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On zeros and growth of solutions of complex difference equations
Advances in Difference Equations volume 2021, Article number: 48 (2021)
Abstract
Let f be an entire function of finite order, let \(n\geq 1\), \(m\geq 1\), \(L(z,f)\not \equiv 0\) be a linear difference polynomial of f with small meromorphic coefficients, and \(P_{d}(z,f)\not \equiv 0\) be a difference polynomial in f of degree \(d\leq n1\) with small meromorphic coefficients. We consider the growth and zeros of \(f^{n}(z)L^{m}(z,f)+P_{d}(z,f)\). And some counterexamples are given to show that Theorem 3.1 proved by I. Laine (J. Math. Anal. Appl. 469:808–826, 2019) is not valid. In addition, we study meromorphic solutions to the difference equation of type \(f^{n}(z)+P_{d}(z,f)=p_{1}e^{\alpha _{1}z}+p_{2}e^{\alpha _{2}z}\), where \(n\geq 2\), \(P_{d}(z,f)\not \equiv 0\) is a difference polynomial in f of degree \(d\leq n2\) with small mromorphic coefficients, \(p_{i}\), \(\alpha _{i}\) (\(i=1,2\)) are nonzero constants such that \(\alpha _{1}\neq \alpha _{2}\). Our results are improvements and complements of Laine 2019, Latreuch 2017, Liu and Mao 2018.
1 Introduction and main results
In this paper, we assume familiarity with the basic results and standard notations of Nevanlinna theory [7, 9, 21]. In addition, we use \(\rho (f)\) to denote the order of growth of f and \(\lambda (f)\) to denote the exponent of convergence of zeros’ sequence of f. For simplicity, we denote by \(S(r,f)\) any quantify satisfying \(S(r,f)=o(T(r,f))\), as \(r\to \infty \), outside of a possible exceptional set of finite logarithmic measure, we use \(S(f)\) to denote the family of all small functions with respect to f.
Nowadays, there has been substantial interest in Nevanlinna theory for differences, as well as meromorphic solutions of difference and functional equations; see, e.g., [1–6, 10–14, 18, 19, 22, 23]. With the establishment of difference analogue of Nevanlinna theory, many outstanding achievements on the complex difference theory are accomplished.
Halburd and Korhonen [5] in 2006 and Chiang and Feng [3] in 2008 presented a difference analogue of the lemma on the logarithmic derivative as follows.
Lemma A
(See [3, Corollary 2.5])
Let f be a meromorphic function of finite order ρ and let η be a nonzero complex number. Then for each \(\varepsilon >0\), we have
Halburd and Korhonen [5] also established the difference analogue of Clunie lemma.
Lemma B
(See [5, Corollary 3.3])
Let f be a nonconstant finiteorder meromorphic solution of
where \(P(z,f)\) and \(Q(z,f)\) are difference polynomials in f with small meromorphic coefficients, and let \(c\in \mathbb{C}\), \(\delta <1\). If the total degree of \(Q(z,f)\), as a polynomial in f and its shifts, is ≤n, then
for all r outside of a possible exceptional set E with finite logarithmic measure \(\int _{E}\frac{dr}{r}<\infty \).
Using the same methods as in the proof of [9, Lemma 2.4.2] and Lemma B, we have a similar conclusion as follows.
Lemma C
Let f be a nonconstant finiteorder meromorphic solution of
where \(P(z,f)\) and \(Q(z,f)\) are differentialdifference polynomials in f with small meromorphic coefficients. If the total degree of \(Q(z,f)\), as a polynomial in f, its derivatives, and its shifts, is ≤n, then
for all r outside of a possible exceptional set with finite logarithmic measure.
This paper is organized as follows. In Sect. 2, we will consider the growth and zeros of certain types of complex difference polynomials and complex difference equations. In Sect. 3, we will study meromorphic solutions of certain type of nonlinear difference equations.
2 Zero distribution of some complex difference polynomials
We now recall the following result proved in [12]; see Theorem 2 therein:
Theorem D
(See [12])
Let f be a transcendental entire function of finite order, and c be a nonzero complex constant. Then for \(n\geq 2\), \(f^{n}f(z+c)\) assumes every nonzero value \(a\in \mathbb{C}\) infinitely often.
