# Some results about functions that share functions with their derivative of higher order

- Jianming Qi
^{1}, - Feng Lü
^{2}and - Wenjun Yuan
^{3, 4}Email author

**2013**:192

https://doi.org/10.1186/1687-1847-2013-192

© Qi et al.; licensee Springer 2013

**Received: **7 January 2013

**Accepted: **29 May 2013

**Published: **1 July 2013

## Abstract

In this paper, we investigate the growth of some functions that share functions with their derivative of higher order. The first main theorem is an improvement of the result obtained by Lü (Bull. Korean Math. Soc. 48: 951-957, 2011), two examples are given to show that the conclusion is sharp. The second main theorem is of estimating, more exactly, the order of an entire function sharing polynomial, which extends the related result of Lü, Xu and Chen (Arch. Math. 92: 593-601, 2009).

**MSC:**30D35, 30D45.

## Keywords

## 1 Introduction and main results

*f*, which are respectively defined as (see [1])

In addition, we say that two meromorphic functions $f(z)$ and $g(z)$ share a finite value *a* IM (ignoring multiplicities) when $f(z)-a$ and $g(z)-a$ have the same zeros. And we say that $f(z)$ and $g(z)$ share a finite value *a* CM (counting multiplicities) when $f(z)-a$ and $g(z)-a$ have the same zeros counting multiplicities. $degP(z)$ denotes the degree of the polynomial $P(z)$. If $R(z)=\frac{{P}_{1}(z)}{{P}_{2}(z)}$ is a rational function (where ${P}_{2}(z)\not\equiv 0$ and ${P}_{1}(z)$, ${P}_{2}(z)$ are two coprime polynomials), then we indicate $degR(z)=max\{deg{P}_{1}(z),deg{P}_{2}(z)\}$ to denote the degree of the rational function.

The subject on sharing values between entire functions and their derivatives was first studied by Rubel and Yang [2]. In 1977, they proved the result that if a nonconstant entire function *f* and its first derivative ${f}^{\prime}$ share two distinct finite numbers *a*, *b* CM, then $f\equiv {f}^{\prime}$. Since then, shared value problems have been studied by many authors and a number of profound results have been obtained (see, *e.g.*, [3, 4]).

In 1982, Bank and Laine [5] investigated the complex oscillation theory of differential equations and obtained the following main result.

**Theorem A** *Let* $A(z)$ *be a nonconstant polynomial of degree* *n*, *and let* ${f}_{1}$ *and* ${f}_{2}$ *be two linearly independent solutions of the equation* ${f}^{\u2033}+A(z)f=0$. *Then at least one of* ${f}_{1}$ *and* ${f}_{2}$ *has the property that the exponent of convergence of its zero*-*sequence is* $\frac{n+2}{2}$.

Since then, to study properties of the exponent of convergence, the order and the hyper-order for the solutions of some differential equations becomes a hot topic and is discussed by many experts.

In 2008, Li and Gao [6] deduced the following result.

**Theorem B**

*Let*${Q}_{1}$

*and*${Q}_{2}$

*be two nonzero polynomials*,

*and let*

*P*

*be a polynomial*.

*If*

*f*

*is a nonconstant solution of the equation*

*then* $\sigma (f)=n$, *where*, *and in the sequel*, *n* *denotes the degree of* *P*.

Recently, Lü [7] obtained the result.

**Theorem C** *Let* *f* *be a transcendental meromorphic function with finitely many poles*, *let* $n\ge 2$ *be an integer*, *and let* $\alpha =P{e}^{Q}$ ($\ne {\alpha}^{\prime}$) *be an entire function such that the order of* *α* *is less than that of* *f*, *where* *P*, *Q* *are two polynomials*. *If* ${f}^{n}$ *and* ${({f}^{n})}^{\prime}$ *share* *α* *CM*, *then* $f=A{e}^{\frac{1}{n}z}$, *where* *A* *is a nonzero constant*.

From Theorem C, we see that ${f}^{n}$ and ${({f}^{n})}^{\prime}$ share a function with finite order. So, it is natural to ask what will happen if they share functions with infinite order and also what will happen if ${({f}^{n})}^{\prime}$ is replaced by ${({f}^{n})}^{(k)}$. In this work, we discuss these problems and derive the following result.

