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Some results of certain types of difference and differential equations
Advances in Difference Equations volume 2012, Article number: 127 (2012)
Abstract
In this article, we shall utilize the value distribution theory and complex oscillation theory to investigate certain types of difference and differential equations. The results we obtain generalize some previous results of Gundersen and Yang.
MSC:30D35, 34M10.
1 Introduction and main results
In this paper, we assume that the reader is familiar with the fundamental results and the standard notations of the Nevanlinna theory (e.g., see [1–3]). In addition, we will use the notation $\sigma (f)$ to denote the order of the meromorphic function $f(z)$. We recall the definition of hyperorder (see [3]), ${\sigma}_{2}(f)$ of $f(z)$ is defined by
Let f and g be two nonconstant meromorphic functions in the complex plane. By $S(r,f)$, we denote any quantity satisfying $S(r,f)=o(T(r,f))$ as $r\to \mathrm{\infty}$, possibly outside a set of r with finite linear measure. Then the meromorphic function β is called a small function of f, if $T(r,\beta )=S(r,f)$. If $f\beta $ and $g\beta $ have the same zeros, counting multiplicity (ignoring multiplicity), then we say f and g share the small function β CM (IM).
Let ${z}_{0}$ be a zero of $f\beta $ with multiplicity p and a zero of $g\beta $ with multiplicity q. We denote by ${N}_{L}(r,\frac{1}{f\beta})$ the counting function of the zeros of $f\beta $ where $p>q\ge 1$, each point counted $pq$ times. In the same way, we also define ${N}_{L}(r,\frac{1}{g\beta})$.
Let $f(z)$ be transcendental meromorphic function in the plane, $c\in \mathbb{C}\setminus \{0\}$ be a constant such that $f(z)\not\equiv f(z+c)$. The forward differences ${\mathrm{\Delta}}^{n}f(z)$ are defined in the standard way [4] by
In 1996, Brück raised the following conjecture:
Conjecture Let f be a nonconstant entire function such that the hyper order ${\sigma}_{2}(f)<\mathrm{\infty}$ and ${\sigma}_{2}(f)$ is not a positive integer. If f and ${f}^{\prime}$ share the finite value a CM, then
where c is a nonzero constant.
The case that $a=0$ had been proved by Brück himself in [5]. From differential equations
we see that when the hyper order ${\sigma}_{2}(f)$ of f is a positive integer or infinite, the conjecture of Brück does not hold.
Gundersen and Yang proved the conjecture holds for entire functions of finite order, see [6], and Yang generalized this finite order for ${f}^{(k)}$ ($k\ge 1$), instead of ${f}^{\prime}$, see [7]. Chen and Shon proved the conjecture holds if ${\sigma}_{2}(f)<\frac{1}{2}$, see [8]. In terms of sharing a small function α IM, recently Wang has generalized Gundersen and Yang’s results, see [9].
In this paper, we consider the uniqueness of entire functions sharing a small function with their linear difference and differential polynomial. Now we present the main theorems.
Theorem 1.1 Let $f(z)$ be a nonconstant entire function, ${\sigma}_{2}(f)$ (<∞) is not a positive integer. Set ($k\ge 2$), where ${a}_{j}(z)$ ($2\le j\le k$) are entire functions of order less than 1 and ${a}_{k}(z)\not\equiv 0$. If $f(z)$ and ${L}_{1}(f)$ share z IM, and
then ${L}_{1}(f)z=h(z)(f(z)z)$, where $h(z)$ is a meromorphic function of order no greater than s.
Remark 1 Note that the term ${f}^{\prime}(z)$ cannot be contained in ${L}_{1}(f)$, otherwise Theorem 1.1 does not hold. For example: Set $f(z)=2{e}^{z}+z$ and ${L}_{1}(f)={f}^{\u2033}(z)+2{f}^{\prime}(z)+f(z)$. Then $f(z)$ and ${L}_{1}(f)$ share z IM, but
Theorem 1.2 Let $f(z)$ be a nonconstant entire function, ${\sigma}_{2}(f)$ (<∞) is not a positive integer. Set ($k\ge 1$), where ${a}_{j}(z)$ ($0\le j\le k$) are entire functions and $max\{\sigma ({a}_{j})j=1,2,\dots ,k\}<\sigma ({a}_{0})\in \mathbb{N}$. If $f(z)$ and ${L}_{2}(f)$ share a IM (a is a constant), then
where $h(z)$ is a meromorphic function of order no less than 1.