This paper prompted many related investigations during the last 12 years, such as [10, 14, 22]. In what follows, we use \(L(z,f):=\sum_{i=1}^{l}a_{i}(z)f(z+\lambda _{i})\) with small meromorphic coefficients, without being otherwise specified.
In 2019, Laine [10, Theorem 2.1] presented an extension to Theorem D as follows:
Theorem E
(See [10])
Let f be a transcendental entire function of finite order ρ, \(b_{0}\) be a nonvanishing small meromorphic function of f, \(L(z,f)\) be nonvanishing and \(n\geq 2\), \(m\geq 1\). Then \(F:=f^{n}L^{m}(z,f)b_{0}\) has sufficiently many zeros to satisfy \(\lambda (F)=\rho \).
Remark 2.1
In Theorem E, a meromorphic function α is said to be small, relative to a given meromorphic function f of finite order ρ, if for any \(\varepsilon >0\), and for some \(\lambda <\rho \), \(T(r,\alpha )=O(r^{\lambda +\varepsilon })+S(r,f)\) outside of a possible exceptional set of finite logarithmic measure.
In this section, our purpose is to improve and extend the results in [10] for an entire function f by considering the zero distribution of \(f^{n}L^{m}(z,f)+P_{d}(z,f)\), where \(n\geq 2\), \(m\geq 1\), \(P_{d}(z,f)\) is a difference polynomial in f of degree \(d\leq n2\), with small meromorphic coefficients. We obtain
Theorem 2.1
Let f be a transcendental entire function of finite order ρ, \(L(z,f)\), \(P_{d}(z,f)\) be nonvanishing and \(n\geq 2\), \(m\geq 1\), \(d\leq n2\). Then \(F:=f^{n}L^{m}(z,f)+P_{d}(z,f)\) has sufficiently many zeros and satisfies \(\lambda (F)=\rho (F)=\rho \).
Remark 2.2
The following examples show that our estimates in Theorem 2.1 are accurate, and the condition \(d\leq n2\) is necessary in Theorem 2.1.
Example 2.1
If \(f(z)=1+e^{z}\), \(P_{1}(z,f)=4f(z+\log 2)\), \(L(z,f)=f(z)+f(z+i\pi )\), then \(F:=f^{2}(z)L^{2}(z,f)+P_{1}(z,f)=4e^{2z}\) has no zeros, and \(0=\lambda (F)<\rho (F)=\rho =1\).
Example 2.2
If \(f(z)=z+e^{z}\), \(P_{0}(z,f)=z(z\log 2)^{m}\), \(L(z,f)=2f(z)f(z+\log 2)\), \(m\geq 1\), then \(F:=f(z)L^{m}(z,f)+P_{0}(z,f)=(z\log 2)^{m}e^{z}\) has finitely many zeros only, and \(0=\lambda (F)<\rho (F)=\rho =1\).
Remark 2.3
Example 2.2 shows that the following result due to Laine [10] may be invalid. Recently, I. Laine and Z. Latreuch [11] gave an extension and a complete version of [10, Theorem 3.1].
Theorem F
(See [10])
Let f be a transcendental entire function of finite order ρ, \(b_{0}\) be a nonvanishing small function of f, and \(L(z,f)\) be nonvanishing. If \(m\geq 2\), then \(F:=fL^{m}(z,f)b_{0}\) satisfies \(\lambda (F)=\rho \).
Examples 2.1 and 2.2 imply that \(F:=f^{n}L^{m}(z,f)+P_{d}(z,f)\) may have no zeros or finitely many zeros only under the condition \(d\leq n1\), where \(n\geq 2\), \(m\geq 1\). Enlightened by Examples 2.1–2.2 and Theorem F, we consider the growth and zeros of entire solutions of the following equation:
where \(n\geq 1\), \(m\geq 1\), \(P_{d}(z,f)\) is a difference polynomial in f of degree \(d\leq n1\), with small meromorphic coefficients, \(\gamma (z)\) is a small function relative to f, and \(h(z)\) is a polynomial. Now we state our result as follows.
Theorem 2.2
Let f be a transcendental entire function of Eq. (2.1) with finite order ρ, where \(L(z,f)\), \(P_{d}(z,f)\) are nonvanishing. Then \(\lambda (f)=\deg h(z)=\rho \).
Before proving Theorems 2.1 and 2.2, we need the following lemmas.