**Theorem 1.1**

*Let*

*f*

*be a meromorphic function with finitely many poles*,

*let*

*R*

*be a rational function*,

*γ*

*be an entire function*.

*If all zeros of*

*f*

*have multiplicity at least*$k+1$

*and*

*where* $\alpha =R{e}^{\gamma}$, *then* $\sigma (f)\le \sigma (\alpha )=\rho (\gamma )$.

**Remark 1** The following examples show that the conclusion $\sigma (f)\le \rho (\gamma )$ is sharp.

**Example 1**Let $f(z)=A{e}^{z}$, where

*A*is a nonzero constant. Let $\alpha (z)={e}^{{e}^{z}+{z}^{2}}$. Noting that $f\equiv {f}^{(k)}$, we have

Thus, $\sigma (f)=0<\sigma (\alpha )=1$.

**Example 2**Let $f(z)=z{e}^{{z}^{2}}+{e}^{z/2}$, $\alpha (z)=(4{z}^{2}-z+2){e}^{{z}^{2}}$, and $\gamma (z)={z}^{2}$. Then

Thus, $\sigma (f)=\rho (\gamma )=0$.

**Remark 2** If *γ* is a polynomial, then the above condition obviously holds.

**Remark 3** In Theorem 1.1, if the order of *γ* is zero, for example, *γ* is a polynomial, then $\sigma (f)=0$.

In 2009, Lü, Xu and Chen [8] obtained the following result.

**Theorem D**

*Let*$f(z)$

*be a nonconstant meromorphic function with finitely many poles*,

*and let*${Q}_{1}$, ${Q}_{2}$ ($\not\equiv {Q}_{1}$)

*be two polynomials*.

*If*

*then* $f(z)$ *is of finite order*.

In this paper, we get the following results which improve Theorem D.

**Theorem 1.2**

*Let*$f(z)$

*be a nonconstant meromorphic function with finitely many poles*,

*and let*${Q}_{1}$, ${Q}_{2}$ ($\not\equiv {Q}_{1}$)

*be two polynomials*.

*If*

*then* $\rho (f)\le 2+2max\{deg{R}_{1}(z),deg{R}_{2}(z)\}$, *where* ${R}_{1}=\frac{{Q}_{2}-{Q}_{1}^{\prime}}{{Q}_{2}-{Q}_{1}}$, ${R}_{2}=\frac{{Q}_{1}-{Q}_{1}^{\prime}}{{Q}_{2}-{Q}_{1}}$ *are rational functions*.

**Corollary 1.3** *Let* $f(z)$ *be a nonconstant entire function and let* ${Q}_{1}$, ${Q}_{2}$ ($\not\equiv {Q}_{1}$) *be two polynomials*. *If* $f(z)={Q}_{1}(z)\Rightarrow {f}^{\prime}(z)={Q}_{1}(z)$ *and* $f(z)={Q}_{2}(z)\Rightarrow {f}^{\prime}(z)={Q}_{2}(z)$, *then* $\rho (f)\le 1+max\{deg{R}_{1}(z),deg{R}_{2}(z)\}$, *where* ${R}_{1}=\frac{{Q}_{2}-{Q}_{1}^{\prime}}{{Q}_{2}-{Q}_{1}}$, ${R}_{2}=\frac{{Q}_{1}-{Q}_{1}^{\prime}}{{Q}_{2}-{Q}_{1}}$ *are rational functions*.

Very recently, Li, Gao and Zhang [9] proved the following result.

**Theorem E** *Let* *f* *be a nonconstant entire function*. *If* *f* *and* ${f}^{\prime}$ *share the value* 1 *CM*, *and if* $N(r,\frac{1}{{f}^{\prime}})<\alpha T(r,f)$, *where* $\alpha \in [0,\frac{1}{4}]$, *then* ${f}^{\prime}-1=c(f-1)$ *for some nonzero constant* *c*.

By the same method of Li, Gao and Zhang [9], we also consider the *k* th derivative and improve the above result as follows.