Theorem 1.3 Let $f(z)$ be a nonconstant entire function of order less than $\frac{1}{2}$ and $a(z)$ be a nonzero small function of $f(z)$. Set $A(f)={a}_{k}(z){\mathrm{\Delta}}^{k}f(z)+\cdots +{a}_{1}(z)\mathrm{\Delta}f(z)+{a}_{0}(z)f(z)$, where ${a}_{j}(z)$ ($j=0,1,\dots ,k$) are polynomial and ${a}_{k}(z)\not\equiv 0$. If $f(z)a(z)=0\to A(f)a(z)=0$, then
where $B(z)$ is a nonzero polynomial.
2 Some lemmas
In order to prove our theorems, we need the following lemmas and notions.
Following Hayman [10], pp.7576], we define an εset to be a countable union of open discs not containing the origin and subtending angles at the origin whose sum is finite. If E is an εset then the set of $r\ge 1$ for which the circle $S(0,r)=\{z\in \mathbb{C}:z=r\}$ meets E has finite logarithmic measure, and for almost all real θ the intersection of E with the ray $argz=\theta $ is bounded.
Lemma 2.1 ([11])
Let $n\in \mathbb{N}$. Let $f(z)$ be transcendental and meromorphic of order less than 1 in the plane. Then there exists an εset ${E}_{n}$ such that
Lemma 2.2 ([12])
Let $w(z)$ be an entire function of order $\rho (w)=\beta <\frac{1}{2}$, $A(r)={inf}_{z=r}logw(z)$ and $B(r)={sup}_{z=r}logw(z)$. If $\beta <\alpha <1$, then
where the lower logarithmic density $\underline{log\hspace{0.17em}dens}H$ of subset $H\subset (1,\mathrm{\infty})$ is defined by
and the upper logarithmic density $\overline{log\hspace{0.17em}dens}H$ of subset $H\subset (1,\mathrm{\infty})$ is defined by
where $\chi H(t)$ is the characteristic function of the set H.
Lemma 2.3 ([9])
Let $f(z)$ be an entire function of finite order. Suppose that α is a nonzero small function of $f(z)$. Then there exists a set $E\subset (1,\mathrm{\infty})$ satisfying $\underline{log\hspace{0.17em}dens}(E)=1$, such that
holds for $z=r\in E$, $r\to \mathrm{\infty}$.
Lemma 2.4 ([13])
Let $f(z)$ be a meromorphic function with $\rho (f)=\eta <\mathrm{\infty}$. Then for any given $\epsilon >0$, there is a set ${E}_{1}\subset (1,+\mathrm{\infty})$ that has finite logarithmic measure such that
holds for $z=r\notin [0,1]\cup {E}_{1}$, $r\to \mathrm{\infty}$.
Applying Lemma 2.4 to $\frac{1}{f}$, it is easy to see that for any given $\epsilon >0$, there is a set ${E}_{2}\subset (1,\mathrm{\infty})$ of finite logarithmic measure such that
holds for $z=r\notin [0,1]\cup {E}_{2}$, $r\to \mathrm{\infty}$.
