# On value distribution and uniqueness of meromorphic function with finite logarithmic order concerning its derivative and *q*-shift difference

- Xiu-Min Zheng
^{1}Email author and - Hong-Yan Xu
^{2}

**2014**:295

https://doi.org/10.1186/1029-242X-2014-295

© Zheng and Xu; licensee Springer. 2014

**Received: **12 April 2014

**Accepted: **30 June 2014

**Published: **19 August 2014

## Abstract

In this paper, we study the value distribution of a meromorphic function $f(z)$ concerning its derivative ${f}^{\prime}(z)$ and *q*-shift difference $f(qz+c)$, where $f(z)$ is of finite logarithmic order. We also investigate the uniqueness of differential-*q*-shift-difference polynomials with more general forms of entire functions of order zero.

## Keywords

*q*-shift-differencemeromorphic functionlogarithmic order

## 1 Introduction and main results

*e.g.*Hayman [1], Yang [2] and Yi and Yang [3]). In addition, for a meromorphic function $f(z)$, we use $S(r,f)$ to denote any quantity satisfying $S(r,f)=o(T(r,f))$ for all

*r*outside a possible exceptional set

*E*of finite logarithmic measure ${lim}_{r\to \mathrm{\infty}}{\int}_{(1,r]\cap E}\frac{dt}{t}<\mathrm{\infty}$ and also use ${S}_{1}(r,f)$ to denote any quantity satisfying ${S}_{1}(r,f)=o(T(r,f))$ for all

*r*on a set

*F*of logarithmic density 1, where the logarithmic density of a set

*F*is defined by

If ${\rho}_{log}(f)<\mathrm{\infty}$, then $f(z)$ is said to be of finite logarithmic order. It is clear that if a meromorphic function $f(z)$ has finite logarithmic order, then $f(z)$ has order zero.

*r*. The logarithmic exponent of convergence of

*a*-points of $f(z)$ is equal to the logarithmic order of $n(r,\frac{1}{f-a})$, which is defined as

We see by [4] that for a meromorphic function $f(z)$ of finite positive logarithmic order ${\rho}_{log}(f)$, the logarithmic order of $N(r,\frac{1}{f-a})$ is ${\lambda}_{log}(a)+1$, where ${\lambda}_{log}(a)$ is the logarithmic order of $n(r,\frac{1}{f-a})$.

Moreover, we assume in the whole paper that *m*, *n*, *k*, ${t}_{m}$, ${t}_{n}$ are positive integers, $q\in C\mathrm{\setminus}\{0\}$, $c\in C$, and $a(z)$ is a non-zero small function with respect to $f(z)$, that is, $a(z)$ is a non-zero meromorphic function of growth $S(r,f)$.

Many mathematicians were interested in the value distribution of different expressions of meromorphic functions and obtained lots of important theorems (see *e.g.* [1, 6–8]). Especially, Hayman [7] discussed Picard’s values of meromorphic functions and their derivatives, and he obtained the following famous theorem in 1959.

**Theorem 1.1** [7]

*Let*$f(z)$

*be a transcendental entire function*.

*Then*

- (a)
*for*$n\ge 2$, ${f}^{\prime}(z)f{(z)}^{n}$*assumes all finite values except possibly zero infinitely often*; - (b)
*for*$n\ge 3$*and*$a\ne 0$, ${f}^{\prime}(z)-af{(z)}^{n}$*assumes all finite values infinitely often*.

Further, for a transcendental meromorphic function $f(z)$, Chen and Fang [9] obtained the following result.

**Theorem 1.2** [[9], Theorem 1]

*Let* $f(z)$ *be a transcendental meromorphic function*. *If* $n\ge 1$ *is a positive integer*, ${f}^{\prime}(z)f{(z)}^{n}-1$ *has infinitely many zeros*.

Recently, with the establishments of difference analogies of the Nevanlinna theory (see *e.g.* [10–12]), many mathematicians focused on studying difference analogies of Theorems 1.1 and 1.2. The main purpose of these results (see *e.g.* [5, 13–16]) is to get the sharp estimation of the value of *n* to make difference polynomials $f(z+c)f{(z)}^{n}-a$ and $f(z+c)-af{(z)}^{n}-b$ admit infinitely many zeros.