Lemma 2.1
(See [20, Theorem 1.51])
Suppose that \(f_{1}, f_{2},\ldots ,f_{n}\) (\(n\geq 2\)) are meromorphic functions and \(g_{1}, g_{2},\ldots ,g_{n}\) are entire functions satisfying the following conditions:
\((1)\) \(\sum_{j=1}^{n}f_{j}e^{g_{j}}\equiv 0\).
\((2)\) \(g_{j}g_{k}\) are not constants for \(1\leq j< k\leq n\).
\((3)\) For \(1\leq j\leq n\), \(1\leq h< k\leq n\),
where \(E\subset [1,\infty )\) has finite linear measure \(\int _{E} dr<\infty \) or finite logarithmic measure \(\int _{E}\frac{dr}{r}<\infty \). Then \(f_{j}\equiv 0\) (\(j=1,\ldots , n\)).
Lemma 2.2
Let f be a transcendental entire function of finite order ρ, \(P_{d}(z,f)\) be difference polynomial in f of degree \(d\leq n1\), \(L(z,f):=\sum_{i=1}^{l}a_{i}(z)f(z+\lambda _{i})\), with small meromorphic coefficients. Let \(L(z,f)\), \(P_{d}(z,f)\) be nonvanishing and \(n\geq 1\), \(m\geq 1\). Then \(F:=f^{n}L^{m}(z,f)+P_{d}(z,f)\) satisfies \(\rho (F)=\rho \).
Proof
Set
where I is a finite set of the index μ, \(t_{\mu }\), \(l_{\mu j}\) (\(\mu \in I\), \(j=1,\dots ,t_{\mu }\)) are natural numbers, \(\delta _{\mu j}\) (\(\mu \in I\), \(j=1,\dots ,t_{\mu }\)) are distinct complex constants. Denoting \(g_{\mu j}(z):=\frac{f(z+\delta _{\mu j})}{f(z)}\) and substituting this equality into (2.2) yields
where \(l_{\mu }=\sum_{j=1}^{t_{\mu }}l_{\mu j}\), \(d=\max_{\mu \in I}\{l_{\mu }\}\), \(c_{k}(z)=\sum_{l_{\mu }=k} (b_{\mu }(z)\prod_{j=1}^{t_{ \mu }}g_{\mu j}^{l_{\mu j}}(z) )\) (\(k=0,\dots ,d\)). By applying Lemma A, we have \(m(r,c_{k}(z))=S(r,f)\) (\(k=0,\dots ,d\)), which gives
From (2.4) and Lemma A, we obtain
namely, \(T(r,F)\leq (2n+m1)T(r,f)+S(r,f)\), and then \(\rho (F)\leq \rho \). If \(\rho (F)<\rho \), then \(T(r,F)=S(r,f)\). Rewrite \(F=f^{n}L^{m}(z,f)+P_{d}(z,f)\) as
By applying Lemma B, we see that
Note that \(L(z,f)\not \equiv 0\), which implies
a contradiction. Thus \(\rho (F)=\rho \). □
Proof of Theorem 2.1
Suppose that \(\lambda (F)=\lambda <\rho \). By the Hadamard representation, we assume that
where \(h(z)\) is a polynomial of degree ≤ρ and \(T(r,\gamma (z))=O(r^{\lambda +\varepsilon })+S(r,f)\), \(\gamma (z)\not \equiv 0\), or else, by using the same reasoning as in the proof of Lemma 2.1, we obtain \(T(r,f)=S(r,f)\), a contradiction. If \(\deg h(z)\leq \mu <\rho \), then
resulting in a contradiction \(\rho \leq \max \{\lambda ,\mu \}<\rho \) by Lemma 2.2. Thus \(\deg h(z)=\rho \). Denote \(L(z,f):=L\) and \(P_{d}(z,f)=P_{d}\). Differentiating (2.5) yields
Eliminating \(e^{h}\) from (2.5) and (2.6), we have
where
and
while \(Q_{d}(z,f)\) is a differentialdifference polynomial in f of degree \(\leq n2\). By applying Lemma C, we get
If \(\psi \not \equiv 0\), we conclude that
which is impossible. If \(\psi \equiv 0\), then \(Q_{d}(z,f)\equiv 0\). Thus \(Q_{d}(z,f)= (\frac{\gamma '}{\gamma }+h' )P_{d}P'_{d} \equiv 0\), and we get \(P_{d}=C\gamma (z)e^{h(z)}\), where \(C\in \mathbb{C}\setminus \{0\}\). If \(C=1\), then \(P_{d}=\gamma (z)e^{h(z)}\). Substituting this equality into (2.5) yields \(f^{n}L^{m}\equiv 0\), a contradiction. If \(C\neq 1\), then \(\gamma (z)e^{h(z)}=\frac{1}{C}P_{d}\). Putting this equality into (2.5), we have
Recalling that \(d\leq n2\) and by applying Lemma B, we have
Making using of the above two equalities and noting that \(L(z,f)\not \equiv 0\), we get
resulting in a contradiction \(T(r,f)=S(r,f)\).
