Abstract
In this paper, we investigate the growth and value distribution of solutions of some kinds of linear difference equations, where there may be more than one coefficient having the same maximal order and the same maximal type.
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1 Introduction
Throughout this paper, we use the standard notations and basic results of Nevanlinna’s value distribution theory (see [1–4]). In addition, we use the notations \(\sigma(f)\), \(\sigma_{2}(f)\) and \(\lambda(f-a)\) to denote, respectively, the order, the hyperorder, and the exponent of convergence of the sequence of a-points of a meromorphic function \(f(z)\) in the complex plane, where \(a\in\mathbb{C}\cup\{\infty\}\). Furthermore, we can get the definition of \(\lambda(f-\varphi)\), when a is replaced by a meromorphic function \(\varphi(z)\).
Recently, with the research and further development of difference analogs of Nevanlinna’s theory, it has been applied more and more widely in the difference field. By this important tool, many scholars investigated the linear difference equation
where \(k\in\mathbb{N}_{+}\) and \(c_{j}\), \(j=1,\ldots,k\) are distinct nonzero complex constants, and obtained many results on the growth and the exponent of convergence of the sequence of zeros of meromorphic solutions of (1.1). For instance, in [5–7], the authors considered the case when there is exactly only one coefficient of (1.1) having the maximal order; in [8–10], the authors considered the case when there is exactly only one coefficient having the type strictly greater than the others among those having the maximal order; and in [5, 6, 11], the authors considered the case when the coefficients of (1.1) are polynomials.
Further, how about the case when there are more than one coefficient having the same maximal order and the same maximal type?
In 2013, Liu [12] considered the growth and the exponent of convergence of the sequence of small function value points of second order linear difference equations and obtained the following theorem.
Theorem A
(see [12])
Let \(c_{1}\), \(c_{2}\) (\(\neq c_{1}\)), a be nonzero constants, \(h_{1}(z)\) be a nonzero meromorphic functions with \(\sigma(h_{1})<1\), \(B(z)\) be a nonzero meromorphic function. If \(B(z)\) satisfies any one of the following three conditions:
-
(i)
\(\sigma(B)>1\) and \(\delta(\infty,B)>0\);
-
(ii)
\(\sigma(B)<1\);
-
(iii)
\(B(z)=h_{0}(z)e^{bz}\), where b is a nonzero constant, \(h_{0}(z)\) (≢0) is a meromorphic function with \(\sigma(h_{0})<1\),
then every meromorphic solution \(f(z)\) (≢0) of the difference equation
satisfies \(\sigma(f)\geq\max\{\sigma(B),1\}+1\). Further, if \(\varphi(z)\) (≢0) is a meromorphic function with \(\sigma(\varphi)<\max\{\sigma(B),1\}+1\), then \(\lambda(f-\varphi)=\sigma(f)\geq\max\{\sigma(B),1\}+1\).
Liu and Mao [13] considered the growth of meromorphic solutions of (1.1), where the coefficients may have the same order and the same type, and obtained the following theorem.
Theorem B
(see [13])
Let \(A_{j}(z)=h_{j}(z)e^{P_{j}(z)}+D_{j}(z)\), \(j=0,1,\ldots,k\), where \(P_{j}(z)=a_{jn}z^{n}+\cdots+a_{j0}\) are polynomials with degree n (≥1), \(h_{j}(z)\) (≢0), \(D_{j}(z)\) are entire functions with order less than n. If \(a_{jn}\), \(j=0,1,\ldots,k\) are distinct complex numbers, then every meromorphic solution \(f(z)\) (≢0) of (1.1) satisfies \(\sigma(f)\geq\max_{0\leq j\leq k}\{\sigma(A_{j})\}+1\).
In this paper, we are concerned with the more general problem than Theorems A and B, and obtain the following results, which extend and improve the previous results.
First, we consider the difference equation (1.1).