**Theorem 1.4** *Let* *f* *be a nonconstant entire function*, *and let* *k* *be a positive integer*. *If* *f* *and* ${f}^{(k)}$ *share the value* 1 *CM and if* $N(r,\frac{1}{{f}^{(k)}})<\alpha T(r,f)$, *where* $\alpha \in [0,\frac{1}{4})$, *then* ${f}^{(k)}-1=c(f-1)$ *for some nonzero constant* *c*.

## 2 Some lemmas

In order to prove our theorems, we need the following lemmas.

**Lemma 2.1** [3]

*Let*ℱ

*be a family of meromorphic functions in the unit disc*△

*with the property that for each*$f(z)\in \mathcal{F}$,

*all zeros of*$f(z)$

*have multiplicity at least*$k+1$.

*If*

*k*

*is a positive integer and*${a}_{n}\to a$, $|a|<1$

*and*${f}_{n}^{\mathrm{\u266f}}({a}_{n})\to \mathrm{\infty}$,

*there exist*

- 1.
*a subsequence of functions*${f}_{n}\in \mathcal{F}$ (*also denoted by*${f}_{n}$); - 2.
*a sequence of complex numbers*${z}_{n}\to {z}_{0}$, $|{z}_{0}|<1$; - 3.
*a positive sequence*${\rho}_{n}\to 0$; - 4.$\frac{{f}_{n}({z}_{n}+{\rho}_{n}\xi )}{{\rho}_{n}^{k}}={g}_{n}(\xi )\to g(\xi )$,
*here**g**is a nonconstant meromorphic*(*entire*)*function satisfying*${g}^{\mathrm{\u266f}}(\xi )\le {g}^{\mathrm{\u266f}}(0)=k+1$*and*${\rho}_{n}\le \frac{M}{\sqrt[k+1]{{f}_{n}^{\mathrm{\u266f}}({a}_{n})}},$

*here* *M*, *n* *are respective positive numbers*.

With a similar method to that in [[3], Lemma 2], we obtain the following Lemma 2.3, which plays an important part in the proof of Theorem 1.1. For the sake of convenience, the detailed proof will be given after Lemma 2.3.

**Lemma 2.2** *Let* *f* *be a meromorphic function of hyper*-*order* $\sigma (f)>0$. *Then*, *for any* $\u03f5>0$, *there exists a sequence* ${z}_{n}\to \mathrm{\infty}$ *such that* ${f}^{\mathrm{\u266f}}({z}_{n})>{e}^{{|{z}_{n}|}^{\sigma (f)-\u03f5}}$ *if* *n* *is large enough*.

*Proof*On the contrary, there exist $\u03f5>0$ and $R>0$ such that $\u03f5<\sigma (f)$ and for all

*z*, $|z|\ge R$ satisfying ${f}^{\mathrm{\u266f}}(z)\le {e}^{{|z|}^{\sigma (f)-\u03f5}}$. Thus,

*f*, we have

*f*is

a contradiction. Thus, the proof is completed. □

**Lemma 2.3** [9]

*Let*

*f*

*be a nonconstant entire function and let*$k\ge 1$

*be positive*.

*Suppose that*

*f*

*and*${f}^{(k)}$

*share the value*1

*CM*.

*Then*$f-1$

*has infinitely many zeros such that each zero of*$f-1$

*is of order at most*

*k*,

*and*

*for any*$\u03f5>0$

*and large enough*

*r*,

*and*

**Remark 4** There exist some mistakes in Theorem 2.2 and its proof and the proof of Theorem 3.1 in [9], we will correct them in another paper.

**Lemma 2.4** [1]

*Let*

*f*

*be a meromorphic function and*$k\ge 1$

*be positive*.

*Then*

**Lemma 2.5** [4]

*Let*$f(z)$

*be a meromorphic function in the complex plane*, $\rho (f)>2$,

*then for each*$0<\mu <\frac{\rho (f)-2}{2}$,

*there exist points*${a}_{n}\to \mathrm{\infty}$ ($n\to \mathrm{\infty}$)

*such that*

**Lemma 2.6** [10]

*Let*$f(z)$

*be an entire function in the complex plane*, $\rho (f)>1$,

*then for each*$0<\mu <\rho (f)-1$,

*there exist points*${a}_{n}\to \mathrm{\infty}$ ($n\to \mathrm{\infty}$)

*such that*

## 3 The proof of Theorem 1.1

*The proof of Theorem 1.1* In a similar way to that of [8, 11], we prove Theorem 1.1 as follows.

