# On a conjecture of R. Brück and some linear differential equations

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

**Received: **6 October 2015

**Accepted: **12 November 2015

**Published: **1 December 2015

## Abstract

In this paper, we mainly investigate the Brück conjecture concerning entire function *f* and its differential polynomial \(L_1(f)=a_k(z)f^{(k)}+\cdots +a_0(z)f\) sharing an entire function \(\alpha (z)\) with \(\sigma (\alpha )\le \sigma (f)\), by using the theory of complex differential equation.

## Keywords

## Mathematics Subject Classification

## Introduction and some results

*f*in the whole complex plane \(\mathbb {C}\), we shall use the following standard notations of the value distribution theory:

*S*(

*r*,

*f*) to denote any quantity satisfying \(S(r,f)=o(T(r,f)),\) as \(r\rightarrow +\infty\), 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 \(\sigma (f), \mu (f)\) to denote the order and the lower order of meromorphic function

*f*(

*z*), which are defined by

*f*(

*z*) with \(0<\sigma (f)=\sigma <+\infty\), which is defined to be (see Hayman 1964)

*f*(

*z*), which is defined to be (see Yi and Yang 2003, 1995)

*f*(

*z*) and

*g*(

*z*) be two nonconstant meromorphic functions, for some \(a\in \mathbb {C}\cup \{\infty \}\), if the zeros of \(f(z)-a\) and \(g(z)-a\) (if \(a=\infty\), zeros of \(f(z)-a\) and \(g(z)-a\) are the poles of

*f*(

*z*) and

*g*(

*z*) respectively) coincide in locations and multiplicities we say that

*f*(

*z*) and

*g*(

*z*) share the value

*a*

*CM*(counting multiplicities) and if coincide in locations only we say that

*f*(

*z*) and

*g*(

*z*) share

*a*

*IM*(ignoring multiplicities).

Rubel and Yang (1977) proved the following result.

###
**Theorem 1.1**

Rubel and Yang (1977). *Let f be a nonconstant entire function. If *
*f*
*and *
\(f'\)
* share two finite distinct values CM, then*
\(f\equiv f'\).

In 1996, Brück proposed the following conjecture Brück (1996):

###
**Conjecture 1.1**

*Let f be a non-constant entire function. Suppose that*\(\sigma _2(f)\)

*is not a positive integer or infinite, if f and*\(f'\)

*share one finite value a*

*CM, then*

*for some non-zero constant c*.

Gundersen and Yang (1998) proved that Brück conjecture holds for entire functions of finite order and obtained the following result.

###
**Theorem 1.2**

[Gundersen and Yang (1998), Theorem 1]. * Let f be a nonconstant entire function of finite order. If *
*f and *
\(f'\)
*share one finite value a*
*CM, then *
\(\frac{f'-a}{f-a}=c\)
*for some non-zero constant c*.

The shared value problems related 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 Chen et al. 2014; Li and Yi 2007; Liao 2015; Mues and Steinmetz 1986; Zhang and Yang 2009; Zhang 2005; Zhao 2012).

Li and Cao (2008) improved the Brück conjecture for entire function and its derivation sharing polynomials and obtained the following result:

###
**Theorem 1.3**

*Let*\(Q_1\)

*and*\(Q_2\)

*be two nonzero polynomials, and let P be a polynomial. If*

*f is a nonconstant entire solution of the equation*

*then*\(\sigma _2(f)=\deg P\)

*, where and in the following,*\(\deg P\)

*is the degree of P*.

Mao (2009) studied the problem on Brück conjecture when \(f^{(k)}\) is replaced by differential polynomial \(L(f)=A_kf^{(k)}+\cdots +A_1f'+A_0f\) in Theorem 1.3.

###
**Theorem 1.4**

*Let*

*P*(

*z*)

*be a polynomial,*\(A_k(z)(\not \equiv 0),\ldots ,A_0(z)\)

*be polynomials, and f be an entire function of order*

*and hyper-order*\(\sigma _2(f)<\frac{1}{2}\).