Theorem 1.1
Let \(k, n\in\mathbb{N}_{+}\), \(A_{j}(z)=B_{j}(z)e^{P_{j}(z)}+D_{j}(z)e^{Q_{j}(z)}+R_{j}(z)\), \(j=0,1,\ldots,k\), where \(P_{j}(z)=a_{jn}z^{n}+\cdots+a_{j0}\), \(Q_{j}(z)=b_{jn}z^{n}+\cdots+b_{j0}\), \(j=0,1,\ldots,k\) are polynomials with degree n and satisfy \(|a_{jn}|\geq|b_{jn}|>0\), \(j=0,1,\ldots,k\), \(B_{j}(z)\), \(D_{j}(z)\), \(R_{j}(z)\), \(j=0,1,\ldots,k\) are meromorphic functions and satisfy \(\max_{0\leq j\leq k}\{\sigma(B_{j}),\sigma(D_{j}),\sigma(R_{j})\}=\omega< n\), \(A_{j}(z)-R_{j}(z)\not\equiv0\), \(j=0,1,\ldots,k\). Let \(c_{j}\), \(j=1,\ldots,k\) be distinct nonzero complex constants. If there exists an \(i\in\{0,1,\ldots,k\}\) such that for all j (≠i), \(|a_{in}|\geq|a_{jn}|\), and
-
(i)
\(\arg a_{in}\neq\arg a_{jn}\), or \(\arg a_{in}=\arg a_{jn}\), \(|a_{in}|>|a_{jn}|\)
and
-
(ii)
\(\arg a_{in}\neq\arg b_{jn}\), or \(\arg a_{in}=\arg b_{jn}\), \(|a_{in}|>|b_{jn}|\)
hold simultaneously, then every meromorphic solution \(f(z)\) (≢0) of (1.1) satisfies \(\sigma(f)\geq n+1\). Further, if \(\varphi(z)\) (≢0) is a meromorphic function with \(\sigma(\varphi)< n+1\), then for every meromorphic solution \(f(z)\) (≢0) of (1.1) with \(\sigma_{2}(f)<1\), we have \(\lambda(f-\varphi)=\sigma(f)\geq n+1\).
Remark 1.1
In Theorem 1.1, if there exist some j (≠i) such that \(A_{j}(z)-R_{j}(z)\equiv0\), or some of \(B_{j}(z)\) (\(j\neq i\)), \(D_{j}(z)\), \(R_{j}(z)\), \(j=0,1,\ldots,k\), are equal to zero, then the corresponding result holds by using a similar proof to the one of Theorem 1.1.
Next, we consider difference operators instead of shift operators in (1.1).
For a nonzero complex constant c, the forward differences \(\Delta^{k}f(z)\), \(k\in\mathbb{N}_{+}\), are defined (see [14]) by
It is shown in [6] that
Then we can obtain the following theorem.
Theorem 1.2
Suppose that \(A_{j}(z)\), \(j=0,1,\ldots,k\), satisfy the hypotheses of Theorem 1.1, and i is also defined as in Theorem 1.1. If \(i=0\), then every meromorphic solution \(f(z)\) (≢0) of the difference equation
satisfies \(\sigma(f)\geq n+1\). Further, if \(\varphi(z)\) (≢0) is a meromorphic function with \(\sigma(\varphi)< n+1\), then for every meromorphic solution \(f(z)\) (≢0) of (1.2) with \(\sigma_{2}(f)<1\), we have \(\lambda(f-\varphi)=\sigma(f)\geq n+1\).
In the end, we can easily get the following corollary.
Corollary 1.1
Let \(a_{j}\in\mathbb{C}\), \(j=0,1,\ldots,k\), such that \(a_{0}\neq a_{j}\), \(|a_{0}|\geq|a_{j}|\geq0\), \(j=1,\ldots,k\), and \(h_{j}(z)\) (≢0), \(j=0,1,\ldots,k\), be meromorphic functions with order less than n, then every meromorphic solution \(f(z)\) (≢0) of the difference equation
satisfies \(\sigma(f)\geq n+1\). Further, if \(\varphi(z)\) (≢0) is a meromorphic function with \(\sigma(\varphi)< n+1\), then for every meromorphic solution \(f(z)\) (≢0) with \(\sigma_{2}(f)<1\), we have \(\lambda(f-\varphi)=\sigma(f)\geq n+1\).
