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# Fixed points of differences of meromorphic functions

*Advances in Difference Equations*
**volume 2019**, Article number: 453 (2019)

## Abstract

Let *f* be a transcendental meromorphic function of finite order and *c* be a nonzero complex number. Define \(\Delta _{c}f=f(z+c)-f(z)\). The authors investigate the existence on the fixed points of \(\Delta _{c}f\). The results obtained in this paper may be viewed as discrete analogues on the existing theorem on the fixed points of \(f'\). The existing theorem on the fixed points of \(\Delta _{c}f\) generalizes the relevant results obtained by (Chen in Ann. Pol. Math. 109(2):153–163, 2013; Zhang and Chen in Acta Math. Sin. New Ser. 32(10):1189–1202, 2016; Cui and Yang in Acta Math. Sci. 33B(3):773–780, 2013) et al.

## 1 Introduction

Let \(f(z)\) be a function meromorphic in the complex plane *C*. We use the general notation of the Nevanlinna theory (see [12, 20, 23]) such as \(m(r, f) \), \(N(r, f)\), \(T(r,f )\), \(m (r, \frac{1}{f-a} )\), \(N (r, \frac{1}{f-a} )\), … , and assume that the reader is familiar with these notations. We also use \(S(r, f)\) to denote any quantity of \(S(r, f)=o(T(r, f))\) (\(r \rightarrow \infty \)), possibly outside a set with finite logarithmic measure. The order and the lower order of \(f(z)\) are denoted by \(\sigma (f)\) and \(\mu (f)\) respectively.

For any \(a\in {C}\), the exponent of convergence of zeros of \(f(z)-a\) (or poles of \(f(z)\)) is denoted by \(\lambda (f, a)\) (or \(\lambda (\frac{1}{f} )\)). Especially, we denote \(\lambda (f, 0)\) by \(\lambda (f)\). If \(\lambda (f, a)<\sigma (f)\) (or \(\lambda (\frac{1}{f} )<\sigma (f)\)), then *a* (or ∞) is said to be a Borel exceptional value of \(f(z)\). Nevanlinna’s deficiency of *f* with respect to complex number \(a\in C\cup \{\infty \}\) is defined by

If \(a=\infty \), then one should replace \(N (r, \frac{1}{f-a} )\) in the above formula by \(N(r, f)\).

A point \(z_{0}\in {C}\cup \{\infty \}\) is said to be a fixed point of \(f(z)\) if \(f(z_{0})=z_{0}\). There is a considerable number of results on the fixed points of meromorphic functions, we refer the reader to Chuang and Yang [7]. It follows Chen and Shon [2, 4], we use the notation \(\tau (f)\) to denote the exponent of convergence of fixed points of *f*, i.e.,

In 1993, Lahiri [13] proved the following theorem.

### Theorem A

*Let*
*f*
*be a transcendental meromorphic function in the plane*. *Suppose that there exists*
\(a\in {C}\)
*with*
\(\delta (a, f)>0\)
*and*
\(\delta (\infty , f)=1\). *Then*
*f*
*has infinitely many fixed points*.

In this paper, we shall study the fixed points of the differences of meromorphic functions. For each \(c\in {C}\backslash \{0 \}\), the forward difference \(\Delta _{c}^{k+1} f(z)\) is defined (see [1]) by

Especially, we denote \(\Delta _{1} f(z)\) by \(\Delta f(z)\).

Recently, some well-known facts of the Nevanlinna theory have been extended for the differences of meromorphic functions (see [5, 6, 9,10,11, 14,15,16,17,18]). For the existence on the fixed points of differences, Cui and Yang [8] have proved the following theorems.

### Theorem B

([8])

*Let*
*f*
*be a function transcendental and meromorphic in the plane with the order*
\(\sigma (f)=1\). *If*
*f*
*has finitely many poles and infinitely many zeros with exponent of convergence of zeros*
\(\lambda (f)\neq 1\), *then* Δ*f*
*has infinitely many zeros and fixed points*.

