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On some complex differential and difference equations concerning sharing function
Advances in Difference Equations volume 2014, Article number: 274 (2014)
Abstract
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 $L(f)={a}_{k}(z){f}^{(k)}+\cdots +{a}_{0}(z)f$ sharing a function $\alpha (z)$ and a constant β. We also study the problem on entire function and its difference polynomials sharing a function.
MSC:39A50, 30D35.
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 [1], Yang [2] and Yi and Yang [3]). We denote by $S(r,f)$ any quantity satisfying $S(r,f)=o(T(r,f))$, as $r\to +\mathrm{\infty}$, possibly outside of a set with finite measure. A meromorphic function $a(z)$ is called a small function with respect to f if $T(r,a)=S(r,f)$. In addition, we will use the notation $\sigma (f)$ to denote the order of meromorphic function $f(z)$, and $\tau (f)$ to denote the type of an entire function $f(z)$ with $0<\sigma (f)=\sigma <+\mathrm{\infty}$, which are defined to be (see [1])
We use ${\sigma}_{2}(f)$ to denote the hyperorder of $f(z)$, ${\sigma}_{2}(f)$ is defined to be (see [3])
In 1976, Rubel and Yang [4] proved the following result.
Theorem 1.1 [4]
Let f be a nonconstant entire function. If f and ${f}^{\prime}$ share two finite distinct values CM, then $f\equiv {f}^{\prime}$.
In 1996, Brück [5] gave the following conjecture.
Conjecture 1.1 [5]
Let f be a nonconstant entire function. Suppose that $\sigma (f)$ is not a positive integer or infinite, if f and ${f}^{\prime}$ share one finite value a CM, then
for some nonzero constant c.
In 1998, Gundersen and Yang [6] proved that Brück’s conjecture holds for entire functions of finite order and obtained the following result.
Theorem 1.2 [[6], Theorem 1]
Let f be a nonconstant entire function of finite order. If f and ${f}^{\prime}$ share one finite value a CM, then $\frac{{f}^{\prime}a}{fa}=c$ for some nonzero constant c.
The shared value problems relative to a meromorphic function f and its derivative ${f}^{(k)}$ 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 [13] 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 [[13], Theorem 1]
Let f be an entire function of finite order and $a(z)$ be a function such that $\sigma (a)<\sigma (f)<\mathrm{\infty}$. If f and ${f}^{\prime}$ share $a(z)$ CM, then $\frac{{f}^{\prime}a}{fa}=c$ for some nonzero constant c.
Thus, there are natural questions to ask:

(i)
What would happen when $\sigma (a)<\sigma (f)<\mathrm{\infty}$ is replaced by $0<\sigma (a)=\sigma (f)<\mathrm{\infty}$ in Theorem 1.3?

(ii)
For Theorems 1.11.3, what would happen when ${f}^{\prime}$ is replaced by differential polynomial
$$L(f)={a}_{k}(z){f}^{(k)}+{a}_{k1}(z){f}^{(k1)}+\cdots +{a}_{1}(z){f}^{\prime}+{a}_{0}(z)f,$$(1)
where ${a}_{k}(z)\phantom{\rule{0.25em}{0ex}}(\not\equiv 0),\dots ,{a}_{0}(z)$ are polynomials?
The main purpose of this article is to study the above questions and obtain the following theorems.
Theorem 1.4 Let $f(z)$ and $\alpha (z)$ be two nonconstant entire functions and satisfy $0<\sigma (\alpha )=\sigma (f)<\mathrm{\infty}$ and $\tau (f)>\tau (\alpha )$, and $L(f)$ be stated as in (1) such that
If $f(z)$ and $L(f(z))$ share $\alpha (z)$ CM, then
for some nonzero constant c.
Theorem 1.5 Let $f(z)$ be a nonconstant transcendental entire function with ${\sigma}_{2}(f)<\mathrm{\infty}$, let ${\sigma}_{2}(f)$ be not an integer, and let $L(f)$ be stated as in (1). If f and $L(f)$ share a nonzero constant a CM and $\delta (0,f)>0$, then
for some nonzero constant c.
