# On the growth of solutions of a class of second-order complex differential equations

- Cai Feng Yi
^{1}Email author, - Xu-Qiang Liu
^{1}and - Hong Yan Xu
^{2}

**2013**:188

https://doi.org/10.1186/1687-1847-2013-188

© Yi et al.; licensee Springer 2013

**Received: **19 March 2013

**Accepted: **29 May 2013

**Published: **27 June 2013

## Abstract

In this paper, we consider the differential equation ${f}^{\u2033}+h(z){e}^{P(z)}{f}^{\prime}+Q(z)f=0$, where $h(z)$ and $Q(z)\not\equiv 0$ are meromorphic functions, $P(z)$ is a non-constant polynomial. Assume that $Q(z)$ has an infinite deficient value and finitely many Borel directions. We give some conditions on $P(z)$ which guarantee that every solution $f\not\equiv 0$ of the equation has infinite order.

**MSC:**34AD20, 30D35.

## Keywords

## 1 Introduction and main results

*Borel*direction in investigating the growth of solutions of the second-order linear differential equation

where $h(z)$ and $Q(z)\not\equiv 0$ are meromorphic functions, $P(z)$ is a non-constant polynomial. We assume that the reader is familiar with the Nevanlinna theory of meromorphic functions and the basic notions such as $N(r,f)$, $m(r,f)$, $T(r,f)$ and $\delta (r,f)$. For the details, see [1] or [2].

*σ*and the hyper-order ${\sigma}_{2}$ are defined as follows:

It is well known that if $A(z)=h(z){e}^{P(z)}$ and $B(z)=Q(z)$ are transcendental entire functions in equation (1) and ${f}_{1}$, ${f}_{2}$ are two linearly independent solutions of equation (1), then at least one of ${f}_{1}$, ${f}_{2}$ must have infinite order. Hence, ‘most’ solutions of equation (1) will have infinite order. On the other hand, there are some equations of the form (1) that possess a solution $f\not\equiv 0$ which has finite order; for example, $f(z)={e}^{z}$ satisfies the equation ${f}^{\u2033}+{e}^{-z}{f}^{\prime}-({e}^{-z}+1)f=0$. Thus the main problem is what condition on $A(z)$ and $B(z)$ can guarantee that every solution $f\not\equiv 0$ of equation (1) has infinite order? There has been much work on this subject. For example, it follows from the work by Gunderson [3], Hellerstein *et al.* [4] that if $A(z)$ and $B(z)$ are entire functions with $\sigma (A)<\sigma (B)$ or $A(z)$ is a polynomial and $B(z)$ is transcendental; or if $\sigma (B)<\sigma (A)\le \frac{1}{2}$, then every solution $f\not\equiv 0$ of equation (1) has infinite order. Furthermore, if *A* is an entire function with finite order having a finite deficient value and $B(z)$ is a transcendental entire function with $\mu (B)<\frac{1}{2}$, then every solution $f\not\equiv 0$ of equation (1) has infinite order [5]. More results can be found in [6–9].

*et al.*discussed the problem correlating with this in [10]. Now we still consider equation (1) with transcendental meromorphic coefficients and discuss the growth of its meromorphic solutions. We shall also involve the deficient value and the Borel direction in the studies of the oscillation of the second-order complex differential equation. We hope that the relations between the orders of coefficient functions will not be restricted. In general, it would not hold that every solution $f\not\equiv 0$ of equation (1) has infinite order; for example, $f(z)=\frac{{e}^{z}}{z}$ satisfies

and $\sigma (f)=1<\mathrm{\infty}$.

To state our theorem, we give some remarks first. Let $P(z)=(\alpha +i\beta ){z}^{n}+\cdots $ ($\alpha ,\beta \in \mathbb{R}$) be a non-constant polynomial. Denote $\delta (P,\theta )=\alpha cosn\theta -\beta sinn\theta $, let deg*P* be the degree of $P(z)$, $\mathrm{\Omega}(\theta ,\epsilon ,r)=\{z:\theta -\epsilon <argz<\theta +\epsilon ,|z|<r\}$. In the following, we give the definition of the *Borel* direction of a meromorphic function $f(z)$.