This completes the proof of Theorem 2.1. □
Proof of Theorem 2.2
Let \(F:=f^{n}L^{m}(z,f)+P_{d}(z,f)\). It follows from Lemma 2.2 that \(\rho (F)=\rho \). On the other hand, f is a transcendental entire function to Eq. (2.1). So as in the beginning of the proof of Theorem 2.1, we obtain \(\deg h(z)=\rho \). We now deduce that \(\lambda (f)=\rho \). On the contrary, suppose that \(\lambda (f)<\rho \). By the Hadamard factorization theorem [20, Theorem 2.5], we assume that
where \(P(z)\) is the canonical product of f formed with the zeros of f, and satisfies \(\lambda (P)=\rho (P)<\rho \), \(Q(z)\) is a polynomial of degree ρ. Substituting (2.8) into \(P_{d}(z,f)\) yields
where \(l_{\mu }=\sum_{j=1}^{t_{\mu }}l_{\mu j}\), \(d=\max_{\mu \in I}\{l_{\mu }\}\), \(c_{k}(z)=\sum_{l_{\mu }=k} (b_{\mu }(z)\prod_{j=1}^{t_{ \mu }} (P(z+\delta _{\mu j})e^{Q(z+\delta _{\mu j})Q(z)})^{l_{\mu j}} )\) (\(k=0,\dots ,d\)). Noting that \(T(r,P(z+\delta _{\mu j}))=S(r,f)\), \(\deg (Q(z+\delta _{\mu j})Q(z))=\deg Q(z)1\), we see that \(T(r,c_{k}(z))=S(r,f) (k=0,\dots ,d)\). Substituting (2.8) into \(L^{m}(z,f)\), we have
Recalling that \(T(r,P(z+\lambda _{i}))=S(r,f)\), \(\deg (Q(z+\lambda _{i})Q(z))=\deg Q(z)1\), we see that \(T(r,C(z))=S(r,f)\). Since \(L(z,f)\not \equiv 0\), one has \(C(z)\not \equiv 0\). Rewrite (2.1) as
Noting that \(\deg h(z)=\rho =\deg Q(z)\), we consider two cases below:
\((i)\) If \(\deg (h(z)jQ(z))=\rho \), \(j=1,2,\dots ,d(\leq n1),n+m\), by applying Lemma 2.1, we obtain
then \(L(z,f)\equiv 0\) or \(f(z)\equiv 0\), which is impossible.
\((ii)\) If \(\deg (h(z)jQ(z))<\rho \), \(j=1,2,\dots ,d(\leq n1),n+m\). We consider two subcases below:
Subcase \((ii1)\). If \(\deg (h(z)(m+n)Q(z))<\rho \), then \(\deg (h(z)jQ(z))=\rho \), \(j=1,2,\dots ,d(\leq n1)\), so rewriting (2.9) as
and by applying Lemma 2.1, we obtain
so then \(P_{d}(z,f)=\sum_{k=0}^{d}c_{k}(z)e^{kQ(z)}\equiv 0\), which is impossible.
Subcase \((ii2)\). If there exists a \(j_{0}, j_{0}\in \{1,2,\dots ,d(\leq n1)\}\) such that \(\deg (h(z)jQ(z))<\rho \), then \(\deg (h(z)(m+n)Q(z))=\rho \), so, by Lemma 2.1 and (2.9), we get \(C(z)P^{n}(z)\equiv 0\), and then \(L(z,f)\equiv 0\) or \(f(z)\equiv 0\), a contradiction.
