Theory and Modern Applications

# Complex oscillation of a second-order linear differential equation with entire coefficients of $\left[p,q\right]-\phi$ order

## Abstract

In this paper, the authors investigate the interaction between the growth, zeros of solutions with the coefficients of second-order linear differential equations in terms of $\left[p,q\right]-\phi$ order and obtain some results in general form.

MSC:30D35, 34A20.

## 1 Introduction and notations

In this paper, we shall assume that readers are familiar with the standard notations of Nevanlinna value distribution theory (see [13]). The theory of complex linear equations has been developed since 1960s. Many authors have investigated the second-order linear differential equation

${f}^{″}+A\left(z\right)f=0,$
(1.1)

where $A\left(z\right)$ is an entire function or a meromorphic function of finite order or finite iterated order, and have obtained many results about the interaction between the solutions and the coefficient of (1.1) (see [47]). What about the case when $A\left(z\right)$ is an entire function of $\left[p,q\right]$-order or more general growth? In the following, we will introduce some notations about $\left[p,q\right]$-order, where p and q are two positive integers and satisfy $p\ge q\ge 1$ throughout this paper (see [811]). Firstly, for $r\in \left[0,+\mathrm{\infty }\right)$, we define ${exp}_{1}r={e}^{r}$ and ${exp}_{i+1}r=exp\left({exp}_{i}r\right)$, $i\in \mathbb{N}$, and for all sufficiently large r, we define ${log}_{1}r=logr$ and ${log}_{i+1}r=log\left({log}_{i}r\right)$, $i\in \mathbb{N}$. Especially, we have ${exp}_{0}r=r={log}_{0}r$ and ${exp}_{-1}r={log}_{1}r$. Secondly, we denote the linear measure and the logarithmic measure of a set $E\subset \left(1,+\mathrm{\infty }\right)$ by $mE={\int }_{E}\phantom{\rule{0.2em}{0ex}}dt$ and ${m}_{l}E={\int }_{E}\frac{dt}{t}$.

Definition 1.1 ([10])

If $f\left(z\right)$ is a meromorphic function, the $\left[p,q\right]$-order of $f\left(z\right)$ is defined by

${\sigma }_{\left[p,q\right]}\left(f\right)=\underset{r\to \mathrm{\infty }}{\overline{lim}}\frac{{log}_{p}T\left(r,f\right)}{{log}_{q}r}.$
(1.2)

Especially, if $f\left(z\right)$ is an entire function, then the $\left[p,q\right]$-order of $f\left(z\right)$ is defined by (see [8, 9, 11, 12])

${\sigma }_{\left[p,q\right]}\left(f\right)=\underset{r\to \mathrm{\infty }}{\overline{lim}}\frac{{log}_{p}T\left(r,f\right)}{{log}_{q}r}=\underset{r\to \mathrm{\infty }}{\overline{lim}}\frac{{log}_{p+1}M\left(r,f\right)}{{log}_{q}r}.$
(1.3)

Remark 1.1 We use ${\sigma }_{\left[1,1\right]}\left(f\right)=\sigma \left(f\right)$ and ${\sigma }_{\left[p,1\right]}\left(f\right)={\sigma }_{p}\left(f\right)$ to denote the order and the iterated order of a function $f\left(z\right)$.

Definition 1.2 ([10, 13])

The growth index (or the finiteness degree) of the iterated order of a meromorphic function $f\left(z\right)$ is defined by

Remark 1.2 By Definition 1.2, we can similarly give the definition of the growth index of the iterated exponent of convergence of the zero-sequence of a meromorphic function $f\left(z\right)$ by ${i}_{\lambda }\left(f,0\right)$.

