{p}\), for \(z\in\mathcal {K}_{p}\), where \(r_{p}=(p+2)^{\frac{1}{p}}\). Since \(p\in\mathbb{N}_{+}\), we have \(z>1\). Since \(\frac{\log(x+2)}{x}\) is decreasing as \(x>1\), we have \(r_{1}>r_{2}>\cdots>1\), and \(r_{p}\rightarrow1\) as \(p\rightarrow\infty\). Based on the above notations, Beardon obtained two main theorems as follows.
Theorem 1.3
([9])
Any transcendental solution f of (1) is of the form
where p is a positive integer, \(b\neq0\) and \(q\in\mathcal {K}_{p}\). In particular, if \(q\notin\mathcal{K}\), then the only formal solutions of (1) are \(\mathcal{O}\) and \(\mathcal{I}\).
Theorem 1.4
([9])
For each positive integer p, there is a unique real entire function
which is a solution of (1) for each \(q\in\mathcal{K}_{p}\). Further, if \(q\in\mathcal{K}_{p}\), then the only transcendental solutions of (1) are the linear conjugates of \(F_{p}\).
Recently, Zhang [10] further studied the growth of solutions of (1) and obtained the following theorem.
Theorem 1.5
([10], TheoremÂ 1.1)
Suppose that f is a transcendental solution of (1) for \(q\in \mathcal{K}\), then we have
where
Regarding TheoremÂ 1.5, Zhang [10] asked the following question: Is the order of transcendental solutions of (1) exactly \(\rho(f)\leq \frac{\log2}{\logq}\)?
In this paper, we further investigate the growth of solution of some class of qdifference differential equation and obtain the following results.
Theorem 1.6
Suppose that f is a solution of equation
where \(q\in\mathcal{K}\) and \(n,s,j\in N_{+}\). If f is a transcendental entire function, then \(n\leq s+1\) and the order of f satisfies
The following example shows that (2) has nontranscendental entire function solution.
Example 1.1
Let \(q=2\), \(n=2\), \(j=1\), and \(s=2\), then \(f(z)=2z^{2}\) satisfies equation
The following example shows that (2) also has a transcendental entire function solution.
Example 1.2
Let \(q=3\), \(n=2\), \(j=1\), and \(s=5\), then \(f(z)=\exp\{3^{\frac {1}{5}}z\}\) satisfies the equation
and
Remark 1.1
Thus, a question arises naturally: Does (2) have a transcendental meromorphic solution?
When the constant q of the right of (2) is replaced by a function, the following example shows that the equation has a transcendental meromorphic solution.
Example 1.3
Let \(f(z)=\frac{e^{z}}{z^{2}}\) and \(q=2\), then \(f(z)\) satisfies the equation
and the order is
Thus, we have the following theorems.
Theorem 1.7
Let f be a transcendental solution of the equation
where q is a nonzero complex number and \(q>1\), n, j, s are positive integers and \(\varphi_{1}(z)\) is a rational function. If f is an entire function, then \(n\leq s+1\) and
Furthermore, if \(n=1\) and f is a meromorphic function with infinitely many poles, then we have
Theorem 1.8
Let f be a transcendental solution of the equation
where q is a complex number and \(q>1\), n, j, s are positive integers and \(\varphi_{2}(z)\) is a small function with respect to f. If f is a meromorphic function with \(\overline{N}(r,f)=S(r,f)\), then \(n< s+1\) and f satisfies
Furthermore, if \(n=1\) and f has infinitely many poles with \(\overline {N}(r,f)=S(r,f)\), and the number of distinct common poles of f and \(\frac{1}{\varphi_{2}}\) is finite, then we have
The following example shows that (4) has a transcendental meromorphic solution f with the order \(\rho(f)=\frac{\log(s+1)}{\logq}\).
Example 1.4
Let \(n=j=s=1\) and \(q=2\), then \(f(z)=\frac{(z1)e^{z}}{z}\) satisfies the equation
where \(\varphi_{2}(z)=\frac{2z^{2}z}{2z1}\) with \(T(r,\varphi_{2})=S(r,f)\) and the order of \(f(z)\) satisfies
Let \(p(z)=p_{k}z^{k}+p_{k1}z^{k1}+\cdots+p_{1}z+p_{0}\), where \(p_{k}(\not\equiv0), \ldots, p_{0}\) are complex constants. Now, we investigate the growth of solutions of such equations, where qz is replaced by \(p(z)\) in (2)(4), and we obtain the following result.