2 Preliminary lemmas
Lemma 2.1
(see [6])
Let \(\eta_{1}\), \(\eta_{2}\) be two arbitrary complex numbers, and \(f(z)\) be a meromorphic function of finite order σ. Let \(\varepsilon>0\) be given, then there exists a set \(E\subset(0,+\infty)\) with finite logarithmic measure such that for all \(r\notin E\cup[0,1]\), we have
Remark 2.1
It follows from Lemma 2.1 that
Lemma 2.2
(see [15])
Let \(f(z)\) be a meromorphic function with \(\sigma(f)=\beta<+\infty\), then for any given \(\varepsilon>0\), there exists a set \(E\subset[0,+\infty)\) with \(m E<+\infty\) such that for all z with \(|z|=r\notin[0,1]\cup E\), \(r\rightarrow\infty\), we have
Lemma 2.3
(see [16])
Suppose that \(P(z)=(\alpha+\beta i)z^{n}+\cdots\) (α, β are real numbers such that \(|\alpha|+|\beta|\neq0\)) is a polynomial with degree n (≥1), and \(\omega(z)\) (≢0) is a meromorphic function with \(\sigma(\omega)< n\). Set \(g(z)=\omega(z)e^{P(z)}\), \(z=re^{i\theta}\), \(\delta(P,\theta)=\alpha\cos n\theta-\beta\sin n\theta\). Then for any given \(\varepsilon>0\), there exists a set \(H_{0}\subset[0,2\pi)\) with linear measure zero, such that for any \(\theta\in[0,2\pi)\backslash(H_{0}\cup H_{1})\), there exists \(r_{0}=r_{0}(\theta, \varepsilon)\) (>0) such that for \(|z|=r>r_{0}\), we have
-
(i)
if \(\delta(P,\theta)>0\), then
$$\exp\bigl\{ (1-\varepsilon)\delta(P,\theta)r^{n}\bigr\} < \bigl\vert g \bigl(re^{i\theta}\bigr)\bigr\vert <\exp\bigl\{ (1+\varepsilon)\delta(P, \theta)r^{n}\bigr\} ; $$ -
(ii)
if \(\delta(P,\theta)<0\), then
$$\exp\bigl\{ (1+\varepsilon)\delta(P,\theta)r^{n}\bigr\} < \bigl\vert g \bigl(re^{i\theta}\bigr)\bigr\vert <\exp\bigl\{ (1-\varepsilon)\delta(P, \theta)r^{n}\bigr\} , $$
where \(H_{1}=\{\theta\in[0,2\pi):\delta(P,\theta)=0\}\) is a finite set.
Remark 2.2
Let \(P(z)=az^{n}+\cdots\) be a polynomial with degree n and \(z=re^{i\theta}\), we denote \(\delta(P,\theta)=|a|\cos(\arg a+n\theta)\).
Lemma 2.4
(see [17])
Let \(f(z)\) be a nonconstant meromorphic function, \(\varepsilon>0\), and \(c\in\mathbb{C}\backslash\{0\}\). If \(\zeta=\sigma_{2}(f)<1\), then
for all r outside of a set of finite logarithmic measure.
Lemma 2.5
Let \(c_{j}\), \(j=1,\ldots,k\) be distinct nonzero complex constants, \(A_{j}(z)\), \(j=0,1,\ldots,k\), \(F(z)\) be meromorphic functions such that \(A_{k}(z)A_{0}(z)F(z)\not\equiv0\). If \(f(z)\) is a meromorphic solution of the difference equation
and satisfies \(\max\{\sigma(F),\sigma(A_{j}),j=0,1,\ldots,k\}=\omega<\sigma(f)=\sigma\) (\(0<\sigma\leq\infty\)), \(\sigma_{2}(f)<1\), then we have
Proof
We use a similar proof to the one in [12] here. We rewrite (2.1) as
By Lemma 2.4, there exists a set \(E\subset(1,+\infty)\) of finite logarithmic measure such that for all z satisfying \(|z|=r\notin E\), we have
Thus, (2.2) implies that
Set \(m_{l}E=\log\delta<\infty\). Since \(\sigma(f)=\sigma\), there exists a sequence \(\{r_{n}'\}_{n=1}^{\infty}\) tending to ∞ such that \((\delta+2)r_{n}'< r_{n+1}'\) and
We may choose \(r_{n}\in[r_{n}',(\delta+2)r_{n}']\backslash E\), \(n=1,2,\ldots\) . Since
we have
that is,
Then for sufficiently small ε (\(0<\varepsilon<\frac{\sigma-\omega}{2}\)) and sufficiently large \(r_{n}\), we have
and
Hence,
and
hold. Then, for sufficiently large \(r_{n}\), we have
It follows from (2.3) that for sufficiently large \(r_{n}\),
that is,
Therefore,
that is, \(\lambda(f)=\sigma(f)\). □
Remark 2.3
Noting that
we see that if \(f(z+c_{i})\), \(i=1,\ldots,k\), in Lemma 2.5 are replaced, respectively, by \(\Delta^{i}f\), \(i=1,\ldots,k\), the corresponding result holds.
