# On some complex differential and difference equations concerning sharing function

- Hua Wang
^{1}Email author, - Lian-Zhong Yang
^{2}and - Hong-Yan Xu
^{1}

**2014**:274

https://doi.org/10.1186/1687-1847-2014-274

© Wang et al.; licensee Springer. 2014

**Received: **24 June 2014

**Accepted: **6 October 2014

**Published: **27 October 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.

## Keywords

## 1 Introduction and main results

*f*be a nonconstant meromorphic function in the whole complex plane ℂ. We shall use the following standard notations of the value distribution theory:

*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])

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* $\frac{{f}^{\prime}-a}{f-a}=c$ *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* $\frac{{f}^{\prime}-a}{f-a}=c$ *for some nonzero constant* *c*.

- (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 1.1-1.3, what would happen when ${f}^{\prime}$ is replaced by differential polynomial$L(f)={a}_{k}(z){f}^{(k)}+{a}_{k-1}(z){f}^{(k-1)}+\cdots +{a}_{1}(z){f}^{\prime}+{a}_{0}(z)f,$(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* ${\eta}_{k},{\eta}_{k-1},\dots ,{\eta}_{1}$ *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*$P(z)={b}_{n}{z}^{n}+{b}_{n-1}{z}^{n-1}+\cdots +{b}_{0}$

*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

*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, 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

*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 left-hand 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.

*ε*satisfying

*M*is a positive constant. Since $-j\sigma (f)+{d}_{k-j}+j+(k-j)\epsilon <-2k\epsilon <0$, it follows from (11) that

which is impossible. Thus, $\gamma (z)$ is not a polynomial.

*c*such that

Thus, this completes the proof of Theorem 1.4.

## 4 The proof of Theorem 1.5

*f*share the constant

*a*

*CM*, then there exists an entire function $\phi (z)$ such that

We will consider two cases as follows.

*K*such that

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.

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.

*f*is transcendental, set

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

*ξ*(≠0) is a Borel exceptional value of $f(z)$, then $f(z)$ can be written in the form

*l*and $p(z)$ is an entire function satisfying $\sigma (p(z))<\sigma (f(z))={deg}_{z}h(z)=l$. Thus, we have

Let $\mathrm{\Phi}:={\sum}_{j=1}^{k}p(z+{\eta}_{j}){e}^{{\mu}_{m-1}^{j}{z}^{m-1}+\cdots}$, it is easy to see that $\mathrm{\Phi}\not\equiv 0$ and $\sigma (\mathrm{\Phi})<\sigma (f)$, that is, $T(r,\mathrm{\Phi})=o(T(r,f))=o(T(r,{e}^{h(z)}))$.

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.

## Declarations

### 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 Jiang-Xi 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).

## Authors’ Affiliations

## References

- Hayman WK:
*Meromorphic Functions*. Clarendon, Oxford; 1964.Google Scholar - Yang L:
*Value Distribution Theory*. Springer, Berlin; 1993.Google Scholar - Yi HX, Yang CC:
*Uniqueness Theory of Meromorphic Functions*. Kluwer Academic, Dordrecht; 2003. Chinese original: Science Press, Beijing (1995)Google Scholar - Rubel L, Yang CC: Values shared by an entire function and its derivative. Lecture Notes in Mathematics 599. In
*Complex Analysis*. Springer, Berlin; 1977:101-103. (Proc. Conf., Univ. Kentucky, Lexington, Ky., 1976)View ArticleGoogle Scholar - Brück R: On entire functions which share one value CM with their first derivative.
*Results Math.*1996, 30: 21-24. 10.1007/BF03322176MathSciNetView ArticleGoogle Scholar - Gundersen GG, Yang LZ: Entire functions that share one value with one or two of their derivatives.
*J. Math. Anal. Appl.*1998, 223: 85-95.MathSciNetView ArticleGoogle Scholar - Mues E, Steinmetz N: Meromorphe funktionen, die mit ihrer ersten und zweiten Ableitung einen endlichen Wert teilen.
*Complex Var. Theory Appl.*1986, 6: 51-71. 10.1080/17476938608814158View ArticleGoogle Scholar - Zhang JL, Yang LZ: A power of a meromorphic function sharing a small function with its derivative.
*Ann. Acad. Sci. Fenn., Math.*2009, 34: 249-260.MathSciNetGoogle Scholar - Zhang QC: Meromorphic function that shares one small function with its derivative.
*J. Inequal. Pure Appl. Math.*2005., 6: Article ID 116Google Scholar - 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.*2014, 38(3):250-253.Google Scholar - 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.*2013, 37(5):449-452.Google Scholar - 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.*2012, 36(6):584-588.MathSciNetGoogle Scholar - Chang JM, Zhu YZ: Entire functions that share a small function with their derivatives.
*J. Math. Anal. Appl.*2009, 351: 491-496. 10.1016/j.jmaa.2008.07.080MathSciNetView ArticleGoogle Scholar - Heittokangas J, Korhonen R, Laine I, Rieppo J, Zhang JL: Value sharing results for shifts of meromorphic functions, and sufficient conditions for periodicity.
*J. Math. Anal. Appl.*2009, 355: 352-363. 10.1016/j.jmaa.2009.01.053MathSciNetView ArticleGoogle Scholar - Li XM, Yi HX: Entire functions sharing an entire function of smaller order with their shifts.
*Proc. Jpn. Acad., Ser. A, Math. Sci.*2013, 89: 34-39.MathSciNetView ArticleGoogle Scholar - Laine I:
*Nevanlinna Theory and Complex Differential Equations*. de Gruyter, Berlin; 1993.View ArticleGoogle Scholar - He YZ, Xiao XZ:
*Algebroid Functions and Ordinary Differential Equations*. Science Press, Beijing; 1988.Google Scholar - Mao ZQ: Uniqueness theorems on entire functions and their linear differential polynomials.
*Results Math.*2009, 55: 447-456. 10.1007/s00025-009-0419-4MathSciNetView ArticleGoogle Scholar - Chiang YM, Feng SJ:On the Nevanlinna characteristic of $f(z+\eta )$ and difference equations in the complex plane.
*Ramanujan J.*2008, 16: 105-129. 10.1007/s11139-007-9101-1MathSciNetView ArticleGoogle Scholar - Barnett DC, Halburd RG, Korhonen RJ, Morgan W: Nevanlinna theory for the
*q*-difference operator and meromorphic solutions of*q*-difference equations.*Proc. R. Soc. Edinb., Sect. A, Math.*2007, 137: 457-474.MathSciNetView ArticleGoogle Scholar - Li XM, Yi HX: An entire function and its derivatives sharing a polynomial.
*J. Math. Anal. Appl.*2007, 330: 66-79. 10.1016/j.jmaa.2006.07.038MathSciNetView ArticleGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.