Theorem 1.9
Let f be a transcendental solution of equation
where \(k\geq2\), n, j, s are positive integers and \(\varphi_{3}(z)\) is a small function with respect to f. If f is a transcendental meromorphic function and \(n< sj+s+1\), then f satisfies
Recently, there were many results on meromorphic solutions of complex functional equations (see [11â€“20]). In 2007, Barnett et al. [21] firstly established an analog of the logarithmic derivative lemma on qdifference operators. In 2010, by applying their theorems, Zheng and Chen [22] considered the growth of meromorphic solutions of qdifference equations and obtained results which extended some theorems given by Heittokangas et al. [23].
Theorem 1.10
([22], TheoremÂ 2)
Suppose that f is a transcendental meromorphic solution of equation
where \(q\in\mathbb{C}\), \(q>1\), the coefficients \(a_{j}(z)\) are rational functions and P, Q are relatively prime polynomials in f over the field of rational functions satisfying \(p=\deg_{f} P\), \(t=\deg_{f} Q\), \(d=pt\geq2\). If f has infinitely many poles, then for sufficiently large r, \(n(r,f)\geq Kd^{\frac{\log r}{n\logq}}\) holds for some constant \(K>0\). Thus, the lower order of f, which has infinitely many poles, satisfies \(\mu(f)\geq\frac{\log d}{n\log q}\), where \(\mu(f)=\liminf_{r\rightarrow+\infty}\frac{\log\log T(r,f)}{\log r}\).
From TheoremÂ 1.10, we further study the growth of the solutions of a class of qdifference differential equation and obtain a result as follows.
Theorem 1.11
Suppose that f is a transcendental meromorphic solution of the equation
where \(q\in\mathbb{C}\), \(q>1\), and P, Q are relatively prime polynomials in f over the field of rational functions satisfying \(p=\deg_{f} P\), \(t=\deg_{f} Q\), \(d=pt\geq4\), where the coefficients of P, Q are rational functions in z. If f has infinitely many poles, then for sufficiently large r, \(n(r,f)\geq K(d1)^{\frac{\log r}{\logq}}\) holds for some constant \(K>0\). Thus, the lower order of f, which has infinitely many poles, satisfies \(\mu(f)\geq\frac{\log(d1)}{\logq}\).
Remark 1.2
Under the conditions of TheoremÂ 1.11, by using the same argument as in TheoremÂ 1.8, we can see that the lower order, the order of f, which has infinitely many poles, satisfies
The following example shows that (6) has a nontranscendental solution.
Example 1.5
Let \(q=2\) and \(d=3\), then \(f(z)=\frac{1}{z^{2}}\) satisfies the equation
The following examples show that (6) has transcendental entire and meromorphic solutions.
Example 1.6
Let \(q=2\) and \(d=3\), then \(f(z)=\sin z\) satisfies the equation
Then we have \(\mu(f)=\rho(f)=1=\frac{\log(31)}{\log2}\).
Example 1.7
Let \(q=2\) and \(d=5\), then \(f(z)=\frac{1}{z}e^{z^{2}}\) satisfies the equation
Then we see that f has finitely many poles and \(\mu(f)=\rho(f)=2=\frac {\log(51)}{\log2}\).
Example 1.8
Let \(q=2\) and \(d=3\), then \(f(z)=\frac{1}{\sin z}\) satisfies the equation
So, \(f(z)\) has infinitely many poles and \(\mu(f)=\rho(f)=1=\frac{\log(31)}{\log 2}\).
Remark 1.3
By comparing ExampleÂ 1.8 and TheoremÂ 1.11, we pose a question as follows: Whether the condition â€˜\(d=pt\geq4\)â€™ may be relaxed to â€˜\(d\geq3\) or \(d\geq2\)â€™ in TheoremÂ 1.11?
2 Some lemmas
Lemma 2.1
(ValironMohonâ€™ko [24])
Let \(f(z)\) be a meromorphic function. Then for all irreducible rational functions in f,
with meromorphic coefficients \(a_{i}(z)\), \(b_{j}(z)\), the characteristic function of \(R(z,f(z))\) satisfies
where \(d=\max\{m,n\}\) and \(\Psi(r)=\max_{i,j}\{T(r,a_{i}),T(r,b_{j})\}\).