### Theorem C

([8])

*Let*
*f*
*be a non*-*periodic function transcendental and meromorphic in the plane with the order*
\(\sigma (f)=1\), \(\max \{\lambda (f), \lambda (\frac{1}{f} ) \} \neq 1\). *If*
*f*
*has infinitely many zeros*, *then* Δ*f*
*has infinitely many zeros and fixed points*.

The conditions of Theorems B and C imply that 0, ∞ are Borel exceptional values. If ∞ and \(d\in {C}\) are Borel exceptional values of *f*, Chen [3] obtains the following theorem.

### Theorem D

([3])

*Let*
*f*
*be a finite order meromorphic function such that*
\(\lambda (\frac{1}{f} )< \sigma (f)\), *and let*
\(c\in {C}\backslash \{0\}\)
*be a constant such that*
\(f(z+c)\not \equiv f(z)\). *If*
\(f(z)\)
*has a Borel exceptional value*
\(d\in {C}\), *then*
\(\tau (\Delta _{c} f)=\sigma (f)\).

In [22], Zhang and Chen showered that the condition \(\lambda (\frac{1}{f} )<\sigma (f)\) in Theorem D cannot be omitted. Moreover, they obtained the following theorem.

### Theorem E

([22])

*Let*
*f*
*be a finite order meromorphic function*, *and let*
\(c\in {C}\backslash \{0 \}\)
*be a constant such that*
\(f(z+c)\not \equiv f(z)\). *If*
\(f(z)\)
*has two Borel exceptional values*, *then*
\(\tau (\Delta _{c} f)=\sigma (f)\).

In [19], Yi and Yang have proved the following theorem.

### Theorem F

([19])

*Let*
*f*
*be a transcendental meromorphic function in*
*C*
*with a positive order*. *If*
*f*
*has two distinct Borel exceptional values*, *say*
\(a_{1}\)
*and*
\(a_{2}\), *then the order of*
*f*
*is a positive integer or* ∞ *and*
\(\sigma (f)=\mu (f)\), \(\delta (a_{1}, f)=\delta (a_{2}, f)=1\).

By Theorem F, we can derive that the order of *f* in Theorems D and E is a positive integer. Is it necessary to ask if the order of *f* is an integer?, i.e., Can we get similar results as those in Theorems B, C, D, and E if the order of *f* is not a positive integer? The main purpose of this paper is to study this question. In fact, we shall prove the following theorems.

### Theorem 1.1

*Let*
*f*
*be a transcendental meromorphic function of finite order in the plane*. *Suppose that*
\(c\in {C}\setminus \{0\}\)
*such that*
\(\Delta _{c} f\not \equiv 0\). *If there is*
\(a\in {C}\)
*with*
\(\delta (a, f)>0\)
*and*
\(\delta (\infty , f)=1\), *then*
\(\Delta _{c} f\)
*have infinitely many fixed points and*
\(\tau ( \Delta _{c} f)=\sigma (f)\).

### Theorem 1.2

*Let*
*f*
*be a transcendental meromorphic function of finite order in the plane*. *Suppose that*
\(c\in {C}\setminus \{0\}\)
*such that*
\(\Delta _{c} f\not \equiv 0\). *If*
\(\delta (\infty , f)=1\), \(\delta (0, f)=1\), *then*

*as*
\(r\rightarrow \infty \), \(r\notin E \), *where*
*E*
*is a possible exception set of*
*r*
*with finite logarithmic measure*.

Let \(f(z)=\frac{e^{z}}{z}\), then \(N(r, f)=\log r=S(r, f)\), \(N (r, \frac{1}{f} )=0\) and \(\Delta _{c} f=\frac{(e^{c}-1)z-1}{z(z+c)}e ^{z}\not \equiv 0\). By the second fundmental theorem, we have

and \(\tau (\Delta _{c} f)=\sigma (f)\).

## 2 Proof of Theorems 1.1 and 1.2

### Lemma 2.1

([6])

*Let*
\(f(z)\)
*be a finite order meromorphic function*, *then*, *for each*
\(k \in {N}\), \(\sigma (\Delta _{c}^{k} f)\leq \sigma (f)\).