3 Proofs of Theorems 1.1 and 1.2
Proof of Theorem 1.1
Suppose that \(f(z)\) (≢0) is a meromorphic solution of (1.1). If \(\sigma(f)=\infty\), then \(\sigma(f)\geq n+1\) holds obviously. Now, we suppose \(\sigma(f)=\sigma<\infty\). Set
It is clear that \(I_{1}\), \(I_{2}\), \(I_{3}\), \(I_{4}\) do not intersect with each other, and \(I=I_{1}\cup I_{2}\cup I_{3}\cup I_{4}=\{0,1,\ldots,k\}\backslash\{i\}\).
Now, we may choose \(\theta_{0}\in(0,2\pi)\) such that \(\cos(\arg a_{in}+n\theta_{0})=1\). (If \(n=1\) and \(\arg a_{in}=0\), then we replace \([0,2\pi)\) in Lemma 2.3 by \([-\frac{\pi}{2},\frac{3\pi}{2})\); if \(n\geq2\), this kind of \(\theta_{0}\) can always be chosen.) Set \(z=re^{i\theta}\), \(\theta\in[0,2\pi)\).
For \(j\in I_{1}\), there exists sufficiently small \(\varepsilon_{1}\) (>0) such that for all \(\theta\in(\theta_{0}-\varepsilon_{1},\theta_{0}+\varepsilon_{1})\subset(0,2\pi)\), we have
Since \(|a_{in}|\geq|a_{jn}|\geq|b_{jn}|>0\), \(j\neq i\), we see that
For \(j\in I_{2}\), there exists sufficiently small \(\varepsilon_{2}\) (>0) such that for all \(\theta\in(\theta_{0}-\varepsilon_{2},\theta_{0}+\varepsilon_{2})\subset(0,2\pi)\), we have
Since \(|a_{in}|>|b_{jn}|\), \(j\neq i\), and \(|a_{in}|\geq|a_{jn}|\), we see that
For \(j\in I_{3}\), there exists sufficiently small \(\varepsilon_{3}\) (>0) such that for all \(\theta\in(\theta_{0}-\varepsilon_{3},\theta_{0}+\varepsilon_{3})\subset(0,2\pi)\), we have
Since \(|a_{in}|>|a_{jn}|\geq|b_{jn}|\), \(j\neq i\), we see that
For \(j\in I_{4}\), there exists sufficiently small \(\varepsilon_{4}\) (>0) such that for all \(\theta\in(\theta_{0}-\varepsilon_{4},\theta_{0}+\varepsilon_{4})\subset(0,2\pi)\), we have
Since \(|a_{in}|>|a_{jn}|\geq|b_{jn}|\), we see that
Set \(\varepsilon_{0}'=\min\{\varepsilon_{1},\varepsilon_{2},\varepsilon _{3},\varepsilon_{4}\}\), then for any \(\theta\in(\theta_{0}-\varepsilon_{0}',\theta_{0}+\varepsilon_{0}')\subset (0,2\pi)\), we have
If \(|b_{in}|<|a_{in}|\) and \(\arg b_{in}=\arg a_{in}\), then for all \(\theta\in(\theta_{0}-\varepsilon_{0}',\theta_{0}+\varepsilon_{0}')\subset (0,2\pi)\), we have
If \(\arg a_{in}\neq\arg b_{in}\) and \(|b_{in}|\leq|a_{in}|\), then there exists \(\varepsilon_{0}\) (\(0<\varepsilon_{0}\leq\varepsilon_{0}'\)) such that for all \(\theta\in(\theta_{0}-\varepsilon_{0},\theta_{0}+\varepsilon_{0})\subset(0,2\pi)\), we have
and, correspondingly,
Therefore, if \(b_{in}\neq a_{in}\), and \(|b_{in}|\leq|a_{in}|\), then there exists \(\varepsilon_{0}\) (>0) such that for all \(\theta\in(\theta_{0}-\varepsilon_{0},\theta_{0}+\varepsilon_{0})\subset(0,2\pi)\), we have
If \(b_{in}=a_{in}\), then \(A_{i}(z)=\tilde{B}_{i}(z)e^{a_{in}z^{n}}+R_{i}(z)\), where \(\tilde{B}_{i}(z)\) is a meromorphic function and satisfies \(\sigma(\tilde{B}_{i})\leq\max\{\omega,n-1\}\). Since \(A_{i}(z)-R_{i}(z)\not\equiv0\), we see that \(\tilde{B}_{i}(z)\not\equiv0\). Here, \(D_{i}(z)\equiv0\), and
By Lemma 2.2, for any given ε (\(0<\varepsilon<\frac{1}{2} \min\{\frac{\delta_{1}-\delta_{2}}{\delta_{1}+\delta_{2}}, \frac{\delta_{1}-\delta_{3}}{\delta_{1}+\delta_{3}}, n-\omega\}\)), there exists a set \(E_{1}\subset(0,+\infty)\) with finite linear measure, such that for all z satisfying \(|z|=r\notin E_{1}\), \(j=0,1,\ldots,k\) and \(r\rightarrow\infty\), we have