Noting that *f* and *R* have at most finitely many poles, there exists a positive number *r* such that *f* and *R* have no poles in $D=\{z:|z|\ge r\}$. Then *f* and *R* are holomorphic in *D*.

Noting that $\alpha =R{e}^{\gamma}$, we have $\sigma (\alpha )=\rho (\gamma )$, and then we just need to prove $\sigma (f)\le \rho (\gamma )$.

*n*. Define ${D}_{1}=\{z:|z|<1\}$ and

Then all ${F}_{n}(z)$ are holomorphic in ${D}_{1}$.

It follows from Marty’s criterion that ${({F}_{n})}_{n}$ is not normal at $z=0$.

*g*is a nonconstant entire function of order at most 1, all zeros of

*g*have multiplicity at least $k+1$, and

for a positive number *M*.

where *A* is a positive constant and *q* is an integer.

*k*. It contradicts with all zeros of

*g*having multiplicity at least $k+1$. Suppose that ${g}^{(k)}({\zeta}_{0})=1$, then by Hurwitz’s theorem there exist ${\zeta}_{n}$, ${\zeta}_{n}\to {\zeta}_{0}$ such that (for sufficiently large

*n*)

Noting that ${f}^{(k)}(z)=\alpha (z)\Rightarrow f(z)=\alpha (z)$ and (3.2), we obtain $g({\zeta}_{0})=\mathrm{\infty}$. It contradicts with the assumption that ${g}^{(k)}({\zeta}_{0})=1$. So, ${g}^{(k)}(\zeta )\ne 1$.

Next, by the famous Hayman inequality for $g(\zeta )$, it is easy to obtain contradiction.

Hence, we complete the proof of Theorem 1.1. □

## 4 Proof of Theorem 1.2

Let $P={Q}_{2}-{Q}_{1}$ and ${P}_{2}={Q}_{2}-{Q}_{1}^{\prime}$. Then $P\not\equiv 0$. We consider the function $F=\frac{H}{P}$, obviously, $\rho (F)=\rho (f)$

*P*is a polynomial, we know that for any $\u03f5>0$, there exists an ${r}_{1}>0$ such that

*F*is holomorphic in

*D*. Without loss of generality, we may assume that $|{w}_{n}|\ge r+1$ for all

*n*. We define ${D}_{1}=\{z:|z|<1\}$ and

Then all ${F}_{n}(z)$ are holomorphic in ${D}_{1}$ and ${F}_{n}^{\mathrm{\u266f}}(0)={F}^{\mathrm{\u266f}}({w}_{n})\to \mathrm{\infty}$ as $n\to \mathrm{\infty}$. It follows from Marty’s criterion that ${({F}_{n})}_{n}$ is not normal at $z=0$.

*g*and

*M*. Let ${G}_{n}(\zeta )={\rho}_{n}\frac{{H}^{\prime}({w}_{n}+{z}_{n}+{\rho}_{n}\zeta )}{P({w}_{n}+{z}_{n}+{\rho}_{n}\zeta )}$, then from (4.2) and $|\frac{{P}^{\prime}({w}_{n}+{z}_{n}+{\rho}_{n}\zeta )}{P({w}_{n}+{z}_{n}+{\rho}_{n}\zeta )}|<1$ as $n\to \mathrm{\infty}$, we get

locally uniformly in ℂ.

*n*sufficiently large)

Thus $g(\zeta )=1\Rightarrow {g}^{\prime}(\zeta )=0$, which yields that the zeros of $g-1$ are of multiplicity at least 2. Similarly, we can prove that the zeros of *g* are of multiplicity at least 2.

*m*(≥2), then ${g}^{(m)}({\xi}_{0})\ne 0$. Thus there exists a positive number

*δ*such that

on ${D}_{\delta}^{o}=\{z:0<|\zeta -{\xi}_{0}|<\delta \}$.

