# Value distribution of *q*-difference differential polynomials of entire functions

- Na Xu
^{1}, - Ting-Bin Cao
^{1}Email author and - Kai Liu
^{1}

**2014**:80

https://doi.org/10.1186/1687-1847-2014-80

© Xu et al.; licensee Springer. 2014

**Received: **26 September 2013

**Accepted: **18 February 2014

**Published: **7 March 2014

## Abstract

For a complex value $q\ne 0,1$, and a transcendental entire function $f(z)$ with order, $0<\sigma (f)<\mathrm{\infty}$, we study the value distribution of *q*-difference differential polynomials ${[{f}^{n}(z)(f(qz)-f(z))]}^{(k)}$ and ${[f(z)f(qz)]}^{(k)}$.

**MSC:**30D35, 39A05.

## Keywords

*q*-differenceentire functions

## 1 Introduction and main results

*f*in $|z|<r$, each pole according to its multiplicity, and that $\overline{n}(r,f)$ counts the number of the distinct poles of

*f*in $|z|<r$, ignoring the multiplicity. The characteristic function of $f(z)$ is defined by

The notation $\overline{N}(r,f)$ is similarly defined with $\overline{n}(r,f)$ instead of $n(r,f)$. More notations and definitions of the Nevanlinna value distribution theory of meromorphic functions can be found in [1, 2].

*E*of finite logarithmic measure. The order and the exponent of convergence of zeros of meromorphic function $f(z)$ is, respectively, defined as

A Borel exceptional value of $f(z)$ is any value $a\in \mathbb{C}\cup \{\mathrm{\infty}\}$ satisfying $\lambda (f-a)<\sigma (f)$, where $\lambda (f-\mathrm{\infty})$ means $\lambda (\frac{1}{f})$.

Recently, the difference variant of the Nevanlinna theory has been established independently in [3–6]. Using these theories, value distributions of difference polynomials have been studied by many papers. For example, Laine and Yang [7] proved that if $f(z)$ is a transcendental entire function of finite order, *c* is a nonzero complex constant and $n\ge 2$, then ${f}^{n}(z)f(z+c)$ takes every nonzero value infinitely often.

Chen [8] considered the value distribution of $f(z)f(z+c)$ and obtained the following theorem.

**Theorem A** [[8], Corollary 1.3]

*Let* $f(z)$ *be a transcendental entire function of finite order*, *and let* *c* *be a nonzero complex constant*. *If* $f(z)$ *has the Borel exceptional value* 0, *then* $H(z)=f(z)f(z+c)$ *takes every nonzero value* $a\in \mathbb{C}$ *infinitely often*.

Chen [9] considered zeros of difference product ${H}_{n}(z)={f}^{n}(z){\mathrm{\Delta}}_{c}f(z)$ and gave some conditions that guarantee ${H}_{n}(z)$ has finitely many zeros or infinitely many zeros.

**Theorem B** [[9], Theorem 1]

*Let*$f(z)$

*be a transcendental entire function of finite order and*$c\in \mathbb{C}\setminus \{0\}$

*be a constant such that*$f(z+c)\not\equiv f(z)$.

*Set*${H}_{n}(z)={f}^{n}(z){\mathrm{\Delta}}_{c}f(z)$

*where*${\mathrm{\Delta}}_{c}f(z)=f(z+c)-f(z)$, $n\ge 2$

*is an integer*.

*Then the following statements hold*.

- (1)
*If*$f(z)$*satisfies*$\sigma (f)\ne 1$,*or has infinitely many zeros*,*then*${H}_{n}(z)$*has infinitely many zeros*. - (2)
*If*$f(z)$*has only finitely many zeros and*$\sigma (f)=1$,*then*${H}_{n}(z)$*has only finitely many zeros*.

The zero distribution of differential polynomials is a classical topic in the theory of meromorphic functions. Hayman [[10], Theorem 10] firstly considered the value distribution of ${f}^{n}{f}^{\prime}-1$, where *f* is a transcendental function.

Recently, Liu, Liu and Cao [11] investigated the zeros of ${[f{(z)}^{n}f(z+c)]}^{(k)}-\alpha (z)$ and ${[f{(z)}^{n}{\mathrm{\Delta}}_{c}f(z)]}^{(k)}-\alpha (z)$, where $\alpha (z)$ is a nonzero small function with respect to $f(z)$.

**Theorem C** [[11], Theorems 1.1 and 1.3]

*Let* $f(z)$ *be a transcendental entire function of finite order and* $\alpha (z)$ *be a nonzero small function with respect to* $f(z)$. *If* $n\ge k+2$, *then* ${[f{(z)}^{n}f(z+c)]}^{(k)}-\alpha (z)$ *has infinitely many zeros*. *If* $f(z)$ *is not a periodic function with period* *c* *and* $n\ge k+3$, *then* ${[f{(z)}^{n}{\mathrm{\Delta}}_{c}f(z)]}^{(k)}-\alpha (z)$ *has infinitely many zeros*.