Lemma 2.2
Let \(f(z)\) be a nonconstant meromorphic function in the complex plane and l be a positive integer. Then
Lemma 2.3
([8])
Let \(\Phi: (1,+\infty)\rightarrow(0,+\infty)\) be a monotone increasing function, and let f be a nonconstant meromorphic function. If for some real constant \(\alpha\in(0,1)\), there exist real constants \(K_{1}>0\) and \(K_{2}\geq1\) such that
then the order of growth of f satisfies
Lemma 2.4
([25])
Let \(f(z)\) be a transcendental meromorphic function and \(p(z)=p_{k}z^{k}+p_{k1}z^{k1}+\cdots+p_{1}z+p_{0}\) be a complex polynomial of degree \(k>0\). For given \(0<\delta<p_{k}\), let \(\lambda=p_{k}+\delta\), \(\mu=p_{k}\delta\), then for given \(\varepsilon>0\) and for r large enough,
Lemma 2.5
Let \(g: (0,+\infty)\rightarrow R\), \(h: (0,+\infty)\rightarrow R\) be monotone increasing functions such that \(g(r)\leq h(r)\) outside of an exceptional set E with finite linear measure, or \(g(r)\leq h(r)\), \(r\notin H\cup(0,1]\), where \(H\subset(1,+\infty)\) is a set of finite logarithmic measure. Then, for any \(\alpha>1\), there exists \(r_{0}\) such that \(g(r)\leq h(\alpha r)\) for all \(r\geq r_{0}\).
Lemma 2.6
([29])
Let \(\psi(r)\) be a function of r (\(r\geq r_{0}\)), positive and bounded in every finite interval.

(i)
Suppose that \(\psi(\mu r^{m})\leq A\psi(r)+B\) (\(r\geq r_{0}\)), where Î¼ (\(\mu>0\)), m (\(m>1\)), A (\(A\geq1\)), B are constants. Then \(\psi(r)=O((\log r)^{\alpha})\) with \(\alpha=\frac{\log A}{\log m}\), unless \(A=1\) and \(B>0\); and if \(A=1\) and \(B>0\), then for any \(\varepsilon>0\), \(\psi(r)=O((\log r)^{\varepsilon})\).

(ii)
Suppose that (with the notation of (i)) \(\psi(\mu r^{m})\geq A\psi(r)\) (\(r\geq r_{0}\)). Then for all sufficiently large values of r, \(\psi(r)\geq K(\log r)^{\alpha}\) with \(\alpha=\frac{\log A}{\log m}\), for some positive constant K.
Lemma 2.7
(see [12])
holds for any meromorphic function f and any nonzero constant q.
3 Proofs of Theorems 1.61.8
3.1 The proof of Theorem 1.6
By LemmaÂ 2.1 and LemmaÂ 2.7, it follows from (2) that
If f is a transcendental entire function, then we have by LemmaÂ 2.2
Since \(q>1\) and f is transcendental, it follows from (8) that \(n\leq s+1\). Set \(\alpha=\frac{1}{q}\), it follows
By LemmaÂ 2.3, we have \(\rho(f)\leq\frac{\log(s+1)\log n}{\log q}\).
3.2 The proof of Theorem 1.7
Since \(\varphi_{1}(z)\) is a rational function, we have \(T(r,\varphi_{1}(z))=O(\log r)\). If f is a transcendental entire function, similar to the argument as in TheoremÂ 1.6, we easily get \(\rho(f)\leq\frac{\log(s+1)\log n}{\logq}\).
If f is a meromorphic function, by LemmaÂ 2.1, LemmaÂ 2.2, and LemmaÂ 2.7, it follows from (3) that
Since \(q>1\), by LemmaÂ 2.3 we have \(\rho(f)\leq\frac{\log(sj+s+1)\log n}{\log q}\).