### Lemma 2.2

([9])

*Let*
*f*
*be a transcendental meromorphic function of finite order*. *Then*, *for any positive integer*
*n*, *we have*

### Lemma 2.3

*Let*
*f*
*be a transcendental meromorphic function of finite order*. *Suppose that*
\(c\in {C}\setminus \{0\}\)
*such that*
\(\Delta _{c} f\not \equiv 0\)
*and*
\(\delta (0, f)>0\). *Then*
\(\Delta _{c} f\)
*is a transcendental and meromorphic function of finite order*.

### Proof

From Lemma 2.1, we know that \(\sigma (\Delta _{c} f)\leq \sigma (f)<+\infty \). If \(\Delta _{c} f\) is not a transcendental meromorphic function, then there is a rational function \(R(z)\) such that \(R(z)\Delta _{c} f\equiv 1\), i.e.,

Applying Lemma 2.2 and noticing that \(f(z)\) is transcendental, we have

This contradicts \(\delta (0, f)>0\). Thus \(\Delta _{c} f\) is a transcendental and meromorphic function of finite order. □

### Lemma 2.4

([11])

*Let*
\(f(z)\)
*be a transcendental meromorphic function of finite order*, *then*

### Lemma 2.5

*Let*
*f*
*be a transcendental meromorphic function of finite order*. *Then*

### Lemma 2.6

*Let*
*f*
*be a finite order transcendental meromorphic function*. *Suppose that*
\(c\in {C}\setminus \{0\}\)
*such that*
\(\Delta _{c} f\not \equiv 0\). *If*
\(\delta (0, f)>0\), *then*

### Proof

By Lemma 2.3, we know that \(\Delta _{c} f\) is a transcendental meromorphic function. Put \(F=\Delta _{c}f\), then there is \(\eta \in {C}\setminus \{0\}\) such that \(z\Delta _{\eta }F-\eta F(z)\not \equiv 0\). If not, then

holds for any \(\eta \in {C}\setminus \{0\}\). Hence \(\frac{F(z)}{z}\) is a constant, which contradicts \(F=\Delta _{c}f\) is a transcendental meromorphic function. Hence there is \(\eta \in \setminus \{0\}\) such that \(z\Delta _{\eta }F-\eta F(z)\not \equiv 0\), i.e.,

Noticing

Combining (1), (2) and Lemmas 2.2, 2.4, we can get

Applying the first fundamental theorem of Nevanlinna theory, we have

and we get

It follows from (1) that

Applying Lemma 2.3 and Lemma 2.5, we know that \((\Delta _{c}f)-z\) is a transcendental meromorphic function of finite order and

Therefore,

It follows from Lemma 2.2 and (6) that

By Lemma 2.5 and (1), we derive

Combining (3)–(5) and (7)–(8), we have

□

### 2.1 Proof of Theorem 1.1

Denoting \(g=f-a\), by Lemma 2.6, we have

Since \(\delta (a, f)>0\) and \(\delta (\infty , f)=1\), then there is a positive number \(\theta <1\) such that

Combining (9)–(11), we can get

Note that *f* is transcendental, we can get that \(\Delta _{c} f\) has infinitely many fixed points and \(\tau (\Delta _{c} f)=\sigma (f)\) from (12).

### 2.2 Proof of Theorem 1.2

Since

By the first fundamental theorem of Nevanlinna theory and (13), we can get

Hence

On the other hand, we have

It follows from Lemma 2.2 and Lemma 2.5 that

As \(\delta (\infty , f)=1\), so

Therefore

Since \(\delta (0, f)=1\) and \(\delta (\infty , f)=1\), then

By (17) and Lemma 2.6, we have

Combining (16) and (18) implies

as \(r\rightarrow \infty \).

## References

Bergweiler, W., Langley, J.K.: Zeros of differences of meromorphic functions. Math. Proc. Camb. Philos. Soc.

**142**, 133–147 (2007)Chen, Z.X.: The fixed points and hyper order of solutions of second order complex differential equations. Acta Math. Sci. Ser. A

**20**(3), 425–432 (2000)Chen, Z.X.: Fixed points of meromorphic functions and of their difference and shifts. Ann. Pol. Math.