Recently, some papers have studied Brück’s conjecture related to difference of entire function (including [14, 15]). In 2009, Heittokangas et al. [14] got the following result.
Theorem 1.6 [[14], Theorem 1]
Let f be a nonconstant meromorphic function of finite order $\sigma (f)<2$, and let η be a nonzero complex number. If $f(z+\eta )$ and $f(z)$ share a finite complex value a CM, then $f(z+\eta )a=c(f(z)a)$ for all $z\in \mathbb{C}$, 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 $f(z)$ be a nonconstant entire function of finite order $0<\sigma (f)<\mathrm{\infty}$, ${L}_{1}(f)$ be difference polynomial of f of the form
where ${\eta}_{k},{\eta}_{k1},\dots ,{\eta}_{1}$ are nonzero complex numbers. If ${L}_{1}(f(z))=cf(z)$ and ξ (≠0) is a Borel exceptional value of $f(z)$, then ${L}_{1}(f)=kf(z)$.
2 Some lemmas
To prove our theorems, we will require some lemmas as follows.
Lemma 2.1 [16]
Let $f(z)$ be a transcendental entire function, $\nu (r,f)$ be the central index of $f(z)$. Then there exists a set $E\subset (1,+\mathrm{\infty})$ with finite logarithmic measure, we choose z satisfying $z=r\notin [0,1]\cup E$ and $f(z)=M(r,f)$, we get
Lemma 2.2 [17]
Let $f(z)$ be an entire function of finite order $\sigma (f)=\sigma <\mathrm{\infty}$, and let $\nu (r,f)$ be the central index of f. Then, for any ε (>0), we have
Lemma 2.3 [18]
Let f be a transcendental entire function, and let $E\subset [1,+\mathrm{\infty})$ be a set having finite logarithmic measure. Then there exists $\{{z}_{n}={r}_{n}{e}^{i{\theta}_{n}}\}$ such that $f({z}_{n})=M({r}_{n},f)$, ${\theta}_{n}\in [0,2\pi )$, ${lim}_{n\to \mathrm{\infty}}{\theta}_{n}={\theta}_{0}\in [0,2\pi )$, ${r}_{n}\notin E$ and if $0<\sigma (f)<\mathrm{\infty}$, then, for any given $\epsilon >0$ and sufficiently large ${r}_{n}$,
Lemma 2.4 [16]
Let $P(z)={b}_{n}{z}^{n}+{b}_{n1}{z}^{n1}+\cdots +{b}_{0}$ with ${b}_{n}\ne 0$ be a polynomial. Then, for every $\epsilon >0$, there exists ${r}_{0}>0$ such that for all $r=z>{r}_{0}$ the inequalities
hold.
Lemma 2.5 Let $f(z)$ and $A(z)$ be two entire functions with $0<\sigma (f)=\sigma (A)=\sigma <\mathrm{\infty}$, $0<\tau (A)<\tau (f)<\mathrm{\infty}$, then there exists a set $E\subset [1,+\mathrm{\infty})$ that has infinite logarithmic measure such that for all $r\in E$ and a positive number $\kappa >0$, we have
Proof By definition, there exists an increasing sequence $\{{r}_{m}\}\to \mathrm{\infty}$ satisfying $(1+\frac{1}{m}){r}_{m}<{r}_{m+1}$ and
For any given β ($\tau (A)<\beta <\tau (f)$), there exists some positive integer ${m}_{0}$ such that for all $m\ge {m}_{0}$ and for any given ε ($0<\epsilon <\tau (f)\beta $), we have
Thus, there exists some positive integer ${m}_{1}$ such that for all $m\ge {m}_{1}$, we have
From (2)(4), for all $m\ge {m}_{2}=max\{{m}_{0},{m}_{1}\}$ and for any $r\in [{r}_{m},(1+\frac{1}{m}){r}_{m}]$, we have
Set $E={\bigcup}_{m={m}_{2}}^{\mathrm{\infty}}[{r}_{m},(1+\frac{1}{m}){r}_{m}]$, then
From the definition of type of entire function, for any sufficiently small $\epsilon >0$, we have
By (5) and (6), set $\kappa =\beta \tau (A)\epsilon $, for all $r\in E$, we have
Thus, this completes the proof of this lemma. □
Lemma 2.6 [[19], Theorem 2.1]
Let $f(z)$ be a meromorphic function of finite order σ, and let η be a fixed nonzero complex number, then, for each $\epsilon >0$, we have
Lemma 2.7 [[19], Corollary 2.5]
Let $f(z)$ be a meromorphic function with order $\sigma =\sigma (f)$, $\sigma <+\mathrm{\infty}$, and let η be a fixed nonzero complex number, then, for each $\epsilon >0$, we have
Let $g:(0,+\mathrm{\infty})\to R$, $h:(0,+\mathrm{\infty})\to R$ be monotone increasing functions such that $g(r)\le h(r)$ outside of an exceptional set E with finite linear measure, or $g(r)\le h(r)$, $r\notin H\cup (0,1]$, where $H\subset (1,\mathrm{\infty})$ is a set of finite logarithmic measure. Then, for any $\alpha >1$, there exists ${r}_{0}$ such that $g(r)\le h(\alpha r)$ for all $r\ge {r}_{0}$.