Since \(\varphi_{1}(z)\) is a rational function, we can choose a sufficiently large constant R (>0) such that \(\varphi_{1}(z)\) has no zeros or poles in \(\{z\in\mathbb{C}: z>R\}\). Since f has infinitely many poles, we can choose a pole \(z_{0}\) of f of multiplicity \(\tau\geq1\) satisfying \(z_{0}>R\). Then the right side of (3) has a pole of multiplicity \(\tau_{1}=(s+1)\tau+sj\) at \(z_{0}\). Then f has a pole of multiplicity \(\tau_{1}\) at \(qz_{0}\). Replacing z by \(qz_{0}\) in (3), we see that f has a pole of multiplicity \(\tau_{2}=(s+1)\tau_{1}+sj\) at \(q^{2}z_{0}\). We proceed to follow the steps above. Since \(\varphi_{1}(z)\) has no zeros or poles in \(\{z\in\mathbb{C}: z>R\}\) and f has infinitely many poles again, we may construct poles \(\zeta_{k}=q^{k}z_{0}\), \(k\in \mathbb{N}_{+}\) of f of multiplicity \(\tau_{k}\) satisfying
as \(k\rightarrow\infty\), \(k\in\mathbb{N}\). Since \(q>1\), \(\zeta_{k}\rightarrow\infty\) as \(k\rightarrow\infty\). For sufficiently large k, we have
Thus, for each sufficiently large r, there exists a \(k\in\mathbb{N}_{+}\) such that
Thus, we have
where
Since, for all \(r\geq r_{0}\),
it follows from (11) that
Thus, this completes the proof of TheoremÂ 1.7.
3.3 The proof of Theorem 1.8
Since \(\varphi_{2}(z)\) is a small function, similar to (7), we have
Since f is a transcendental meromorphic function and \(\overline{N}(r,f)=S(r,f)\), by LemmaÂ 2.2 we have
Thus, from (12) and (13), by using the same argument as in TheoremÂ 1.6, we can get
If \(n=1\) and f has infinitely many poles, since the number of distinct common poles of f and \(\frac{1}{\varphi_{2}}\) is finite, we can choose a sufficiently large constant R (>0) such that f and \(\frac{1}{\varphi_{2}(z)}\) have no common poles in \(\{z\in \mathbb{C}: z>R\}\). Thus, we can take a pole \(z_{0}\) of f of multiplicity \(\tau\geq1\) satisfying \(z_{0}>R\). By using the same argument as in TheoremÂ 1.7, we can see that
Hence, from (14) and (15), we complete the proof of TheoremÂ 1.8.
4 The proof of Theorem 1.9
Since f is a transcendental meromorphic solution of (5), and \(\varphi_{3}(z)\) is a small function with respect to f, similar to the proof of (12), and by LemmaÂ 2.2, we have
Then, by LemmaÂ 2.5, for any \(\beta>1\) and for all \(r>r_{0}\), we have
Since \(p(z)\) is a polynomial with \(\deg_{z}p(z)=k\geq2\), by LemmaÂ 2.4, for given \(0<\delta<p_{k}\), let \(\mu=p_{k}\delta\), for given \(\varepsilon>0\) and for sufficiently large r, it follows for (16) that
Set \(R=\beta r\), then we have
Since \(n< s+sj+1\) and \(\beta>1\), \(\mu>0\), we have \(\frac{s+sj+1}{n}>1\) and \(\mu\beta^{k}>0\). Thus, by LemmaÂ 2.6, letting \(\varepsilon\rightarrow0\) and \(\beta\rightarrow1\), we have
Thus, this completes the proof of TheoremÂ 1.9.
5 The proof of Theorem 1.11
Suppose that f is a transcendental meromorphic solution of (6). Since f has infinitely many poles, we can take a pole \(z_{0}\) of f of multiplicity \(\tau\geq1\). Since \(d\geq4\), we see that the right side of (6) has a pole of multiplicity dÏ„ at \(z_{0}\). Then it follows that \(qz_{0}\) is a pole of f of multiplicity \(\tau_{1}= d\tau\tau1\). Since \(d\geq4\) and \(\tau\geq1\), we have \(\tau_{1}\geq1\). Replacing z by \(qz_{0}\) in (6), we have
Thus the right side of (18) has a pole of multiplicity \(d\tau_{1}\) at \(qz_{0}\). Then we see that \(q^{2}z_{0}\) is a pole of f of multiplicity \(\tau_{2}=d\tau_{1}\tau_{1}1=(d1)^{2}\tau(d1)1\).
We proceed to follow the steps above. Since f has infinitely many poles, we may construct poles \(\zeta_{k}=q^{k}z_{0}\), \(k\in N_{+}\) of f of multiplicity \(\tau_{k}\) satisfying
Since \(\tau\geq1\) and \(d\geq4\), \(\tau\frac{1}{d2}>0\). Thus, since \(\zeta_{k}\rightarrow\infty\) as \(k\rightarrow\infty\), for sufficiently largeÂ k, we have
Thus, for each sufficiently large r, there exists a \(k\in \mathbb{N}_{+}\) such that \(r\in[q^{k}z_{0} , q^{k+1}z_{0})\). By using the same method as in the proof of TheoremÂ 1.7, from (20), we have
where
Since for all \(r\geq r_{0}\), we have
Thus, it follows that
Thus, this completes the proof of TheoremÂ 1.11.