Definition 1.3 ([10, 11])

The $\left[p,q\right]$ exponent of convergence of the (distinct) zero-sequence of a meromorphic function $f\left(z\right)$ is respectively defined by

${\lambda }_{\left[p,q\right]}\left(f\right)=\underset{r\to \mathrm{\infty }}{\overline{lim}}\frac{{log}_{p}n\left(r,\frac{1}{f}\right)}{{log}_{q}r}=\underset{r\to \mathrm{\infty }}{\overline{lim}}\frac{{log}_{p}N\left(r,\frac{1}{f}\right)}{{log}_{q}r},$
(1.4)
${\overline{\lambda }}_{\left[p,q\right]}\left(f\right)=\underset{r\to \mathrm{\infty }}{\overline{lim}}\frac{{log}_{p}\overline{n}\left(r,\frac{1}{f}\right)}{{log}_{q}r}=\underset{r\to \mathrm{\infty }}{\overline{lim}}\frac{{log}_{p}\overline{N}\left(r,\frac{1}{f}\right)}{{log}_{q}r}.$
(1.5)

Definition 1.4 ([10])

The $\left[p,q\right]$ exponent of convergence of the (distinct) pole-sequence of a meromorphic function $f\left(z\right)$ is respectively defined by

${\lambda }_{\left[p,q\right]}\left(\frac{1}{f}\right)=\underset{r\to \mathrm{\infty }}{\overline{lim}}\frac{{log}_{p}n\left(r,f\right)}{{log}_{q}r},$
(1.6)
${\overline{\lambda }}_{\left[p,q\right]}\left(\frac{1}{f}\right)=\underset{r\to \mathrm{\infty }}{\overline{lim}}\frac{{log}_{p}\overline{n}\left(r,f\right)}{{log}_{q}r}.$
(1.7)

Remark 1.3 We use ${\lambda }_{\left[1,1\right]}\left(f\right)=\lambda \left(f\right)$, ${\lambda }_{\left[p,1\right]}\left(f\right)={\lambda }_{p}\left(f\right)$ and ${\lambda }_{\left[1,1\right]}\left(\frac{1}{f}\right)=\lambda \left(\frac{1}{f}\right)$, ${\lambda }_{\left[p,1\right]}\left(\frac{1}{f}\right)={\lambda }_{p}\left(\frac{1}{f}\right)$ to denote the (iterated) exponent of convergence of the zero-sequence and pole-sequence of a meromorphic function $f\left(z\right)$.

Recently, some authors have investigated the exponent of convergence of the zero-sequence and pole-sequence of the solutions of second-order linear differential equations (see [1315]) and have obtained the following results.

Theorem A ([5])

Let A be a transcendental meromorphic function of order $\sigma \left(A\right)$, where $0<\sigma \left(A\right)\le \mathrm{\infty }$, and assume that $\overline{\lambda }\left(A\right)<\sigma \left(A\right)$. Then, if $f\not\equiv 0$ is a meromorphic solution of (1.1), we have

$\sigma \left(A\right)\le max\left\{\overline{\lambda }\left(f\right),\overline{\lambda }\left(\frac{1}{f}\right)\right\}.$

Theorem B ([13])

Let $A\left(z\right)$ be an entire function with $i\left(A\right)=p\in {\mathbb{N}}_{+}$. Let ${f}_{1}$, ${f}_{2}$ be two linearly independent solutions of (1.1) and denote $F={f}_{1}{f}_{2}$. Then ${i}_{\lambda }\left(F,0\right)\le p+1$ and

${\lambda }_{p+1}\left(F,0\right)={\sigma }_{p+1}\left(F\right)=max\left\{{\lambda }_{p+1}\left({f}_{1},0\right),{\lambda }_{p+1}\left({f}_{2},0\right)\right\}\le {\sigma }_{p}\left(A\right).$

If ${i}_{\lambda }\left(F,0\right)\le p$, then ${i}_{\lambda }\left(f,0\right)=p+1$ holds for all solutions of type $f={c}_{1}{f}_{1}+{c}_{2}{f}_{2}$, where ${c}_{1}{c}_{2}\ne 0$.

Theorem C ([13])

Let $A\left(z\right)$ be an entire function with $0, let f be any non-trivial solution of (1.1), and assume ${\overline{\lambda }}_{p}\left(A,0\right)<{\sigma }_{p}\left(A\right)\ne 0$. Then ${\lambda }_{p+1}\left(f,0\right)\le {\sigma }_{p}\left(A\right)\le {\lambda }_{p}\left(f,0\right)$.