This completes the proof of Theorem 2.2. □
3 Meromorphic solutions of certain difference equations
Recently, there has been a renewed interest in the existence of entire or meromorphic solutions for nonlinear differential or difference equations, or differentialdifference equations in the complex plane, see [1, 13, 15–19, 23].
In 2011, Li proved the following result, see [16], Theorem 2.
Theorem G
(See [16])
Let \(n\geq 2\) be an integer, \(Q_{d}(z,f)\) be a differential polynomial in f of degree \(d\leq n2\), and \(p_{1}\), \(p_{2}\), \(\alpha _{1}\), \(\alpha _{2}\) be nonzero constants and \(\alpha _{1}\neq \alpha _{2}\). If f is a transcendental meromorphic solution of the following equation:
and satisfying \(N(r,f)=S(r,f)\), then one of the following holds:
 \((i)\):

\(f(z)=c_{0}+c_{1}e^{\alpha _{1} z/n}\);
 \((ii)\):

\(f(z)=c_{0}+c_{2}e^{\alpha _{2} z/n}\);
 \((iii)\):

\(f(z)=c_{1}e^{\alpha _{1} z/n}+c_{2}e^{\alpha _{2} z/n}\) and \(\alpha _{1}+\alpha _{2}=0\),
where \(c_{0}\) is a small function of f and \(c_{1}\), \(c_{2}\) are constants satisfying \(c_{i}^{2}=p_{i}\), \(i=1,2\).
Replacing the differential polynomial \(Q_{d}(z,f)\) in Eq. (3.1) by a difference, or differentialdifference polynomial \(P_{d}(z,f)\), many scholars considered the existence of solutions of the following equation:
where \(n\geq 2\), \(P_{d}(z,f)\) is a difference or differentialdifference polynomial in f of degree \(d\leq n1\), with small meromorphic coefficients.
In 2018, Liu and Mao [18] investigated the entire solutions of finite order of difference Eq. (3.2) and obtained the following result corresponding to Theorem G.
Theorem H
(See [18])
Let \(n\geq 2\) be an integer, \(P_{d}(z,f)\) be a difference polynomial in f of degree \(d\leq n2\) such that \(P_{d}(z,0)\not \equiv 0\), \(p_{1}\), \(p_{2}\) be nonzero small functions of \(e^{z}\), and let \(\alpha _{1}\), \(\alpha _{2}\) be nonzero constants. If \(\frac{\alpha _{1}}{\alpha _{2}}<0\), and Eq. (3.2) has an entire solution f of finite order, then \(\alpha _{1}+\alpha _{2}=0\) and \(f(z)=\gamma _{1}e^{\alpha _{1} z/n}+\gamma _{2}e^{\alpha _{2} z/n}\), where \(\gamma _{1}\), \(\gamma _{2}\) are constants satisfying \(\gamma _{i}^{2}=p_{i}\), \(i=1,2\).
It is natural to ask whether the conditions \(P_{d}(z,0)\not \equiv 0\) and \(\frac{\alpha _{1}}{\alpha _{2}}<0\) in Theorem H can be omitted or not. In this section, we give an affirmative answer to this question by proving the following result.
Theorem 3.1
Let \(n\geq 2\) be an integer, \(P_{d}(z,f)\) be a difference polynomial in f of degree \(d\leq n2\), and \(p_{1}\), \(p_{2}\), \(\alpha _{1}\), \(\alpha _{2}\) be nonzero constants and \(\alpha _{1}\neq \alpha _{2}\). If f is a transcendental meromorphic solution of Eq. (3.2), and satisfying \(N(r,f)=S(r,f)\), then \(\rho (f)=1\), and one of the following holds:
 \((i)\):

\(f(z)=\gamma _{0}+\gamma _{1}e^{\alpha _{1} z/n}\);
 \((ii)\):

\(f(z)=\gamma _{0}+\gamma _{2}e^{\alpha _{2} z/n}\);
 \((iii)\):

\(f(z)=\gamma _{1}e^{\alpha _{1} z/n}+\gamma _{2}e^{\alpha _{2} z/n}\) and \(\alpha _{1}+\alpha _{2}=0\);
where \(\gamma _{0}\), \(\gamma _{1}\), \(\gamma _{2}\) are constants satisfying \(\gamma _{i}^{2}=p_{i}\), \(i=1,2\).