Meantime, *q*-difference analogies of the Nevanlinna theory and their applications on the value distribution of *q*-difference polynomials and *q*-shift-difference equations are also studied (see *e.g.* [17–19]). Especially, for a transcendental meromorphic (resp. entire) function $f(z)$ of order zero, Zhang and Korhonen [20] studied the value distribution of *q*-difference polynomials of $f(z)$ and found that if $n\ge 6$ (resp. $n\ge 2$), then $f(qz)f{(z)}^{n}$ assumes every non-zero value $a\in C$ infinitely often (see [[20], Theorem 4.1]).

Further, Xu and Zhang [21] investigated the zeros of *q*-shift difference polynomials of meromorphic functions of finite logarithmic order and obtained the following result in 2012.

**Theorem 1.3** [[21], Theorem 2.1]

*If* $f(z)$ *is a transcendental meromorphic function of finite logarithmic order* ${\rho}_{log}(f)$, *with the logarithmic exponent of convergence of poles less than* ${\rho}_{log}(f)-1$ *and* *q*, *c* *are non*-*zero complex constants*, *then for* $n\ge 2$, $f{(z)}^{n}f(qz+c)$ *assumes every value* $b\in C$ *infinitely often*.

One main aim of this paper is to investigate the zeros of differential-*q*-shift-difference polynomials about $f(z)$, ${f}^{\prime}(z)$, and $f(qz+c)$, where $f(z)$ is of finite positive logarithmic order.

**Theorem 1.4**

*Let*$f(z)$

*be a transcendental meromorphic function of finite logarithmic order*${\rho}_{log}(f)$,

*with the logarithmic exponent of convergence of poles less than*${\rho}_{log}(f)-1$.

*Set*

*If* $n\ge 3$, *then* ${F}_{1}(z)-a(z)$ *has infinitely many zeros*.

We also deal with the value distribution of a differential-*q*-shift-difference polynomial with another form about $f(z)$, ${f}^{\prime}(z)$ and $f(qz+c)$, and we obtain the following result.

**Theorem 1.5**

*Let*$f(z)$

*be a transcendental meromorphic function of finite logarithmic order*${\rho}_{log}(f)$,

*with the logarithmic exponent of convergence of poles less than*${\rho}_{log}(f)-1$.

*Set*

*If* $n\ge 3$, *then* ${F}_{2}(z)-a$ *has infinitely many zeros*, *where* $a\in C$.

Some more general differential-*q*-shift-difference polynomials are investigated in the following.

**Theorem 1.6**

*Let*$f(z)$

*be a transcendental meromorphic function of finite logarithmic order*${\rho}_{log}(f)$,

*with the logarithmic exponent of convergence of poles less than*${\rho}_{log}(f)-1$.

*Set*

*If* *m*, *n* *satisfy* $m\ge n+2$ *or* $n\ge m+2$, *then* ${F}_{3}(z)-a(z)$ *has infinitely many zeros*.

be a non-zero polynomial, where ${a}_{0},{a}_{1},\dots ,{a}_{n}$ (≠0) are complex constants and ${t}_{n}$ is the number of the distinct zeros of ${P}_{n}(z)$. Then we also obtain the following results.

**Theorem 1.7**

*Let*$f(z)$

*be a transcendental meromorphic function of finite logarithmic order*${\rho}_{log}(f)$,

*with the logarithmic exponent of convergence of poles less than*${\rho}_{log}(f)-1$.

*Set*

*If* $m\ge n+k+1$, *then* ${F}_{4}(z)-a(z)$ *has infinitely many zeros*.

**Theorem 1.8**

*Let*$f(z)$

*be a transcendental meromorphic function of finite logarithmic order*${\rho}_{log}(f)$,

*with the logarithmic exponent of convergence of poles less than*${\rho}_{log}(f)-1$.

*Set*

*If* $m\ge n+k+1$, *then* ${F}_{5}(z)-a(z)$ *has infinitely many zeros*.