Theorem D ([13])

Let $A\left(z\right)$ be an entire function with $i\left(A\right)=p$ and ${\sigma }_{p}\left(A\right)=\sigma <\mathrm{\infty }$. Let ${f}_{1}$ and ${f}_{2}$ be two linearly independent solutions of (1.1) such that $max\left\{{\lambda }_{p}\left({f}_{1},0\right),{\lambda }_{p}\left({f}_{2},0\right)\right\}<\sigma$. Let $\mathrm{\Pi }\left(z\right)\not\equiv 0$ be any entire function for which either $i\left(\mathrm{\Pi }\right) or $i\left(\mathrm{\Pi }\right)=p$ and ${\sigma }_{p}\left(\mathrm{\Pi }\right)<\sigma$. Then any two linearly independent solutions ${g}_{1}$ and ${g}_{2}$ of the differential equation ${y}^{″}+\left(A\left(z\right)+\mathrm{\Pi }\left(z\right)\right)y=0$ satisfy $max\left\{{\lambda }_{p}\left({g}_{1}\right),{\lambda }_{p}\left({g}_{2}\right)\right\}\ge \sigma$.

Theorem E ([14])

Let A be a meromorphic function with $i\left(A\right)=p\in {\mathbb{N}}_{+}$, and assume that ${\overline{\lambda }}_{p}\left(A\right)<{\sigma }_{p}\left(A\right)$. Then, if f is a nonzero meromorphic solution of (1.1), we have

${\sigma }_{p}\left(A\right)\le max\left\{{\overline{\lambda }}_{p}\left(f\right),{\overline{\lambda }}_{p}\left(\frac{1}{f}\right)\right\}.$

In the special case where either $\delta \left(\mathrm{\infty },f\right)>0$ or the poles of f are of uniformly bounded multiplicities, we can conclude that

$max\left\{{\lambda }_{p+1}\left(f\right),{\lambda }_{p+1}\left(\frac{1}{f}\right)\right\}\le {\sigma }_{p}\left(f\right)\le \left\{{\overline{\lambda }}_{p}\left(f\right),{\overline{\lambda }}_{p}\left(\frac{1}{f}\right)\right\}.$

In [16], Chyzhykov and his co-authors introduced the definition of φ-order of $f\left(z\right)$, where $f\left(z\right)$ is a meromorphic function in the unit disc and used it to investigate the interaction between the analytic coefficients and solutions of

${f}^{\left(k\right)}+{A}_{k-1}\left(z\right){f}^{\left(k-1\right)}+\cdots +{A}_{0}\left(z\right)f=0$

in the unit disc, where the definition of φ-order of $f\left(z\right)$ is given as follows.

Definition 1.5 ([16])

Let $\phi :\left[0,1\right)\to \left(0,+\mathrm{\infty }\right)$ be a non-decreasing unbounded function, the φ-order of a meromorphic function $f\left(z\right)$ in the unit disc is defined by

$\sigma \left(f,\phi \right)=\underset{r\to {1}^{-}}{\overline{lim}}\frac{{log}^{+}T\left(r,f\right)}{log\phi \left(r\right)}.$
(1.8)

On the basis of Definition 1.5, it is natural for us to give the $\left[p,q\right]-\phi$ order of a meromorphic function $f\left(z\right)$ in the complex plane.