Remark 3.1
The following examples show that the condition \(P_{d}(z,0)\not \equiv 0\) is not necessary in Theorem 3.1.
Example 3.1
The difference equation
has an entire solution \(f(z)=e^{z}\), where \(P_{1}(z,f)=f(z+\log 2)f(z)\) and \(P_{1}(z,0)=0\).
Example 3.2
The difference equation
has an entire solution \(f(z)=e^{z}+e^{z}\), where \(P_{1}(z,f)=3f(z)\) and \(P_{1}(z,0)=0\).
The following lemmas will be used in the proof of Theorem 3.1.
Lemma 3.1
(See [16, Lemma 6])
Suppose that f is a transcendental meromorphic function, a, b, c, d are small functions with respect to f and \(acd\not \equiv 0\). If
then
In particular, if a, b, c, d are constants and \(b^{2}4ac\neq 0\), then \(b=0\) and
where \(\gamma _{1}\), \(\gamma _{2}\) and λ are nonzero constants.
Remark 3.2
The condition \(acd\not \equiv 0\) in Lemma 3.1 is not necessary and it can be replaced with \(cd\not \equiv 0\).
Lemma 3.2
(See [8, p. 247])
Suppose that \(f(z)\) is a transcendental meromorphic function and \(K>1\). Then there exists a set \(M(K)\) of upper logarithmic density at most \(\delta (K)=\min \{(2e^{K1}1)^{1},(1+e(K1))\exp (e(1K))\}\) such that for every positive integer k, we have
Remark 3.3
By Lemma 3.2, we see that if f is a transcendental meromorphic function, and if φ satisfies \(T(r,\varphi ^{(k)})=S_{1}(r,f)\), then \(T(r,\varphi )=S_{1}(r,f)\), where \(S_{1}(r,f)\) is defined to be any quantity such that for any positive number ε there exists a set \(E(\varepsilon )\) whose upper logarithmic density is less than ε, and \(S_{1}(r,f)=o(T(r,f))\) as \(r\to \infty \), \(r\notin E(\varepsilon )\).
The proof given below for Theorem 3.1 is different from the proof previously given for the preceding result in [18].
Proof of Theorem 3.1
Clearly, \(\rho (p_{1}e^{\alpha _{1}z}+p_{2}e^{\alpha _{2}z})=1\), where \(\alpha _{1}\neq \alpha _{2}\). Similar as in the beginning of the proof of Lemma 2.2, we have
Noting that \(N(r,f)=S(r,f)\), by (3.2) and the above inequality, we have
and
From the above two inequalities, we derive
thus \(\rho (f)=1\). Denote \(P_{d}:=P_{d}(z,f)\). Suppose that f is a transcendental meromorphic solution of Eq. (3.2) which satisfies \(N(r,f)=S(r,f)\). By differentiating (3.2), we have
Eliminating \(e^{\alpha _{1}z}\) and \(e^{\alpha _{2}z}\) from (3.2) and (3.3), respectively, we have
Differentiating (3.5) yields
Eliminating \(e^{\alpha _{1}z}\) from (3.5) and (3.6), we get
where
and
while \(Q_{d}(z,f)\) is a differentialdifference polynomial in f of degree \(\leq n2\). By (3.7) and Lemma C, we have \(m(r,\varphi )=S(r,f)\). Note that \(N(r,f)=S(r,f)\), thus \(T(r,\varphi )=S(r,f)\). We distinguish two cases below:
Case 1. If \(\varphi \equiv 0\), then \(Q_{d}(z,f)\equiv 0\). Since \(\alpha _{1}\neq \alpha _{2}\), we see that \(\alpha _{1}P_{d}P_{d}'\equiv 0\) and \(\alpha _{2}P_{d}P_{d}'\equiv 0\) cannot hold simultaneously. Suppose that \(\alpha _{2}P_{d}P_{d}'\not \equiv 0\). By (3.9), we have
where A is a nonzero constant. Substituting (3.10) into (3.5), we have
Since the righthand side of (3.11) is a differentialdifference polynomial in f of degree \(\leq n2\), by Lemma C, we have \(m(r,\alpha _{2}fnf')=S(r,f)\) and \(m(r,f(\alpha _{2}fnf'))=S(r,f)\), and then \(\alpha _{2}fnf'\equiv 0\). Otherwise, \(\alpha _{2}fnf'\not \equiv 0\), thus we have
resulting in a contradiction \(T(r,f)\leq S(r,f)\). Since \(\alpha _{2}fnf'\equiv 0\), one gets \(f^{n}=\widetilde{p}_{2}e^{\alpha _{2}z}\), \(\widetilde{p}_{2}\in \mathbb{C}\setminus \{0\}\). Substituting this equality and (3.10) into (3.2) yields
If \(p_{2}\neq \widetilde{p}_{2}\), by Lemma C, we have \(T(r,f)=m(r,f)+N(r,f)=S(r,f)\), a contradiction. Therefore \(p_{2}=\widetilde{p}_{2}\), \(f(z)=\gamma _{2}e^{\frac{\alpha _{2}}{n}z}\), \(\gamma _{2}^{n}=p_{2}\). Similarly, if \(\alpha _{1}P_{d}P_{d}'\not \equiv 0\), then we obtain \(f(z)=\gamma _{1}e^{\frac{\alpha _{1}}{n}z}\), \(\gamma _{1}^{n}=p_{1}\).