**Definition 1.1** [11]

*Borel*direction of order

*σ*of $f(z)$ if the following equality:

holds for any real number $\epsilon >0$ and every complex number $a\in \mathbb{C}\cup \{\mathrm{\infty}\}$ with at most two exceptions.

The main results in this article are stated as follows.

**Theorem 1.1** *Let* $P(z)$ *be a non*-*constant polynomial with* $degP=n$, *let* $h(z)$ *be a meromorphic function with* $\sigma (h)<n$. *Suppose that* $Q(z)$ *is a finite*-*order meromorphic function having an infinite deficient value*, $Q(z)$ *has only finitely many Borel directions*: ${B}_{j}:argz={\theta}_{j}$ ($j=1,2,\dots ,q$). *Denote that* ${\mathrm{\Omega}}_{j}=\{z:{\theta}_{j}<argz<{\theta}_{j+1}\}$, $j=1,2,\dots ,q$ . *Suppose that there exists* ${\phi}_{j}$ (${\theta}_{j}<{\phi}_{j}<{\theta}_{j+1}$) *such that* $\delta (P,{\phi}_{j})<0$ *for each angular domain* ${\mathrm{\Omega}}_{j}$. *Then every meromorphic solution* $f\not\equiv 0$ *of equation* (1) *has infinite order and* ${\sigma}_{2}(f)\ge \sigma (Q)$.

**Remark 1.1**We apply the theorem to some particular equations. For example, when $Q(z)=g(z){e}^{bz}$, where $g(z)$ is a non-constant polynomial and $b\ne -1$. Chen proved [7] that every meromorphic solution $f\not\equiv 0$ of the equation

has infinite order with ${\sigma}_{2}(f)=1$. Except the case of $argb=0,\pi $, by the theorem, we can get the part of the results above. But this theorem includes more general forms.

From the structure of ${E}_{1}=\{\phi :\delta (P,\phi )<0\}$ and ${E}_{2}=\{\phi :\delta (P,\phi )>0\}$ in $[0,2\pi )$, we can easily get the following conclusion.

**Corollary 1.2**

*Let*$P(z)$

*be a non*-

*constant polynomial with*$degP=n$,

*let*$h(z)$

*be a meromorphic function with*$\sigma (h)<n$.

*Let*$Q(z)$

*be a transcendental meromorphic function with finite order*.

*If*$Q(z)$

*has a deficient value*∞

*and has only*

*q*

*Borel directions*${B}_{j}:argz={\theta}_{j}$ ($j=1,2,\dots ,q$)

*that satisfy*${\theta}_{1}<{\theta}_{2}<\cdots <{\theta}_{q}<{\theta}_{q+1}$ (${\theta}_{q+1}={\theta}_{1}+2\pi $)

*and*

*then every meromorphic solution* $f\not\equiv 0$ *of equation* (1) *has infinite order and* ${\sigma}_{2}(f)\ge \sigma (Q)$.

has infinite order with ${\sigma}_{2}(f)\ge n$.

## 2 Some lemmas

To prove our theorem, we need the following lemmas.

**Lemma 2.1** [12]

*Let*$(f,\mathrm{\Gamma})$

*denote a pair that consists of a transcendental meromorphic function*$f(z)$

*and a finite set*

*of distinct pairs of integers that satisfy*${k}_{i}>{j}_{i}\ge 0$

*for*$i=1,2,\dots ,q$.

*Let*$\alpha >1$

*and*$\epsilon >0$

*be given real constants*.

*Then the following three statements hold*.

- (i)
*There exists a set*${E}_{1}\subset [0,2\pi )$*that has linear measure zero*,*and there exists a constant*$c>0$*that depends only on**α**and*Γ*such that if*${\phi}_{0}\in [0,2\pi )-{E}_{1}$,*then there is a constant*${R}_{0}={R}_{0}({\phi}_{0})>1$*such that for all**z**satisfying*$argz={\phi}_{0}$*and*$|z|=r\ge {R}_{0}$,*and for all*$(k,j)\in \mathrm{\Gamma}$,*we have*$\left|\frac{{f}^{(k)}(z)}{{f}^{(j)}(z)}\right|\le c{(\frac{T(\alpha r,f)}{r}{log}^{\alpha}rlogT(\alpha r,f))}^{k-j}.$(4)

*In particular*,

*if*$f(z)$

*has finite order*$\sigma (f)$,

*then*(4)

*is replaced by*(5).