Definition 1.6 Let $\phi :\left[0,+\mathrm{\infty }\right)\to \left(0,+\mathrm{\infty }\right)$ be a non-decreasing unbounded function, the $\left[p,q\right]-\phi$ order and $\left[p,q\right]-\phi$ lower order of a meromorphic function $f\left(z\right)$ are respectively defined by

${\sigma }_{\left[p,q\right]}\left(f,\phi \right)=\underset{r\to \mathrm{\infty }}{\overline{lim}}\frac{{log}_{p}T\left(r,f\right)}{{log}_{q}\phi \left(r\right)},$
(1.9)
${\mu }_{\left[p,q\right]}\left(f,\phi \right)=\underset{r\to \mathrm{\infty }}{\underline{lim}}\frac{{log}_{p}T\left(r,f\right)}{{log}_{q}\phi \left(r\right)}.$
(1.10)

Similar to Definition 1.6, we can also define the $\left[p,q\right]-\phi$ exponent of convergence of the (distinct) zero-sequence of a meromorphic function $f\left(z\right)$.

Definition 1.7 The $\left[p,q\right]-\phi$ exponent of convergence of the (distinct) zero-sequence of a meromorphic function $f\left(z\right)$ is respectively defined by

${\lambda }_{\left[p,q\right]}\left(f,\phi \right)=\underset{r\to \mathrm{\infty }}{\overline{lim}}\frac{{log}_{p}n\left(r,\frac{1}{f}\right)}{{log}_{q}\phi \left(r\right)},$
(1.11)
${\overline{\lambda }}_{\left[p,q\right]}\left(f,\phi \right)=\underset{r\to \mathrm{\infty }}{\overline{lim}}\frac{{log}_{p}\overline{n}\left(r,\frac{1}{f}\right)}{{log}_{q}\phi \left(r\right)}.$
(1.12)

Proposition 1.1 If ${f}_{1}\left(z\right)$, ${f}_{2}\left(z\right)$ are meromorphic functions satisfying ${\sigma }_{\left[p,q\right]}\left({f}_{1},\phi \right)=a$, ${\sigma }_{\left[p,q\right]}\left({f}_{2},\phi \right)=b$, then

1. (i)

${\sigma }_{\left[p,q\right]}\left({f}_{1}+{f}_{2},\phi \right)\le max\left\{a,b\right\}$, ${\sigma }_{\left[p,q\right]}\left({f}_{1}\cdot {f}_{2},\phi \right)\le max\left\{a,b\right\}$;

2. (ii)

If $a\ne b$, ${\sigma }_{\left[p,q\right]}\left({f}_{1}+{f}_{2},\phi \right)=max\left\{a,b\right\}$, ${\sigma }_{\left[p,q\right]}\left({f}_{1}\cdot {f}_{2},\phi \right)=max\left\{a,b\right\}$.

In this paper, we add two conditions on $\phi \left(r\right)$ as follows: $\phi \left(r\right):\left[0,+\mathrm{\infty }\right)\to \left(0,+\mathrm{\infty }\right)$ is a non-decreasing unbounded function and satisfies (i) ${lim}_{r\to \mathrm{\infty }}\frac{{log}_{p+1}r}{{log}_{q}\phi \left(r\right)}=0$, (ii) ${lim}_{r\to \mathrm{\infty }}\frac{{log}_{q}\phi \left(\alpha r\right)}{{log}_{q}\phi \left(r\right)}=1$ for some $\alpha >1$. Throughout this paper, we assume that $\phi \left(r\right)$ always satisfies the above two conditions without special instruction.

Proposition 1.2 Let $\phi \left(r\right)$ satisfy the above two conditions (i)-(ii).

1. (i)

If $f\left(z\right)$ is an entire function, then

$\begin{array}{r}{\sigma }_{\left[p,q\right]}\left(f,\phi \right)=\underset{r\to \mathrm{\infty }}{\overline{lim}}\frac{{log}_{p}T\left(r,f\right)}{{log}_{q}\phi \left(r\right)}=\underset{r\to \mathrm{\infty }}{\overline{lim}}\frac{{log}_{p+1}M\left(r,f\right)}{{log}_{q}\phi \left(r\right)},\\ {\mu }_{\left[p,q\right]}\left(f,\phi \right)=\underset{r\to \mathrm{\infty }}{\underline{lim}}\frac{{log}_{p}T\left(r,f\right)}{{log}_{q}\phi \left(r\right)}=\underset{r\to \mathrm{\infty }}{\underline{lim}}\frac{{log}_{p+1}M\left(r,f\right)}{{log}_{q}\phi \left(r\right)}.\end{array}$
2. (ii)