*If f and*

*L*(

*f*)

*share P CM, then*

*for some constant*\(c\ne 0\)

*, where, and in the sequel,*\(\deg A_j\)

*denotes the degree of*\(A_j(z)\),

*k is a positive integer.*

Chang and Zhu (2009) further investigated the problem related to Brück conjecture and proved that Theorem 1.2 remains valid if the value *a* is replaced by a function *a*(*z*).

###
**Theorem 1.5**

[
Chang and Zhu (2009), Theorem 1] *. Let f be an entire function of finite order and *
*a*(*z*) *be a function such that*
\(\sigma (a)<\sigma (f)<+\infty\). * If f and *
\(f'\)
*share a*(*z*) *CM, then *
\(\frac{f'-a}{f-a}=c\)
*for some non-zero constant c*.

Thus, an interesting subject arises naturally about this problem: *what would happen when*
\(\sigma (a)<\sigma (f)<+\infty\)
*is replaced by*
\(0<\sigma (a)=\sigma (f)<+\infty\)
*in Theorems 1.2–1.5?*

## Conclusions

*f*and its linear differential polynomial

*k*is a positive integer, \(a_k(z)(\not \equiv 0),a_{k-1}(z),\ldots ,a_1(z)\) and \(a_0(z)\) are polynomials, and obtain the following theorems.

###
**Theorem 2.1**

*Let f*(

*z*)

*and*\(\alpha (z)\)

*be two nonconstant entire functions and satisfy*\(0<\sigma (\alpha )=\sigma (f)<+\infty\)

*and*\(\tau (f)>\tau (\alpha )\)

*, and let P*(

*z*)

*be a polynomial such that*

*If f is a nonconstant entire solution of the following differential equation*

*where*\(L_1(f)\)

*is stated as in*(1).

*Then P*(

*z*)

*is a constant.*

*If*\(L_1(f)\)

*is replaced by the following linear differential polynomial*\(L_2(f)\)

*k*is a positive integer, \(a_k(z)(\not \equiv 0),a_{k-1}(z),\ldots ,a_1(z)\) and \(a_0(z)\) are polynomials, and \(\beta\) is an entire function satisfying either \(\sigma (\beta )<\mu (f)\) or \(0<\sigma (\beta )=\sigma (f)<+\infty\) and \(\tau (\beta )<\tau (f)\), then we obtain the following results.

###
**Theorem 2.2**

*Let f*(

*z*)

*and*\(\alpha (z)\)

*be two nonconstant entire functions and satisfy*\(0<\sigma (\alpha )=\sigma (f)<+\infty\)

*and*\(\tau (f)>\tau (\alpha )\)

*, and let P*(

*z*

*) be a polynomial satisfying*(2).

*If f is a nonconstant entire solution of the following differential equation*

*where*\(L_2(f)\)

*is stated as in*(4)

*and*\(\beta\)

*is an entire function satisfying*\(0<\sigma (\beta )=\sigma (f)<+\infty\)

*and*\(\tau (\beta )<\tau (f)\).

*Then*

*P*(

*z*)

*is a constant.*

###
**Theorem 2.3**

*Let f*(*z*) *and *
\(\alpha (z)\)
*be two nonconstant entire functions and satisfy *
\(\sigma (\alpha )<\mu (f)\)
*, and let P*(*z*) *be a polynomial satisfying* (2). *If f is a nonconstant entire solution of Eq. * (5), *where *
\(L_2(f)\)
*is stated as in* (4) *and*
\(\beta\)
*is an entire function satisfying*
\(\sigma (\beta )<\mu (f)\)
*. Then *
\(\sigma _2(f)=\deg P\).

## Some Lemmas

To prove our theorems, we will require some lemmas as follows.

###
**Lemma 3.1**

*Let f*(

*z*)

*be a transcendental entire function,*\(\nu (r,f)\)

*be the central index of f*(

*z*).