Next, we investigate the uniqueness of differential-*q*-shift-difference polynomials of entire functions of order zero and obtain the following results.

**Theorem 1.9** *Let* $f(z)$ *and* $g(z)$ *be transcendental entire functions of order zero and* $n\ge 5$. *If *$f{(qz+c)}^{n}{f}^{\prime}(z)$ *and* $g{(qz+c)}^{n}{g}^{\prime}(z)$ *share a non*-*zero polynomial* $p(z)$ *CM*, *then* $f{(qz+c)}^{n}{f}^{\prime}(z)=g{(qz+c)}^{n}{g}^{\prime}(z)$.

**Theorem 1.10** *Let* $f(z)$ *and* $g(z)$ *be transcendental entire functions of order zero and* $m\ge n+2{t}_{n}+5$. *If* $f{(z)}^{m}{P}_{n}(f(qz+c)){f}^{\prime}(z)$ *and* $g{(z)}^{m}{P}_{n}(g(qz+c)){g}^{\prime}(z)$ *share a non*-*zero polynomial* $p(z)$ *CM*, *then* $f{(z)}^{m}{P}_{n}(f(qz+c)){f}^{\prime}(z)=g{(z)}^{m}{P}_{n}(g(qz+c)){g}^{\prime}(z)$.

**Theorem 1.11** *Let* $f(z)$ *and* $g(z)$ *be transcendental entire functions of order zero and* $n\ge m+2{t}_{m}+5$. *If* ${P}_{m}(f(z))f{(qz+c)}^{n}{f}^{\prime}(z)$ *and* ${P}_{m}(g(z))g{(qz+c)}^{n}{g}^{\prime}(z)$ *share a non*-*zero polynomial* $p(z)$ *CM*, *then* ${P}_{m}(f(z))f{(qz+c)}^{n}{f}^{\prime}(z)={P}_{m}(g(z))g{(qz+c)}^{n}{g}^{\prime}(z)$.

## 2 Some lemmas

To prove the above theorems, we need some lemmas as follows.

**Lemma 2.1** [3]

*Let*$f(z)$

*be a non*-

*constant meromorphic function and*$P(f)={a}_{0}+{a}_{1}f+\cdots +{a}_{n}{f}^{n}$,

*where*${a}_{0},{a}_{1},\dots ,{a}_{n}$

*are complex constants and*${a}_{n}\ne 0$,

*then*

**Lemma 2.2** [21]

*Let*$f(z)$

*be a transcendental meromorphic function of finite logarithmic order and*

*q*,

*η*

*be two non*-

*zero complex constants*.

*Then we have*

**Lemma 2.3** [[22], Theorem 2.1]

*Let*$f(z)$

*be a non*-

*constant zero*-

*order meromorphic function and*$q\in C\setminus \{0\}$.

*Then*

**Lemma 2.4** [[3], p.37]

*Let*$f(z)$

*be a non*-

*constant meromorphic function in the complex plane and*

*l*

*be a positive integer*.

*Then*

**Lemma 2.5** [4]

*If*$f(z)$

*is a transcendental meromorphic function of finite logarithmic order*${\rho}_{log}(f)$,

*then for any two distinct small functions*$a(z)$

*and*$b(z)$

*with respect to*$f(z)$,

*we have*

*where*$U(r,f)={(logr)}^{{\rho}_{log}(f)}$

*is a logarithmic*-

*type function of*$T(r,f)$.

*Furthermore*,

*if*$T(r,f)$

*has a finite lower logarithmic order*

*with*${\rho}_{log}(f)-\mu <1$,

*then*

**Remark 2.1** From the proof of Lemma 2.5 (see [[4], Theorem 7.1]), we can easily see that complex values *a* and *b* can be changed into $a(z)$ and $b(z)$, where $a(z)$ and $b(z)$ are two distinct small functions with respect to $f(z)$.

**Lemma 2.6**

*Let*$f(z)$

*be a transcendental meromorphic function of order zero*, ${F}_{1}(z)=f{(qz+c)}^{n}{f}^{\prime}(z)$.