If $f\left(z\right)$ is a meromorphic function, then

$\begin{array}{r}{\lambda }_{\left[p,q\right]}\left(f,\phi \right)=\underset{r\to \mathrm{\infty }}{\overline{lim}}\frac{{log}_{p}n\left(r,\frac{1}{f}\right)}{{log}_{q}\phi \left(r\right)}=\underset{r\to \mathrm{\infty }}{\overline{lim}}\frac{{log}_{p}N\left(r,\frac{1}{f}\right)}{{log}_{q}\phi \left(r\right)},\\ {\overline{\lambda }}_{\left[p,q\right]}\left(f,\phi \right)=\underset{r\to \mathrm{\infty }}{\overline{lim}}\frac{{log}_{p}\overline{n}\left(r,\frac{1}{f}\right)}{{log}_{q}\phi \left(r\right)}=\underset{r\to \mathrm{\infty }}{\overline{lim}}\frac{{log}_{p}\overline{N}\left(r,\frac{1}{f}\right)}{{log}_{q}\phi \left(r\right)}.\end{array}$

Proof (i) By the inequality $T\left(r,f\right)\le {log}^{+}M\left(r,f\right)\le \frac{R+r}{R-r}T\left(R,f\right)$ ($0), set $R=\alpha r$ ($\alpha >1$), we have

$T\left(r,f\right)\le {log}^{+}M\left(r,f\right)\le \frac{\alpha +1}{\alpha -1}T\left(\alpha r,f\right).$
(1.13)

By (1.13) and ${lim}_{r\to \mathrm{\infty }}\frac{{log}_{q}\phi \left(\alpha r\right)}{{log}_{q}\phi \left(r\right)}=1$, it is easy to see that conclusion (i) holds.

1. (ii)

Without loss of generality, assume that $f\left(0\right)\ne 0$, then $N\left(r,\frac{1}{f}\right)={\int }_{0}^{r}\frac{n\left(t,\frac{1}{f}\right)}{t}\phantom{\rule{0.2em}{0ex}}dt$. Since

$N\left(r,\frac{1}{f}\right)-N\left({r}_{0},\frac{1}{f}\right)={\int }_{{r}_{0}}^{r}\frac{n\left(t,\frac{1}{f}\right)}{t}\phantom{\rule{0.2em}{0ex}}dt\le n\left(r,\frac{1}{f}\right)log\frac{r}{{r}_{0}}\phantom{\rule{1em}{0ex}}\left(0<{r}_{0}
(1.14)

then by (1.14) and ${lim}_{r\to \mathrm{\infty }}\frac{{log}_{p+1}r}{{log}_{q}\phi \left(r\right)}=0$, we have

$\underset{r\to \mathrm{\infty }}{\overline{lim}}\frac{{log}_{p}N\left(r,\frac{1}{f}\right)}{{log}_{q}\phi \left(r\right)}\le max\left\{\underset{r\to \mathrm{\infty }}{\overline{lim}}\frac{{log}_{p}n\left(r,\frac{1}{f}\right)}{{log}_{q}\phi \left(r\right)},\underset{r\to \mathrm{\infty }}{\overline{lim}}\frac{{log}_{p+1}r}{{log}_{q}\phi \left(r\right)}\right\}=\underset{r\to \mathrm{\infty }}{\overline{lim}}\frac{{log}_{p}n\left(r,\frac{1}{f}\right)}{{log}_{q}\phi \left(r\right)}.$
(1.15)

On the other hand, since $\alpha >1$, we have

$N\left(\alpha r,\frac{1}{f}\right)={\int }_{0}^{\alpha r}\frac{n\left(t,\frac{1}{f}\right)}{t}\phantom{\rule{0.2em}{0ex}}dt\ge {\int }_{r}^{\alpha r}\frac{n\left(t,\frac{1}{f}\right)}{t}\phantom{\rule{0.2em}{0ex}}dt\ge n\left(r,\frac{1}{f}\right)log\alpha .$
(1.16)