Case 2. If \(\varphi \not \equiv 0\), by applying \(T(r,\varphi )=S(r,f)\) and the lemma on the logarithmic derivative, we have
By (3.8), if \(z_{0}\) is a multiple zero of f, then \(z_{0}\) must be a zero of φ. Hence \(N_{(2} (r,\frac{1}{f} )=S(r,f)\), where \(N_{(2} (r,\frac{1}{f} )\) denotes the counting function of multiple zeros of f, which implies
where \(N_{1)} (r,\frac{1}{f} )\) denotes the counting function of simple zeros of f, and we deduce that f has infinitely many simple zeros. Differentiating (3.8) gives
If \(z_{0}\) is a simple zero of f, it follows from (3.8) and (3.14) that \(z_{0}\) is a zero of \((2n1)\varphi f''[(n1)\varphi '+(\alpha _{1}+\alpha _{2})\varphi ]f'\). Define
then we have \(T(r,\alpha )=S(r,f)\). It follows that
Substituting (3.16) into (3.8) yields
where \(a=\alpha _{1}\alpha _{2}+\frac{n\alpha }{(2n1)\varphi }\), \(b=\frac{n(n1)}{2n1} [\frac{\varphi '}{\varphi }2(\alpha _{1}+ \alpha _{2}) ]\), \(c=n(n1)\). By Lemma 3.1, we have
We consider two subcases as follows.
Subcase 2.1. Suppose that \(4acb^{2}\not \equiv 0\). It follows from (3.18) that
By integration, we see that there exists a \(C\in \mathbb{C}\setminus \{0\}\) such that
which implies \(e^{2(\alpha _{1}+\alpha _{2})z}\in S(f)\), then \(\alpha _{1}+\alpha _{2}=0\). It follows from (3.4) and (3.5) that
where \(\psi =\alpha _{1}\alpha _{2}f^{2}+n^{2}(f')^{2}\), \(R(z,f)\) is a differentialdifference polynomial of degree \(\leq 2n2\). By applying Lemma C, we have \(m(r,\psi )=S(r,f)\), then \(T(r,\psi )=S(r,f)\). We deduce that \(\psi \not \equiv 0\). Otherwise, we assume that \(\psi =\alpha _{1}\alpha _{2}f^{2}+n^{2}(f')^{2}\equiv 0\). Then \((\frac{f'}{f} )^{2}\equiv  \frac{\alpha _{1}\alpha _{2}}{n^{2}}\), which implies that
combining with (3.13), we have \(T(r,f)=S(r,f)\), a contradiction. By Lemma 3.1 and (3.2), we see that ψ must be constant and \(f(z)=\gamma _{1}e^{\alpha _{1} z/n}+\gamma _{2}e^{\alpha _{2} z/n}\), \(\alpha _{1}+\alpha _{2}=0\), where \(\gamma _{1}\), \(\gamma _{2}\) are constants satisfying \(\gamma _{i}^{2}=p_{i}\), \(i=1,2\).