- (ii)
*There exists a set*${E}_{2}\subset (1,\mathrm{\infty})$*that has finite logarithmic measure*,*and there exists a constant*$c>0$*that depends only on**α**and*Γ*such that for all**z**satisfying*$|z|=r\notin {E}_{2}\cup [0,1]$*and for all*$(k,j)\in \mathrm{\Gamma}$,*the inequality*(4)*holds*.

*In particular*,

*if*$f(z)$

*has finite order*$\sigma (f)$,

*then the inequality*(5)

*holds*.

- (iii)
*There exists a set*${E}_{3}\subset [0,\mathrm{\infty})$*that has finite linear measure*,*and there exists a constant*$c>0$*that depends only on**α**and*Γ*such that for all**z**satisfying*$|z|=r\notin {E}_{3}$*and for all*$(k,j)\in \mathrm{\Gamma}$,*we have*$\left|\frac{{f}^{(k)}(z)}{{f}^{(j)}(z)}\right|\le c{(T(\alpha r,f){r}^{\epsilon}logT(\alpha r,f))}^{k-j}.$(6)

*In particular*,

*if*$f(z)$

*has finite order*$\sigma (f)$,

*then*(6)

*is replaced by*(7)

**Lemma 2.2** [13]

*Suppose that*$g(z)=h(z){e}^{P(z)}$,

*where*$P(z)$

*is a non*-

*constant polynomial with*$degP=n$,

*and*$h(z)$

*is a meromorphic function with*$\sigma (h)<n$.

*There exists a set*${E}_{1}\subset [0,2\pi )$

*that has linear measure zero such that for all*$\phi \in [0,2\pi )\mathrm{\setminus}{E}_{1}$,

*we have*

- (i)
*If*$\delta (P,\phi )<0$,*then there is a constant*${R}_{0}={R}_{0}(\phi )>0$*such that the inequality*$\left|g\left(r{e}^{i\phi}\right)\right|<exp\{\frac{1}{2}\delta (P,\phi ){r}^{n}\}$(8)

*holds for*$r>{R}_{0}$.

- (ii)
*If*$\delta (P,\phi )>0$,*then there is a constant*${R}_{0}^{\prime}={R}_{0}^{\prime}(\phi )>0$*such that the inequality*$\left|g\left(r{e}^{i\phi}\right)\right|>exp\{\frac{1}{2}\delta (P,\phi ){r}^{n}\}$(9)

*holds for* $r>{R}_{0}^{\prime}$.

**Lemma 2.3** [2]

*Let*$f(z)$

*be a transcendental meromorphic function with finite order*

*σ*,

*then there exists a function*$\lambda (r)$

*with the following properties*:

- (i)
$\lambda (r)$

*is a non*-*negative and continuous function for*$r\ge 0$*with*${lim}_{r\u27f6\mathrm{\infty}}\lambda (r)=\sigma $. - (ii)
$\lambda (r)$

*is a differentiable function for all**r**in*$(0,\mathrm{\infty})$*with at most countable exceptions and*${lim}_{r\u27f6\mathrm{\infty}}{\lambda}^{\prime}(r)logr=0$ . - (iii)
*The inequality*${r}^{\lambda (r)}\ge T(r,f)$*holds for all sufficiently large**r*,*and there exists a sequence*${r}_{n}$*with*${r}_{n}\to \mathrm{\infty}$*satisfying*${r}_{n}^{\lambda ({r}_{n})}=T({r}_{n},f)$.

*We shall call the function* $\lambda (r)$ *the proximate order of* $f(z)$, *and the function* $U(r)={r}^{\lambda (r)}$ *the type function of* $f(z)$.