By Lemma 2.3 and (3.1)-(3.4), for the above \(\varepsilon>0\), there exists \(H_{0}\subset[0,2\pi)\) with linear measure zero and a finite set \(H_{1}=\bigcup_{i=0}^{k}\{\theta\in[0,2\pi)| \delta(P_{i},\theta)=0\mbox{ or }\delta(Q_{i},\theta)=0\}\), such that for all \(\theta\in(\theta_{0}-\varepsilon_{0},\theta_{0}+\varepsilon_{0})\backslash (H_{0}\cup H_{1})\), there exists \(r_{0}=r_{0}(\theta,\varepsilon)>0\), such that for \(r>r_{0}\), we have
and
By Lemma 2.1, for the above \(\varepsilon>0\), there exists a set \(E_{2}\subset(1,+\infty)\) with finite logarithmic measure, such that for all z satisfying \(|z|=r\notin E_{2}\cup[0,1]\), we have
Equation (1.1) gives
and, correspondingly,
It follows from (3.5)-(3.8) that for all \(z=re^{i\theta }\), where \(\theta\in(\theta_{0}-\varepsilon_{0},\theta_{0}+\varepsilon_{0})\backslash (H_{0}\cup H_{1})\), \(r\notin[0,1]\cup E_{1}\cup E_{2}\) and \(r\rightarrow\infty\), we have
Since \(0<\varepsilon<\frac{1}{2} \min\{\frac{\delta_{1}-\delta_{2}}{\delta_{1}+\delta_{2}}, \frac{\delta_{1}-\delta_{3}}{\delta_{1}+\delta_{3}}, n-\omega\}\), we have
and
Then from (3.9), for sufficiently large r, we have
i.e.,
i.e.,
Then (3.10) implies \(n\leq\sigma-1+\varepsilon\). Since ε is arbitrary, we have \(\sigma(f)=\sigma\geq n+1\).
Therefore, every meromorphic solution \(f(z)\) (≢0) satisfies \(\sigma(f)\geq n+1\).
Set \(g(z)=f(z)-\varphi(z)\), then \(g(z)\) solves the equation
Since \(\sigma(\varphi)< n+1\), \(\varphi(z)\) (≢0) does not solve (1.1), that is,
and
Therefore, by Lemma 2.5, we have \(\lambda(g)=\sigma(g)\), i.e., \(\lambda(f-\varphi)=\sigma(f)\).
The proof of Theorem 1.1 is completed. □
Proof of Theorem 1.2
Equation (1.2) gives
By Remark 2.1, we have
By combining (3.5), (3.6), (3.12) with (3.11), for all \(z=re^{i\theta}\), where \(\arg z=\theta\in(\theta_{0}-\varepsilon_{0},\theta_{0}+\varepsilon_{0})\backslash (H_{0}\cup H_{1})\), \(|z|=r\notin[0,1]\cup E_{1}\cup E_{2}\) and for sufficiently large r, we have
By using a similar method to the one in the proof of Theorem 1.1, we have \(\sigma(f)=\sigma\geq n+1\).
Further, set \(g(z)=f(z)-\varphi(z)\), then \(g(z)\) solves the equation
Since \(\sigma(\varphi)< n+1\), \(\varphi(z)\) (≢0) does not solve (1.2), that is,
and
Therefore, by Lemma 2.5 and Remark 2.3, we have \(\lambda(g)=\sigma(g)\), i.e., \(\lambda(f-\varphi)=\sigma(f)\).
The proof of Theorem 1.2 is completed. □
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Acknowledgements
This work was supported by the National Natural Science Foundation of China (11301233, 11171119), the Natural Science Foundation of Jiangxi Province in China (20151BAB201004), and the Youth Science Foundation of Education Bureau of Jiangxi Province in China (GJJ14271).
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Wu, SZ., Zheng, XM. Growth of solutions of some kinds of linear difference equations. Adv Differ Equ 2015, 142 (2015). https://doi.org/10.1186/s13662-015-0485-8
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DOI: https://doi.org/10.1186/s13662-015-0485-8