**109**(2), 153–163 (2013)Chen, Z.X., Shon, K.H.: Fixed points of meromorphic solutions for some difference equations. Abstr. Appl. Anal.

**2013**, Article ID 496096 (2013)Chiang, Y.M., Feng, S.J.: On the Nevanlinna characteristic of \(f(z+\eta )\) and difference equations in the complex plane. Ramanujan J.

**16**, 105–129 (2008)Chiang, Y.M., Feng, S.J.: On the growth of logarithmic difference, difference equations and logarithmic derivatives of meromorphic functions. Transl. Am. Math. Soc.

**361**, 3767–3791 (2009)Chuang, C.T., Yang, C.C.: Theory of Fix Points and Factorization of Meromorphic Functions. Mathematical Monograph Series. Peking University Press, Beijing (1986)

Cui, W.W., Yang, L.Z.: Zeros and fixed points of difference operators of meromorphic functions. Acta Math. Sci.

**33B**(3), 773–780 (2013)Halburd, R.G., Korhonen, R.J.: Nevanlinna theory for the difference operator. Ann. Acad. Sci. Fenn., Math.

**31**, 463–478 (2006)Halburd, R.G., Korhonen, R.J.: Difference analogue of the lemma on the logarithmic derivative with applications to difference equations. J. Math. Anal. Appl.

**314**, 477–487 (2006)Halburd, R.G., Korhonen, R.J.: Meromorphic solutions of difference equations, integrability and the discrete Painlev’e equations. J. Phys. A, Math. Theor.

**40**, 1–38 (2007)Hayman, W.K.: Meromorphic Functions. Oxford Mathematical Monographs. Clarendon, Oxford (1964)

Lahiri, I.: Milloux theorem, deficiency and fix-points for vector-valued meromorphic functions. J. Indian Math. Soc.

**59**, 45–60 (1993)Li, N., Yang, L.Z.: Value distribution of difference and q-difference polynomials. Adv. Differ. Equ.

**2013**, 98, 1–9 (2013)Liu, K., Cao, H.Z., Cao, T.B.: Entire solutions of Fermat type differential difference equations. Arch. Math.

**99**, 147–155 (2012)Wu, Z.J.: Value distribution for difference operator of meromorphic functions with maximal deficiency sum. J. Inequal. Appl.

**2013**, 530, 1–9 (2013)Xu, H.Y., Cao, T.B., Liu, B.X.: The growth of solutions of systems of complex q-shift difference equations. Adv. Differ. Equ.

**2012**, 216, 1–22 (2012)Xu, J.F., Zhang, X.B.: The zeros of q-shift difference polynomials of meromorphic functions. Adv. Differ. Equ.

**2012**, 200, 1–10 (2012)Yang, C.C., Yi, H.X.: Uniqueness Theory of Meromorphic Functions. Mathematics and Its Application, vol. 557. Kluwer Academic, Dordrecht (2003)

Yang, L.: Value Distribution Theory. Springer, Berlin (1993) Translated and revised from the 1982 Chinese original

Zhang, R.R., Chen, Z.X.: Value distribution of difference polynomials of meromorphic functions. Sci. Sin., Math.

**42**(11), 1115–1130 (2012) (in Chinese)Zhang, R.R., Chen, Z.X.: Fixed points of meromorphic functions and of their difference, divided differences and shifts. Acta Math. Sin. New Ser.

**32**(10), 1189–1202 (2016)Zheng, J.H.: Value Distribution of Meromorphic Functions. Tsinghua University Press, Beijing (2010)

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This work was supported by the Natural Science Foundation of Hubei Provincial Department of Education(Grant No. D20182801) and by Hubei Key Laboratory of Applied Mathematics (Hubei University).

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Wu, Z., Wu, J. Fixed points of differences of meromorphic functions.
*Adv Differ Equ* **2019**, 453 (2019). https://doi.org/10.1186/s13662-019-2386-8

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DOI: https://doi.org/10.1186/s13662-019-2386-8