By (1.16) and ${lim}_{r\to \mathrm{\infty }}\frac{{log}_{q}\phi \left(\alpha r\right)}{{log}_{q}\phi \left(r\right)}=1$, we have

$\underset{r\to \mathrm{\infty }}{\overline{lim}}\frac{{log}_{p}N\left(r,\frac{1}{f}\right)}{{log}_{q}\phi \left(r\right)}\ge \underset{r\to \mathrm{\infty }}{\overline{lim}}\frac{{log}_{p}n\left(r,\frac{1}{f}\right)}{{log}_{q}\phi \left(r\right)}.$
(1.17)

By (1.15) and (1.17), it is easy to see that ${\lambda }_{\left[p,q\right]}\left(f,\phi \right)={\overline{lim}}_{r\to \mathrm{\infty }}\frac{{log}_{p}n\left(r,\frac{1}{f}\right)}{{log}_{q}\phi \left(r\right)}={\overline{lim}}_{r\to \mathrm{\infty }}\frac{{log}_{p}N\left(r,\frac{1}{f}\right)}{{log}_{q}\phi \left(r\right)}$. By the same proof above, we can obtain the conclusion ${\overline{\lambda }}_{\left[p,q\right]}\left(f,\phi \right)={\overline{lim}}_{r\to \mathrm{\infty }}\frac{{log}_{p}\overline{n}\left(r,\frac{1}{f}\right)}{{log}_{q}\phi \left(r\right)}={\overline{lim}}_{r\to \mathrm{\infty }}\frac{{log}_{p}\overline{N}\left(r,\frac{1}{f}\right)}{{log}_{q}\phi \left(r\right)}$. □

Remark 1.4 If $\phi \left(r\right)=r$, Definitions 1.1 and 1.3 are special cases of Definitions 1.6 and 1.7.

## 2 Main results

In this paper, our aim is to make use of the concept of $\left[p,q\right]-\phi$ order of entire functions to investigate the growth, zeros of the solutions of equation (1.1).

Theorem 2.1 Let $A\left(z\right)$ be an entire function satisfying ${\sigma }_{\left[p,q\right]}\left(A,\phi \right)>0$. Then ${\sigma }_{\left[p+1,q\right]}\left(f,\phi \right)={\sigma }_{\left[p,q\right]}\left(A,\phi \right)$ holds for all non-trivial solutions of (1.1).

Theorem 2.2 Let $A\left(z\right)$ be an entire function satisfying ${\sigma }_{\left[p,q\right]}\left(A,\phi \right)>0$, let ${f}_{1}$, ${f}_{2}$ be two linearly independent solutions of (1.1) and denote $F={f}_{1}{f}_{2}$. Then $max\left\{{\lambda }_{\left[p+1,q\right]}\left({f}_{1},\phi \right),{\lambda }_{\left[p+1,q\right]}\left({f}_{2},\phi \right)\right\}={\lambda }_{\left[p+1,q\right]}\left(F,\phi \right)={\sigma }_{\left[p+1,q\right]}\left(F,\phi \right)\le {\sigma }_{\left[p,q\right]}\left(A,\phi \right)$. If ${\sigma }_{\left[p+1,q\right]}\left(F,\phi \right)<{\sigma }_{\left[p,q\right]}\left(A,\phi \right)$, then ${\lambda }_{\left[p+1,q\right]}\left(f,\phi \right)={\sigma }_{\left[p,q\right]}\left(A,\phi \right)$ holds for all solutions of type $f={c}_{1}{f}_{1}+{c}_{2}{f}_{2}$, where ${c}_{1}{c}_{2}\ne 0$.

Theorem 2.3 Let $A\left(z\right)$ be an entire function satisfying ${\overline{\lambda }}_{\left[p,q\right]}\left(A,\phi \right)<{\sigma }_{\left[p,q\right]}\left(A,\phi \right)$. Then ${\lambda }_{\left[p+1,q\right]}\left(f,\phi \right)\le {\sigma }_{\left[p,q\right]}\left(A,\phi \right)\le {\lambda }_{\left[p,q\right]}\left(f,\phi \right)$ holds for all non-trivial solutions of (1.1).