**Lemma 2.4** [2]

*Let*$f(z)$

*be a transcendental meromorphic function with order*

*σ*($0<\sigma <\mathrm{\infty}$). ${B}_{1}:argz={\phi}_{1}$

*and*${B}_{2}:argz={\phi}_{2}$ ($0\le {\phi}_{1}<{\phi}_{2}\le 2\pi +{\phi}_{1}$)

*are two half rays starting from the origin*,

*and*

*f*

*has no Borel direction in the angular domain*${\phi}_{1}<argz<{\phi}_{2}$.

*Suppose that there exists a sequence*${r}_{n}$

*with*${r}_{n}\to \mathrm{\infty}$ ($n\to \mathrm{\infty}$)

*and a complex number*${a}_{0}$ (${a}_{0}\in \mathbb{C}\cup \mathrm{\infty}$)

*such that the following inequality*:

*holds for any given constant*$\epsilon >0$

*and all sufficiently large*

*n*

*in some rays*$argz=\phi $,

*where*${\phi}_{1}<\phi <{\phi}_{2}$.

*We denote the arc*${A}_{n}=\{{r}_{n}{e}^{i\phi}:{\phi}_{1}<\phi <{\phi}_{2}\}$

*and the angular set*${E}_{{n}^{\prime}}$

*such that any*$\phi \in {E}_{{n}^{\prime}}$

*satisfies the inequality*(10).

*If there exists a constant*${K}_{1}>0$ (

*not dependent on*

*ε*)

*such that*$meas{E}_{{n}^{\prime}}>{K}_{1}$,

*then we can get a list of curve segment*${L}_{n}$

*satisfying the following two conditions for any given*${K}_{2}$ (${K}_{2}>0$)

*and sufficiently small*$\alpha >0$:

- (i)${L}_{n}$
*lies in the area of*${\phi}_{1}+8\alpha \le argz\le {\phi}_{2}-8\alpha $, ${r}_{n-1}\le |z|\le {r}_{n}$,*whose end points respectively for*${r}_{n}{e}^{i({\phi}_{1}+{\phi}_{j}^{\prime})}$*and*${r}_{n}{e}^{i({\phi}_{2}-{\phi}_{j}^{\prime})}$ ($8\alpha \le {\phi}_{j}^{\prime}\le 9\alpha $),*and we have the following inequality*:$meas\{\phi :{r}_{n}{e}^{i\phi}\in {A}_{n}-{L}_{n}\}<{K}_{2}.$(11) - (ii)
*For any positive number*$\eta >0$,*the inequality*$\{\begin{array}{ll}log\frac{1}{|f(z)-{a}_{0}|}>{r}_{n}^{\sigma -\eta},& {a}_{0}\ne \mathrm{\infty},\\ log|f(z)|>{r}_{n}^{\sigma -\eta},& {a}_{0}=\mathrm{\infty},\end{array}$(12)

*holds for sufficiently large* *n* *and* $z\in {L}_{n}$.

**Lemma 2.5**

*Let*$f(z)$

*be a transcendental meromorphic function with order*

*σ*

*having an infinite deficient value*.

*If*$f(z)$

*has*

*q*

*Borel directions*, ${B}_{j}:argz={\theta}_{j}$ ($j=1,2,\dots ,q$),

*and these half*-

*rays divide the whole complex plane into*

*q*

*angular domains*, ${\mathrm{\Omega}}_{j}=\{z:{\theta}_{j}<argz<{\theta}_{j+1}\}$, $j=1,2,\dots ,q$, ${\theta}_{q+1}={\theta}_{1}+2\pi $,

*then for any given constant*$\eta >0$

*and*$\xi >0$,

*there exists an angular domain*${\mathrm{\Omega}}_{{j}_{0}}$

*at least and a sequence*${R}_{n}$

*with*${R}_{n}\to \mathrm{\infty}$ ($n\to \mathrm{\infty}$)

*such that the following inequality*:

*holds for all sufficiently large*

*n*,

*where*

*Proof*Let $\lambda (r)$ be a proximate order of $f(z)$ with a type function $U(r)={r}^{\lambda (r)}$. According to the properties of $\lambda (r)$ of Lemma 2.3, there exists a sequence ${r}_{n}$ with ${r}_{n}\to \mathrm{\infty}$ satisfying ${lim}_{{r}_{n}\u27f6\mathrm{\infty}}\frac{T({r}_{n},f)}{U({r}_{n})}=1$. Let ${b}_{\nu}$ ($\nu =1,2,\dots ,n$ ($3{r}_{n},f=\mathrm{\infty}$)) be all the poles of $f(z)$ in $|z|\le 3{r}_{n}$. For every ${r}_{n}$, by the Boutroux-Cartan theorem [2], we have

*n*, we can choose ${R}_{n}\in [{r}_{n},2{r}_{n}]$ satisfying $\{z:|z|={R}_{n}\}\cap ({\gamma}_{n})=\mathrm{\varnothing}$. By the Poisson-Jensen formula and (14), for any

*z*satisfying $|z|={R}_{n}$, we have

where $K=5+\frac{log25e}{log\frac{4}{3}}$.

holds for all $n>{N}_{0}$.

Since the whole complex plane is divided into *q* angular domains and there is no *Borel* direction in them, the circle $|z|={R}_{n}$ is also divided into *q* arcs: ${A}_{nj}:\{{R}_{n}{e}^{i\phi}:{\theta}_{j}<\phi <{\theta}_{j+1}\}$ ($j=1,2,\dots ,q$).

holds for all sufficiently large *n*.

*n*, there exists a curve ${L}_{n,{j}_{0}}$ such that (11) and (12) hold. So, for any given $\eta >0$, we have

The proof of Lemma 2.5 is completed. □

## 3 Proof of Theorem 1.1

*Proof*Suppose that $f\not\equiv 0$ is a meromorphic solution of equation (1) with $\sigma (f)<\mathrm{\infty}$. We shall seek for a contradiction. From equation (1), we have the following inequality:

holds for all $z=|z|{e}^{i\phi}$ with $\phi \notin {E}_{1}$ and $|z|>{R}_{0}$.

holds for all $z=r{e}^{i\phi}$ satisfying $r>{R}_{0}^{\prime}$ and $\phi \in E\mathrm{\setminus}{E}_{2}$.

Denote that ${\mathrm{\Omega}}_{j}=\{z:{\theta}_{j}<argz<{\theta}_{j+1}\}$, $j=1,2,\dots ,q$. Applying Lemma 2.5 to $Q(z)$, then for any given constants $\eta >0$ and $\xi >0$, there exists an angular domain ${\mathrm{\Omega}}_{{j}_{0}}$ and a sequence ${r}_{m}$ with ${r}_{m}\to \mathrm{\infty}$ ($m\to \mathrm{\infty}$) such that (13) holds for all sufficiently large *m*.

*m*, we can choose ${\phi}_{m}\in [{\theta}_{1}^{\prime},{\theta}_{2}^{\prime}]\mathrm{\setminus}({E}_{1}\cup {E}_{2})$ such that (19), (20) and the inequality

Obviously, when *m* is sufficiently large, this is a contradiction.

Next, we will prove ${\sigma}_{2}(f)\ge \sigma (Q)$.

*z*satisfying $|z|=r>{R}_{0}^{\u2033}$ and $argz\notin {E}_{3}$, the following inequality holds:

*m*, we can choose ${\phi}_{m}^{\prime}\in [{\theta}_{1}^{\prime},{\theta}_{2}^{\prime}]\mathrm{\setminus}({E}_{2}\cup {E}_{3})$ such that (20), (21) and (23) hold for ${z}_{m}={r}_{m}{e}^{i{\phi}_{m}^{\prime}}$. From (18), (20), (21) and (23), we get

As *η* can be arbitrary small, we have ${\sigma}_{2}(f)\ge \sigma (Q)$.

The proof of the theorem is completed. □

## Declarations

### Acknowledgements

The authors thank the referee for his/her valuable suggestions to improve the present article. This work was supported by the NSFC (11171170, 61202313), the Natural Science Foundation of Jiang-Xi Province in China (Grant No. 2010GQS0119, No. 20132BAB211001 and No. 20122BAB201016).

## Authors’ Affiliations

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