References
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)
Ritt, J: Transcendental transcendency of certain functions of PoincarÃ©. Math. Ann. 95, 671682 (1925/26)
Rubel, L: Some research problems about algebraic differential equations. Trans. Am. Math. Soc. 280(1), 4352 (1983)
Ishizaki, K: Hypertranscendency of meromorphic solutions of a linear functional equation. Aequ. Math. 56(3), 271283 (1998)
Wittich, H: Bemerkung zu einer Funktionalgeichungen von H. PoincarÃ©. Arch. Math. (Basel) 2, 9095 (1949/50)
Gundersen, GG, Heittokangas, J, Laine, I, Rieppo, J, Yang, D: Meromorphic solutions of generalized SchrÃ¶der equations. Aequ. Math. 63, 110135 (2002)
Beardon, AF: Entire solutions of \(f (kz) = kf (z)f'(z)\). Comput. Methods Funct. Theory 12(1), 273278 (2012)
Zhang, GW: On a question of Beardon. J. Inequal. Appl. 2013, Art. 331 (2013)
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. 38(3), 250253 (2014)
Bergweiler, W, Ishizaki, K, Yanagihara, N: Growth of meromorphic solutions of some functional equations I. Aequ. Math. 63(1/2), 140151 (2002)
Halburd, R, Korhonen, R: Finiteorder meromorphic solutions and the discrete PainlevÃ© equations. Proc. Lond. Math. Soc. 94, 443474 (2007)
Halburd, R, Korhonen, R: Nevanlinna theory for the difference operator. Ann. Acad. Sci. Fenn., Math. 31, 463478 (2006)
Laine, I, Rieppo, J, Silvennoinen, H: Remarks on complex difference equations. Comput. Methods Funct. Theory 1, 7788 (2005)
Silvennoinen, H: Meromorphic solutions of some composite functional equations. Ann. Acad. Sci. Fenn., Math. Diss. 13, 1420 (2003)
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. 37(5), 449452 (2013)
Wang, J: Growth and poles of meromorphic solutions of some difference equations. J. Math. Anal. Appl. 379, 367377 (2011)
Zhang, JL, Korhonen, R: On the Nevanlinna characteristic of \(f(qz)\) and its applications. J. Math. Anal. Appl. 369, 537544 (2010)
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. 36(6), 584588 (2012)
Barnett, DC, Halburd, RG, Korhonen, RJ, Morgan, W: Nevanlinna theory for the qdifference operator and meromorphic solutions of qdifference equations. Proc. R. Soc. Edinb., Sect. A, Math. 137, 457474 (2007)
Zheng, XM, Chen, ZX: Some properties of meromorphic solutions of qdifference equations. J. Math. Anal. Appl. 361, 472480 (2010)
Heittokangas, J, Korhonen, R, Laine, I, Rieppo, J, Tohge, K: Complex difference equations of Malmquist type. Comput. Methods Funct. Theory 1(1), 2739 (2001)
Laine, I: Nevanlinna Theory and Complex Differential Equations. de Gruyter, Berlin (1993)
Goldstein, R: Some results on factorization of meromorphic functions. J. Lond. Math. Soc. 4(2), 357364 (1971)
Bank, S: A general theorem concerning the growth of solutions of firstorder algebraic differential equations. Compos. Math. 25, 6170 (1972)
Gundersen, GG: Finite order solutions of second order linear differential equations. Trans. Am. Math. Soc. 305, 415429 (1988)
Chen, ZX, Huang, ZB, Zheng, XM: On properties of difference polynomials. Acta Math. Sci. 31B(2), 627633 (2011)
Goldstein, R: On meromorphic solutions of certain functional equations. Aequ. Math. 18, 112157 (1978)
Acknowledgements
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 Jiangxi Province in China (20132BAB211001, 20151BAB201008), and the Foundation of Education Department of Jiangxi (GJJ14644) of China. The second author was supported by the NSF of China (11371225, 11171013, and 11041005).
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authorsâ€™ contributions
HYX completed the main part of this article, HYX, LZY, and HW corrected the main theorems. All authors read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Xu, HY., Yang, LZ. & Wang, H. Growth of the solutions of some qdifference differential equations. Adv Differ Equ 2015, 172 (2015). https://doi.org/10.1186/s1366201505209
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s1366201505209