*n*large enough,

so each ${\zeta}_{n,j}$ is a simple zero of $H({w}_{n}+{z}_{n}+{\rho}_{n}\zeta )$, that is, ${\zeta}_{n,j}\ne {\zeta}_{n,i}$ ($1\le i\ne j\le m$).

here ${l}_{2}=deg{R}_{2}=deg\frac{{Q}_{1}-{Q}_{1}^{\prime}}{{Q}_{2}-{Q}_{1}}$.

Noting (4.8), (4.10) and that ${K}_{n}(\zeta )$ has *m* zeros ${\zeta}_{n,j}$ ($j=1,2,\dots ,m$) in ${D}_{\delta /2}$, we obtain from Hurwitz’s theorem that ${\xi}_{0}$ is a zero of ${g}^{\prime}(\zeta )$ with multiplicity *m*, and thus ${g}^{(m)}({\xi}_{0})=0$. This is a contradiction. Hence $g(\zeta )\ne 0$.

We have shown that *g* is a nonvanishing entire function that takes the value 1 always with multiplicity at least 2. But this contradicts Nevanlinna’s second fundamental theorem that the sum of the defects is at most 2.

This completes the proof of Theorem 1.2.

## 5 The proof of Corollary 1.3

By Lemma 2.6, the reader could give the proof of Corollary 1.3 with almost the same argument as that in the proof of Theorem 1.2. Here we omit it.

## 6 The proof of Theorem 1.4

By a similar way to that of Li, Gao and Zhang [9], we prove Theorem 1.4 as follows.

*f*is an entire function, by Lemma 2.4 we have

*F*is a meromorphic function and hence

*F*appear to the zeros of ${f}^{(k)}$ and ${f}^{(k+1)}$, by assumption and (6.1), we have

Combining (6.3) with (6.4), we see that $T(r,F)<2\alpha T(r,f)+S(r,f)$.

*k*multiples. Let ${z}_{0}$ be a common zero of $f-1$ and ${f}^{(k)}-1$. Then ${f}^{(k+1)}({z}_{0})\ne 0$, and it is easy to see that

*F*is holomorphic at ${z}_{0}$, and $F({z}_{0})=0$. Thus we have

*r*. However, by Lemma 2.3, we have

for large enough *r*. (6.6) contradicts to (6.5).

*A*is a nonzero constant. Since

*f*and ${f}^{(k)}$ share 1 CM, for any point ${z}_{1}$ satisfying that $f({z}_{1})={f}^{(k)}({z}_{1})=1$, we obtain that ${f}^{\prime}({z}_{1}){f}^{(k+1)}({z}_{1})\ne 0$, and then $A=\frac{{f}^{(k+1)}({z}_{1})}{{[{f}^{\prime}({z}_{1})]}^{2}}$. Thus if we assume that $\frac{{f}^{(k+1)}}{{f}^{(k)}}$ is not a constant function, we see from (6.7) that ${f}^{(k)}(z)\ne 0$ and ${f}^{(k+1)}(z)\ne 0$. Noting that $f(z)$ is an entire function, we see that $\frac{{f}^{(k+1)}}{{f}^{(k)}}$ has no both pole and zero. So $A\frac{{f}^{(k+1)}}{{f}^{(}k)}=:{e}^{h(z)}$ is a small function of ${f}^{(k)}$, where $h(z)$ is an entire function. Now we may change (6.7) into

a contradiction.

*c*such that

This completes the proof of Theorem 1.4.

## Declarations

### Acknowledgements

This work was supported by the Visiting Scholar Program of Chern Institute of Mathematics at Nankai University when the authors worked as visiting scholars. The authors would like to express their hearty thanks to Chern Institute of Mathematics that provided very comfortable research environment to them. Project supported by the funding scheme for training young teachers in colleges and universities in Shanghai (ZZSDJ12020), also supported by the NNSF of China (No. 11271090, 11171184, 11001057), the NSF of Guangdong Province (S2012010010121) and by project 10XKJ01, 12C401 and 12C104 from the Leading Academic Discipline Project of Shanghai Dianji University. The authors wish to thank the referee and managing editor for their very helpful comments and useful suggestions.

## Authors’ Affiliations

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