Lemma 2.5 ([14])
Let $f(z)$ be a transcendental meromorphic function, and $\alpha >1$ be a given constant. Then

(i)
there exists a set $E\subset (1,\mathrm{\infty})$ with finite linear measure zero and a constant $B>0$ that depends only on α and $j=1,\dots ,k$, such that if ${\phi}_{0}\in [0,2\pi )\setminus E$, then there is a constant $R=R({\phi}_{0})>1$ so that for all z satisfying $argz={\phi}_{0}$ and $z=r\ge R$, we have
$$\left\frac{{f}^{(j)}(z)}{f(z)}\right\le B{(\frac{T(\alpha r,f)}{r}\left({log}^{\alpha}r\right)logT(\alpha r,f))}^{j},$$(2.1)
for all $j=1,\dots ,k$;

(ii)
there exists a set $E\subset (1,\mathrm{\infty})$ with finite logarithmic measure and a constant $B>0$ that depends only on α and $j=1,\dots ,k$, such that for all z satisfying $z=r\notin [0,1]\cup E$, we have (2.1) holds.
Lemma 2.6 ([15])
Let $f(z)$ be an entire function of infinite order with ${\sigma}_{2}(f)=\sigma $, and $\mu (r)$ be the central index of $f(z)$. Then
3 Proof of Theorem 1.1
Under the hypothesis of Theorem 1.1, see [3], it is easy to get that
where $\theta (z)$ is an entire function, entire functions ${h}_{1}(z)$ and ${h}_{2}(z)$ satisfy
Therefore, by (3.1), we see that $h(z)=\frac{{h}_{1}(z)}{{h}_{2}(z)}$ is a meromorphic function of order no greater than s (<1). If $f(z)$ is a polynomial, then $\sigma (\frac{{L}_{1}(f)z}{f(z)z})=0$. Theorem 1.1 holds under this condition. Next we suppose that $f(z)$ is transcendental. Set $F(z)=f(z)z$. Then $F(z)$ is transcendental, $\sigma (F)=\sigma (f)$, ${\sigma}_{2}(F)={\sigma}_{2}(f)<\mathrm{\infty}$ and ${\sigma}_{2}(F)\notin \mathbb{N}$.
Substituting $f(z)=F(z)+z$ into (3.1), we get
If $\theta (z)$ is a constant, then Theorem 1.1 holds. Otherwise, $\theta (z)$ is a polynomial or a transcendental entire function, rewritten (3.2), we have
Set ${E}_{1}=\{z\frac{{h}_{1}(z)}{{h}_{2}(z)}=0\text{or}\frac{{h}_{1}(z)}{{h}_{2}(z)}=\mathrm{\infty}\}$, obviously, ${E}_{1}$ has finite linear measure. From Lemma 2.4, for any given $\epsilon >0$, there exists a set ${E}_{2}\subset (1,\mathrm{\infty})$ that has finite logarithmic measure such that
holds for $z=r\notin [0,1]\cup {E}_{2}$, $r\to \mathrm{\infty}$, where $\alpha =max\{\sigma ({a}_{j})j=2,\dots ,k\}<1$.
By Lemma 2.5, there exists a set ${E}_{3}\subset (1,\mathrm{\infty})$ with finite logarithmic measure and a constant $B>0$ such that for all z satisfying $z=r\notin [0,1]\cup {E}_{3}$, we have
By (3.3)(3.5), for $z=r\notin [0,1]\cup ({E}_{1}\cup {E}_{2}\cup {E}_{3})$, $r\to \mathrm{\infty}$, we have
Let $M(r,\frac{{h}_{1}(z)}{{h}_{2}(z)}{e}^{\theta (z)}1)=\frac{{h}_{1}(z)}{{h}_{2}(z)}{e}^{\theta (z)}1$, we obtain
From (3.6), we obtain
By WimanValiron theory, there exists a set ${E}_{4}\subset (1,\mathrm{\infty})$ having finite logarithmic measure, we choose z satisfying $z=r\notin [0,1]\cup {E}_{4}\cup {E}_{1}$, and $F(z)=M(r,F(z))$, then
where $\upsilon (r)$ is the central index of $F(z)$. By (3.3), (3.5), (3.8), we obtain
Hence
By Lemma 2.6, (3.9), $\alpha <1$, and ${\sigma}_{2}(F)\ge 1$, we obtain
By combining (3.7) and (3.10), we obtain
If $\theta (z)$ is polynomial, we know ${\sigma}_{2}(f)\in \mathbb{N}$; If $\theta (z)$ is transcendental, we get ${\sigma}_{2}(f)=\mathrm{\infty}$. Hence this contradicts the hypothesis of Theorem 1.1. Theorem 1.1 is thus proved.