3 The proof of Theorem 1.4
Since $f(z)$ is an entire function, and $f(z)$ and $L(f(z))$ share $\alpha (z)$ CM, then there is an entire function $\gamma (z)$ such that
Next, we will claim that $\gamma (z)$ is a constant.
Suppose that $\gamma (z)$ is transcendental. It follows that $\sigma ({e}^{\gamma (z)})=\mathrm{\infty}$. However, since $0<\sigma (f)=\sigma (\alpha )<\mathrm{\infty}$, it follows from the lefthand side of (7) that $\sigma (\frac{L(f(z))\alpha (z)}{f(z)\alpha (z)})<\mathrm{\infty}$, a contradiction. Thus, $\gamma (z)$ is not transcendental.
Suppose that $\gamma (z)$ is a nonconstant polynomial, let
where ${b}_{m},\dots ,{b}_{0}$ are constants and ${b}_{m}\ne 0$, $m\ge 1$. Thus, it follows from (7) and Lemma 2.4 that
Since $L(f)={a}_{k}{f}^{k}+{a}_{k1}{f}^{(k1)}+\cdots +{a}_{0}f$, from Lemma 2.1, then there exists a subset ${E}_{1}\subset (1,+\mathrm{\infty})$ with finite logarithmic measure such that for some point $z=r{e}^{i\theta}$ ($\theta \in [0,2\pi )$), $r\notin {E}_{1}$ and $M(r,f)=f(z)$, we have
Thus, it follows that
From Lemma 2.3, there exists $\{{z}_{n}={r}_{n}{e}^{i{\theta}_{n}}\}$ such that $f({z}_{n})=M({r}_{n},f)$, ${\theta}_{n}\in [0,2\pi )$, ${lim}_{n\to \mathrm{\infty}}{\theta}_{n}={\theta}_{0}\in [0,2\pi )$, ${r}_{n}\notin {E}_{1}$, then, for any given ε satisfying
where ${d}_{kj}={deg}_{z}{a}_{kj}{deg}_{z}{a}_{k}$, and sufficiently large ${r}_{n}$, we have
Since ${a}_{j}(z)$, $j=0,1,\dots ,k$, are polynomials, let ${a}_{j}(z)={\sum}_{t=0}^{{s}_{j}}{l}_{jt}{z}^{t}$, where ${s}_{j}={deg}_{z}{a}_{j}$, $j=0,1,\dots ,k$. Then, from Lemma 2.4 and (10), we have
where ${d}_{kj}={s}_{kj}{s}_{k}$ and M is a positive constant. Since $j\sigma (f)+{d}_{kj}+j+(kj)\epsilon <2k\epsilon <0$, it follows from (11) that
Since $0<\sigma (\alpha )=\sigma (f)<\mathrm{\infty}$ and $\tau (\alpha )<\tau (f)<\mathrm{\infty}$, from Lemma 2.5, there exists a set $E\subset [1,+\mathrm{\infty})$ that has infinite logarithmic measure such that for a sequence ${\{{r}_{n}\}}_{1}^{\mathrm{\infty}}\in {E}_{2}=E{E}_{1}$, we have
From (8), (9), (12), (13) and Lemma 2.2, we can get that
which is impossible. Thus, $\gamma (z)$ is not a polynomial.