*Then there exists a set*\(E\subset (1,+\infty )\)

*with finite logarithmic measure, we choose z satisfying*\(|z|=r\not \in [0,1]\cup E\)

*and*\(|f(z)|=M(r,f)\)

*, we get*

###
**Lemma 3.2**

*Let f*(

*z*)

*be an entire function of finite order*\(\sigma (f)=\sigma <+\infty\)

*, and let*\(\nu (r,f)\)

*be the central index of*

*f*

*. Then*

*And if f is a transcendental entire function of hyper order*\(\sigma _2(f)\)

*, then*

###
**Lemma 3.3**

*Let f be a transcendental entire function and let*\(E\subset [1,+\infty )\)

*be a set having finite logarithmic measure. Then there exists*\(\{z_n=r_ne^{i\theta _n}\}\)

*such that*\(|f(z_n)|=M(r_n,f),\theta _n\in [0,2\pi )\), \(\lim _{n\rightarrow +\infty }\theta _n=\theta _0\in [0,2\pi ), r_n\not \in E\)

*and if*\(0<\sigma (f)<+\infty\)

*, then for any given*\(\varepsilon >o\)

*and sufficiently large*\(r_n\),

*If*\(\sigma (f)=+\infty\)

*, then for any given large*\(M>0\)

*and sufficiently large*\(r_n\), \(\nu (r_n,f)>r_n^M\).

###
**Lemma 3.4**

*Let*\(P(z)=b_nz^n+b_{n-1}z^{n-1}+\cdots +b_0\)

*with*\(b_n\ne 0\)

*be a polynomial. Then, for every*\(\varepsilon >0\)

*, there exists*\(r_0>0\)

*such that for all*\(r=|z|>r_0\)

*the inequalities*

*hold.*

###
**Lemma 3.5**

*Let f*(

*z*)

*and A*(

*z*)

*be two entire functions with*\(0<\sigma (f)=\sigma (A)=\sigma <+\infty , 0<\tau (A)<\tau (f)<+\infty\)

*, then there exists a set*\(E\subset [1,+\infty )\)

*that has infinite logarithmic measure such that for all*\(r\in E\)

*and a positive number*\(\kappa >0\),

*we have*

###
*Proof*

## The proof of Theorem 2.1

###
*Proof*

*P*(

*z*) is a polynomial, assume that \(\deg P=m\ge 1\). Let

*M*is a positive constant. Since \(-j\sigma (f)+d_{k-j}+\deg P+(k-j)\varepsilon <-2k\varepsilon <0\), it follows from (14) that

*P*(

*z*) is not a polynomial, that is,

*P*(

*z*) is a constant.

Thus, this completes the proof of Theorem 2.1. \(\square\)

## The proof of Theorem 2.2

###
*Proof*

*P*(

*z*) is a constant.

This completes the proof of Theorem 2.2. \(\square\)

## The proof of Theorem 2.3.

###
*Proof*

From *P*(*z*) is a polynomial, we will consider two cases (i) \(\sigma (f)<+\infty\) and (ii) \(\sigma (f)=+\infty\).

*Case 1.*Suppose that \(\sigma (f)<+\infty\). Then \(\sigma _2(f)=0\). Since \(\sigma (\alpha )<\mu (f), \sigma (\beta )<\mu (f)\), from Definitions of the order and the lower order, there exists infinite sequence \(\{z_n\}_1^\infty\), we have

*P*(

*z*) is a constant, that is, \(\deg P=0\). Therefore, \(\sigma _2(f)=\deg P\).

*Case 2.*Suppose that \(\sigma (f)=+\infty\). Set \(F(z)=f(z)-\alpha (z)\). Since \(\sigma (\alpha )<\mu (f)\), it follows from (2) that

*z*satisfying \(|z|=r\not \in [0,1]\cup E_4\) and \(|F(z)|=M(r,F)\), we get

*K*and for sufficiently large \(r_n\) we have

*P*(

*z*) is a polynomial, then \(\sigma (e^{P})=\deg P=m\). By combining (29), we have \(\sigma _2(f)=\deg P\).

Therefore, this completes the proof of Theorem 2.3. \(\square\)

## Declarations

### Authors’ contributions

HYX and LZY completed the main part of this article. Both authors read and approved the final manuscript.

### Acknowledgements

This first author was supported by the NSF of China (11561033, 11301233), the Natural Science Foundation of Jiangxi Province in China (20132BAB211001,20151BAB201008), and the Foundation of Education Department of Jiangxi (GJJ14644) of China. The second author was supported by the NSF of China (11371225, 11171013 and 11041005).

### Competing interests

The authors declare that they have no competing interests.

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

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