4 Proof of Theorem 1.2
Under the hypothesis of Theorem 1.2, it is easy to get that
where $h(z)$ is a meromorphic function. If $\sigma (h)<1$, by the proof of Theorem 1.1, similarly, we can prove
which contradicts the fact that ${\sigma}_{2}(f)\notin \mathbb{N}$. This completes the proof of Theorem 1.2.
5 Proof of Theorem 1.3
Proof: By the Hadamard factorization theorem, we have
where $B(z)$ is an entire function of order less than $\frac{1}{2}$. If $f(z)$ is polynomial, Theorem 1.3 holds. Next, we consider that $f(z)$ is transcendental. Set $F(z)=f(z)a(z)$. Then $F(z)$ is transcendental, and $\sigma (F)<\frac{1}{2}$. Substituting $f(z)=F(z)+a(z)$ into (5.1), we have
where $b(z)={a}_{k}(z){\mathrm{\Delta}}^{k}a(z)+\cdots +{a}_{1}(z)\mathrm{\Delta}a(z)+a(z)({a}_{0}(z)1)$. By Lemma 2.1, there exists an εset ${E}_{n}$ such that
as $z\to \mathrm{\infty}$ in $\mathbb{C}\setminus {E}_{n}$. By WimanValiron theory, there is a subset ${E}_{5}\subset (1,\mathrm{\infty})$ with finite logarithmic measure. We choose z satisfying $z=r\notin {E}_{5}$ and $F(z)=M(r,f(z))$, then we have
where $\upsilon (r)$ is the central index of $F(z)$. By (5.2)(5.4), we have
By Lemma 2.3, there exists a set ${E}_{6}\subset (1,\mathrm{\infty})$ satisfying $\underline{log\hspace{0.17em}dens}({E}_{6})=1$ such that
Together with $\sigma (F)<1$, we get
By (5.5)(5.7), we have
where C is a constant. By Lemma 2.2, for any α satisfying $\mu <\alpha <\frac{1}{2}$, there exists a set ${E}_{7}$ with $\underline{log\hspace{0.17em}dens}({E}_{7})\ge 1\frac{\mu}{\alpha}$, such that
for $z=r\in {E}_{7}$, where $\lambda =cos\pi \alpha $.
Note the characteristic function of ${E}_{6}$ and ${E}_{7}$ such that the relation
Obviously, $\overline{log\hspace{0.17em}dens}({E}_{6}\cup {E}_{7})\le 1$. Hence we obtain
Thus, the upper logarithmic density of $({E}_{6}\cap {E}_{7})\setminus ({E}_{n}\cup {E}_{4}\cup {E}_{5}\cup [0,1])$ is also more than $1\frac{\mu}{\alpha}$. By (5.8) and (5.9), for $z\in ({E}_{6}\cap {E}_{7})\setminus ({E}_{n}\cup {E}_{4}\cup {E}_{5}\cup [0,1])$, we have
If $B(z)$ is transcendental, we get $\sigma (f)=\mathrm{\infty}$, which contradicts our assumption that $\sigma (f)<\frac{1}{2}$. So $B(z)$ is polynomial. This proves Theorem 1.3.
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Acknowledgements
The work was supported by the NNSF of China (No. 10771121), the NSF of Shangdong Province, China (No. Z2008A01) and Shandong university graduate student independent innovation fund (yzc11024).
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Authors’ contributions
YL and YH completed the main parts of the paper. YL, YH and HX corrected the main theorems. All authors read and approved the final manuscript.
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Liu, Y., Cao, Y.H. & Yi, H.X. Some results of certain types of difference and differential equations. Adv Differ Equ 2012, 127 (2012). https://doi.org/10.1186/168718472012127
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Keywords
 entire functions
 uniqueness
 complex difference
 share value