The main purpose of this paper is to consider a transcendental entire function $f(z)$ with positive and finite order and obtain some results on the value distributions of the *q*-difference differential polynomials ${[{f}^{n}(z)(f(qz)-f(z))]}^{(k)}$ and ${[f(z)f(qz)]}^{(k)}$. The first theorem will consider what conditions guarantee that ${[{f}^{n}(z){\mathrm{\nabla}}_{q}f(z)]}^{(k)}$ has infinitely many zeros.

**Theorem 1.1** *Let* $f(z)$ *be a transcendental entire function of finite and positive order* $\sigma (f)$, $q\in \mathbb{C}\setminus \{0,1\}$ *be a constant such that* ${q}^{\sigma (f)}\ne 1$ *and* $f(z)\not\equiv f(qz)$. *Set* $H(z)={[{f}^{n}(z){\mathrm{\nabla}}_{q}f(z)]}^{(k)}$, $n\ge 1$ *is an integer*. *If* $f(z)$ *has finitely many zeros*, *then* $H(z)-\alpha (z)$ *has infinitely many zeros*, *where* $\alpha (z)$ *is a nonzero small entire function with respect to* $f(z)$.

In the following, we will study the value distribution of ${[f(z)f(qz)]}^{(k)}$.

**Theorem 1.2**

*Let*$f(z)$

*be a transcendental entire function of finite and positive order*$\sigma (f)$,

*a*

*be a finite Borel exceptional value of*$f(z)$, $q\in \mathbb{C}\setminus \{0,1\}$

*be a constant such that*${q}^{\sigma (f)}\ne \pm 1$.

*Set*$H(z)={[f(z)f(qz)]}^{(k)}$,

*then the following statements hold*.

- (1)
*If*$a=0$,*then*0*is also the Borel exceptional value of*$H(z)$.*So that*$H(z)$*has no nonzero finite Borel exceptional value*. - (2)
*If*$a\ne 0$,*then*$H(z)$*has no finite Borel exceptional value*. - (3)
$H(z)$

*takes every nonzero value*$c\in \mathbb{C}$*infinitely often and satisfies*$\lambda (H-c)=\sigma (f)$.

Using the similar method of the proof of Theorem 1.2(1), we get the following result immediately.

**Corollary 1.1** *Let* $f(z)$ *be a transcendental entire function of finite and positive order* $\sigma (f)$, $q\in \mathbb{C}\setminus \{0,1\}$ *be a constant such that* ${q}^{\sigma (f)}\ne -n$. *If* 0 *is a Borel exceptional value of* $f(z)$, *then* 0 *is also the Borel exceptional value of* ${[{f}^{n}(z)f(qz)]}^{(k)}$.

## 2 Some lemmas

The following are the well-known Weierstrass factorization and Hadamard factorization theorems.

**Lemma 2.1** [12]

*If an entire function*$f(z)$

*has a finite exponent of convergence*$\lambda (f)$

*for its zero*-

*sequence*,

*then*$f(z)$

*has a representation in the form*

*satisfying* $\lambda (Q)=\sigma (Q)=\lambda (f)$. *Further*, *if* $f(z)$ *is of finite order*, *then* $g(z)$ *in the above form is a polynomial of degree less or equal to the order of* $f(z)$.

**Lemma 2.2** [13]

*Suppose that*${f}_{1}(z),{f}_{2}(z),\dots ,{f}_{n}(z)$ ($n\ge 2$)

*are meromorphic functions and*${g}_{1}(z),{g}_{2}(z),\dots ,{g}_{n}(z)$

*are entire functions satisfying the following conditions*:

- (1)
${\sum}_{j=1}^{n}{f}_{j}(z){e}^{{g}_{j}(z)}\equiv 0$;

- (2)
${g}_{j}(z)-{g}_{k}(z)$

*are not constants for*$1\le j<k\le n$; - (3)
*For*$1\le j\le n$, $1\le h<k\le n$, $T(r,{f}_{j})=o(T(r,{e}^{{g}_{h}-{g}_{k}}))$ ($r\to \mathrm{\infty}$, $r\notin E$).

*Then* ${f}_{j}(z)\equiv 0$ ($j=1,2,\dots ,n$).

## 3 The proofs

### 3.1 Proof of Theorem 1.1

This is a contradiction.

*c*is a constant, then (3.1) can be rewritten as

This is a contradiction.