Subcase 2.2. Suppose that \(4acb^{2}\equiv 0\). Differentiating (3.17) yields
Suppose \(z_{0}\) is a simple zero of f which is not a zero of a, b. It follows from (3.17) and (3.22) that \(z_{0}\) is a zero of \(2\varphi f'' (\varphi '\frac{b}{c}\varphi )f'\). Denote
then we have \(T(r,\beta )=S(r,f)\). It follows that
Substituting (3.24) into (3.22) yields
where \(d=a'+\frac{b\beta }{2\varphi }\), \(h=2a+b'+\frac{b}{2}\frac{\varphi '}{\varphi }\frac{b^{2}}{2c}+ \frac{c\beta }{\varphi }\). Eliminating \((f')^{2}\) from (3.17) and (3.25), we have
where
Note that A and B are small functions of f. If \(z_{0}\) is a simple zero of f and not a zero of A, B, it follows from (3.26) that \(A=B\equiv 0\). By (3.24), we have
where \(b=\frac{n(n1)}{2n1} [\frac{\varphi '}{\varphi }2(\alpha _{1}+ \alpha _{2}) ]\not \equiv 0\). Otherwise, \(b\equiv 0\), by \(4acb^{2}\equiv 0\), and then \(a\equiv 0\). It follows from (3.17) that \(c(f')^{2}\equiv \varphi \), then \(T(r,f')=S(r,f)\). By Lemma 3.2 and Remark 3.3, we obtain that \(T(r,f)=S_{1}(r,f)\), a contradiction. Substituting \(4acb^{2}\equiv 0\) into (3.27) yields
It follows from (3.16) and (3.28) that
If \(\varphi '\equiv 0\), then \(\frac{\alpha }{\varphi }\equiv 0\). Substituting this identity into \(4acb^{2}\equiv 0\) yields
which implies that \(\frac{\alpha _{1}}{\alpha _{2}}=\frac{n1}{n}\) or \(\frac{\alpha _{1}}{\alpha _{2}}=\frac{n}{n1}\). By substituting \(\varphi '\equiv 0\) into (3.28), we obtain
then
where \(C_{1},C_{2}\in \mathbb{C}\setminus \{0\}\). Otherwise, one of \(C_{1}\) and \(C_{2}\) is equal to zero, then \(N(r,1/f)=S(r,f)\). It follows from (3.13) that \(T(r,f)=S(r,f)\), a contradiction. As \(\frac{\alpha _{1}}{\alpha _{2}}=\frac{n1}{n}\), then
As \(\frac{\alpha _{1}}{\alpha _{2}}=\frac{n}{n1}\), then
If \(\varphi '\not \equiv 0\), differentiating (3.29) gives
It follows from \(4acb^{2}\equiv 0\) that \(bb'=2ca'\), namely,
Denoting \(\gamma :=\frac{\varphi '}{\varphi }\) and combining (3.30) with (3.31) yields
If \(\gamma '\equiv 0\), then \(\varphi =C_{3}e^{C_{4}z}\), \(C_{3},C_{4}\in \mathbb{C}\). It follows from \(\varphi '\not \equiv 0\) that \(C_{3}C_{4}\neq 0\), which implies that \(\varphi \notin S(f)\), a contradiction. If \(\gamma '\not \equiv 0\), it follows from (3.32) that
which implies that \(e^{(\alpha _{1}+\alpha _{2})z}\in S(f)\), and then \(\alpha _{1}+\alpha _{2}=0\). Using similar reasoning as in Subcase 2.1, we obtain \(f(z)=\gamma _{1}e^{\alpha _{1} z/n}+\gamma _{2}e^{\alpha _{2} z/n}\), \(\alpha _{1}+\alpha _{2}=0\), where \(\gamma _{1}\), \(\gamma _{2}\) are constants satisfying \(\gamma _{i}^{2}=p_{i}\), \(i=1,2\).
This completes the proof of Theorem 3.1. □
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The authors would like to thank the referee for his/her reading of the original version of the manuscript with valuable suggestions and comments.
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The first author was supported by the NNSF of China (Nos: 12001117, 12001503), and by the National Natural Science Foundation of Guangdong Province (Nos: 2018A030313267, 2018A030313508). The second author was supported by the NNSF of China (No: 11701524).
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Chen, MF., Cui, N. On zeros and growth of solutions of complex difference equations. Adv Differ Equ 2021, 48 (2021). https://doi.org/10.1186/s1366202003211w
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DOI: https://doi.org/10.1186/s1366202003211w