Therefore, $\gamma (z)$ 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 $L(f)$ and f share the constant a CM, then there exists an entire function $\phi (z)$ such that
We will consider two cases as follows.
Case 1. If $a=0$, it follows from (15) that
Since $L(f(z))={a}_{k}(z){f}^{(k)}(z)+\cdots +{a}_{1}(z){f}^{\prime}(z)+{a}_{0}(z)$ and ${a}_{j}(z)$, $j=0,1,\dots ,k$, are polynomials, it follows from (16) that
outside of an exceptional set ${E}_{3}$ with finite linear measure. Thus, there exists a constant K such that
By Lemma 2.8, there exists an ${r}_{0}>0$, and for all $r\ge {r}_{0}$, we have
Thus, we can deduce from (17) that $\sigma ({e}^{\phi})\le {\sigma}_{2}(f)<\mathrm{\infty}$, that is, $\phi (z)$ is a polynomial.
By using the same argument as in [[21], Theorem 1.1], we can get that ${\sigma}_{2}(f)={deg}_{z}\phi $, which is a contradiction to ${\sigma}_{2}(f)$ is not a positive integer. Thus, $\phi (z)$ is only a constant, it follows from (15) that $L(f(z))=cf(z)$, where c is a nonzero constant.
Case 2. If $a\ne 0$, from the derivation of (15) and eliminating ${e}^{\phi}$, we can get
If ${\phi}^{\prime}(z)\equiv 0$, that is, $\phi (z)\equiv c$, c is a constant. Thus, we can prove the conclusion of Theorem 1.5 easily.
If ${\phi}^{\prime}(z)\not\equiv 0$, then it follows from (18) that
We can rewrite (18) in the following form:
Since ${\phi}^{\prime}\not\equiv 0$ and f is transcendental, set
then we have $m(r,\mathrm{\Psi})=S(r,f)$. Thus, it follows from (20) and (21) that
Since $\phi (z)$ 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 $f(z)$ is an entire function of finite order $0<\sigma (f)<\mathrm{\infty}$ and ξ (≠0) is a Borel exceptional value of $f(z)$, then $f(z)$ can be written in the form
where $h(z)$ is a polynomial of degree l and $p(z)$ is an entire function satisfying $\sigma (p(z))<\sigma (f(z))={deg}_{z}h(z)=l$. Thus, we have
From Lemma 2.7, we have $\sigma (p(z+{\eta}_{j}))<\sigma (f(z+{\eta}_{j}))=\sigma (f(z))$ and ${deg}_{z}h(z+{\eta}_{j})={deg}_{z}h(z)=l$ for $j=1,2,\dots ,k$. Since ${L}_{1}(f(z))=cf(z)$, it follows from (23) and (24) that
Set $h(z)={\mu}_{l}{z}^{l}+\cdots $ and ${\mu}_{l}\ne 0$, then we can deduce from (25) that
Let $\mathrm{\Phi}:={\sum}_{j=1}^{k}p(z+{\eta}_{j}){e}^{{\mu}_{m1}^{j}{z}^{m1}+\cdots}$, it is easy to see that $\mathrm{\Phi}\not\equiv 0$ and $\sigma (\mathrm{\Phi})<\sigma (f)$, that is, $T(r,\mathrm{\Phi})=o(T(r,f))=o(T(r,{e}^{h(z)}))$.
Suppose that $c\ne k$. Since $\xi \ne 0$, it follows from (26) that
By the second fundamental theorem concerning small functions, for any $\epsilon >0$, we have
Since ε is arbitrary, we can get a contradiction from the above inequality. Thus, we can get that $c=k$.
Therefore, we prove that ${L}_{1}(f(z))=kf(z)$, that is, the conclusion of Theorem 1.7 holds.
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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), and the Foundation of Education Department of Jiangxi (GJJ14644) of China. The second author was supported by the NSF of China (11371225, 11171013).
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HW, LZY and HYX completed the main part of this article, HW, HYX 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/10.1186/168718472014274
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Keywords
 entire function
 Brück’s conjecture
 difference equation