If $(n+1)h(z)-{h}_{1}(z)=c$, where *c* is a constant, then using the same method as above, we also obtain a contradiction.

Hence $H(z)-\alpha (z)$ has infinitely many zeros.

### 3.2 Proof of Theorem 1.2

*a*, then by Lemma 2.1, $f(z)$ can be written as

*s*is a positive integer,

*α*is a nonzero constant, $g(z)$ is a nonzero entire function satisfying $\sigma (g)<\sigma (f)=s$, thus

Here ${g}_{1}(z)$, ${g}_{2}(z)$, ${g}_{3}(z)$ are differential polynomials of $g(z)$, $g(qz)$ and $\sigma ({g}_{i}(z))<s$, $i=1,2,3$.

Since ${q}^{\sigma (f)}={q}^{s}\ne -1$ and $\sigma ({g}_{3}(z))<s$, this implies that 0 is the Borel exceptional value of $H(z)$.

*b*, then by Lemma 2.1, $H(z)$ can be written as

*β*is a nonzero constant, $h(z)$ is a nonzero entire function satisfying $\sigma (h)<\sigma (H)=s$. Combing the above equalities, we obtain

Since ${q}^{\sigma (f)}={q}^{s}\ne \pm 1$, we have $\alpha {q}^{s}\ne \alpha \ne \alpha (1+{q}^{s})$.

This is a contradiction.

Which is a contradiction.

If $\alpha =\beta $ or $\alpha (1+{q}^{s})=\beta $, then using the same method as above, we also obtain a contradiction.

Case 3. From Case 1 and Case 2, we get that if $f(z)$ has a finite Borel exceptional value, then any nonzero finite value *c* must not be the Borel exceptional value of $H(z)$, so $H(z)$ takes every nonzero value $c\in \mathbb{C}$ infinitely often, since $\sigma (H)=s$, then $\lambda (H-c)=\sigma (H)=\sigma (f)$.

The proof of Theorem 1.2 is completed.

## Declarations

### Acknowledgements

This research was partly supported by the NSFC (no. 11101201, 11301260), Foundation of Post Ph.D. of Jiangxi, the NSF of Jiangxi (no. 20122BAB211001, 20132BAB211003) and NSF of education department of Jiangxi (no. GJJ13077, GJJ13078) of China.

## Authors’ Affiliations

## References

- Hayman W:
*Meromorphic Functions*. Clarendon, Oxford; 1964.Google Scholar - Yang L:
*Value Distribution Theory*. Science Press, Beijing; 1982. Springer, Berlin (1993)Google 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*2007, 137: 457–474.MathSciNetView ArticleGoogle Scholar - Chiang YM, Feng SJ:On the Nevanlinna characteristic $f(z+\eta )$ and difference equations in the complex plane.
*Ramanujan J.*2008, 16: 105–129. 10.1007/s11139-007-9101-1MathSciNetView ArticleGoogle Scholar - Halburd RG, Korhonen RJ: Nevanlinna theory for the difference operator.
*Ann. Acad. Sci. Fenn., Math.*2006, 31: 463–478.MathSciNetGoogle Scholar - Halburd RG, Korhonen RJ: Difference analogue of the lemma on the logarithmic derivative with applications to difference equations.
*J. Math. Anal. Appl.*2006, 314: 477–487. 10.1016/j.jmaa.2005.04.010MathSciNetView ArticleGoogle Scholar - Laine I, Yang CC: Value distribution of difference polynomials.
*Proc. Jpn. Acad., Ser. A, Math. Sci.*2007, 83: 148–151. 10.3792/pjaa.83.148MathSciNetView ArticleGoogle Scholar - Chen ZX, Huang ZB, Zheng XM: On properties of difference polynomials.
*Acta Math. Sci.*2011, 31B(2):627–633.MathSciNetGoogle Scholar - Chen ZX: Value distribution of products of meromorphic functions and their differences.
*Taiwan. J. Math.*2011, 15: 1411–1421.Google Scholar - Hayman WK: Picard values of meromorphic functions and their derivatives.
*Ann. Math.*1959, 70: 9–42. 10.2307/1969890MathSciNetView ArticleGoogle Scholar - Liu K, Liu XL, Cao TB: Some results on zeros and uniqueness of difference-differential polynomials.
*Appl. Math. J. Chin. Univ. Ser. B*2012, 27: 94–104. 10.1007/s11766-012-2795-xMathSciNetView ArticleGoogle Scholar - Ash R:
*Complex Variables*. Academic Press, New York; 1971.Google Scholar - Yi HX, Yang CC:
*Uniqueness Theory of Meromorphic Functions*. Science Press, Beijing; 1995. Kluwer Academic, Dordrecht (2003)Google Scholar

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