*Then we have*

*Proof*If $f(z)$ is a meromorphic function of order zero, from Lemmas 2.2 and 2.4, we have

Thus, we get (1). □

**Lemma 2.7**

*Let*$f(z)$

*be a transcendental meromorphic function of zero order*, ${F}_{3}(z)=f{(z)}^{m}f{(qz+c)}^{n}{f}^{\prime}(z)$.

*Then we have*

*and*

*Proof*If $f(z)$ is a meromorphic function of order zero, from Lemmas 2.2 and 2.4, we have

that is, we have (3), where we assume $m\ge n$ without loss of generality. □

**Lemma 2.8**

*Let*$f(z)$

*be a transcendental meromorphic function of order zero*, ${F}_{4}(z)=f{(z)}^{m}{P}_{n}(f(qz+c)){\prod}_{j=1}^{k}{f}^{(j)}(z)$.

*Then we have*

*Proof*Since $f(z)$ is a transcendental meromorphic function of order zero, by Lemmas 2.1, 2.2, and 2.4, we can easily get the second inequality. On the other hand, it follows by Lemmas 2.1, 2.2, and 2.4 that

Thus, this completes the proof of Lemma 2.8. □

Similar to Lemma 2.8, we have the following lemma.

**Lemma 2.9**

*Let*$f(z)$

*be a transcendental meromorphic function of zero order*, ${F}_{5}(z)={P}_{m}(f(z))f{(qz+c)}^{n}{\prod}_{j=1}^{k}{f}^{(j)}(z)$.

*Then we have*

## 3 Proofs of Theorems 1.4-1.8

### 3.1 Proof of Theorem 1.4

*r*on a set of logarithmic density 1. Since $f(z)$ is transcendental and $n\ge 3$, ${F}_{1}(z)$ is transcendental by Lemma 2.6 again. Since the logarithmic exponent of convergence of poles of $f(z)$ less than ${\rho}_{log}(f)-1$, we have

which contradicts the fact that $T(r,f)$ has finite logarithmic order ${\rho}_{log}(f)$. Thus, ${F}_{1}(z)-a(z)$ has infinitely many zeros, that is, $f{(qz+c)}^{n}{f}^{\prime}(z)-a(z)$ has infinitely many zeros.

This completes the proof of Theorem 1.4.

### 3.2 Proof of Theorem 1.5

*A*is a non-zero complex constant. Thus, we have $\rho (f)=1$, which contradicts the fact that $f(z)$ is of order zero. Thus, set

*r*on a set of logarithmic density 1. By Lemma 2.2, we have

which contradicts that $T(r,f)$ has finite logarithmic order ${\rho}_{log}(f)$. Thus, $f{(qz+c)}^{n}+{f}^{\prime}(z)+f(z)-a$ has infinitely many zeros.

This completes the proof of Theorem 1.5.

### 3.3 Proofs of Theorems 1.6, 1.7, and 1.8

Similar to the argument as in Theorem 1.4, by applying Lemmas 2.7, 2.8, and 2.9 instead, we can easily prove Theorems 1.6, 1.7, and 1.8 respectively.

## 4 Proofs of Theorems 1.9-1.11

Here, we only give the proof of Theorem 1.10 because the methods of the proofs of Theorems 1.9, 1.10, and 1.11 are very similar.

### 4.1 Proof of Theorem 1.10

*CM*, we have

where *η* is a non-zero constant. If $\eta =1$, then we have $F(z)=G(z)$, that is, $f{(z)}^{m}{P}_{n}(f(qz+c)){f}^{\prime}(z)=g{(z)}^{m}{P}_{n}(g(qz+c)){g}^{\prime}(z)$.

which contradicts $m\ge n+2{t}_{n}+5$.

This completes the proof of Theorem 1.10.

## Declarations

### Acknowledgements

This work was supported by the National Natural Science Foundation of China (11301233, 61202313, 11171119), the Natural Science Foundation of Jiangxi Province in China (20132BAB211001) and Sponsored Program for Cultivating Youths of Outstanding Ability in Jiangxi Normal University of China, and the Foundation of Education Department of Jiangxi (GJJ14271, GJJ14644) of China.

## Authors’ Affiliations

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