Theorem 2.4 Let $A\left(z\right)$ be an entire function satisfying ${\sigma }_{\left[p,q\right]}\left(A,\phi \right)={\sigma }_{1}>0$, let ${f}_{1}$ and ${f}_{2}$ be two linearly independent solutions of (1.1) such that $max\left\{{\lambda }_{\left[p,q\right]}\left({f}_{1},\phi \right),{\lambda }_{\left[p,q\right]}\left({f}_{2},\phi \right)\right\}<{\sigma }_{1}$. Let $\mathrm{\Pi }\left(z\right)\not\equiv 0$ be any entire function satisfying ${\sigma }_{\left[p,q\right]}\left(\mathrm{\Pi },\phi \right)<{\sigma }_{1}$. Then any two linearly independent solutions ${g}_{1}$ and ${g}_{2}$ of the differential equation ${f}^{″}+\left(A\left(z\right)+\mathrm{\Pi }\left(z\right)\right)f=0$ satisfy $max\left\{{\lambda }_{\left[p,q\right]}\left({g}_{1},\phi \right),{\lambda }_{\left[p,q\right]}\left({g}_{2},\phi \right)\right\}\ge {\sigma }_{1}$.

## 3 Some lemmas

Lemma 3.1 ([1719])

Let $f\left(z\right)$ be a transcendental entire function, and let z be a point with $|z|=r$ at which $|f\left(z\right)|=M\left(r,f\right)$. Then, for all $|z|$ outside a set ${E}_{1}$ of r of finite logarithmic measure, we have

$\frac{{f}^{\left(j\right)}\left(z\right)}{f\left(z\right)}={\left(\frac{{v}_{f}\left(r\right)}{z}\right)}^{j}\left(1+o\left(1\right)\right)\phantom{\rule{1em}{0ex}}\left(j\in \mathbb{N}\right),$
(3.1)

where ${v}_{f}\left(r\right)$ is the central index of $f\left(z\right)$.

Lemma 3.2 ([7, 19, 20])

Let $g:\left[0,+\mathrm{\infty }\right)⟶\mathbb{R}$ and $h:\left[0,+\mathrm{\infty }\right)⟶\mathbb{R}$ be monotone non-decreasing functions such that $g\left(r\right)\le h\left(r\right)$ outside of an exceptional set ${E}_{2}$ of finite linear measure or finite logarithmic measure. Then, for any $d>1$, there exists ${r}_{0}>0$ such that $g\left(r\right)\le h\left(dr\right)$ for all $r>{r}_{0}$.

Lemma 3.3 ([18, 21])

Let $f\left(z\right)={\sum }_{n=0}^{\mathrm{\infty }}{a}_{n}{z}^{n}$ be an entire function, $\mu \left(r\right)$ be the maximum term, i.e., $\mu \left(r\right)=max\left\{|{a}_{n}|{r}^{n};n=0,1,\dots \right\}$, and let ${v}_{f}\left(r\right)$ be the central index of f.

1. (i)

If $|{a}_{0}|\ne 0$, then

$log\mu \left(r\right)=log|{a}_{0}|+{\int }_{0}^{r}\frac{{v}_{f}\left(t\right)}{t}\phantom{\rule{0.2em}{0ex}}dt.$
(3.2)
2. (ii)

For $r, we have

$M\left(r,f\right)<\mu \left(r\right)\left\{{v}_{f}\left(R\right)+\frac{R}{R-r}\right\}.$
(3.3)

Lemma 3.4 Let $f\left(z\right)$ be an entire function satisfying ${\sigma }_{\left[p,q\right]}\left(f,\phi \right)={\sigma }_{2}$ and ${\mu }_{\left[p,q\right]}\left(f,\phi \right)={\mu }_{1}$, and let ${v}_{f}\left(r\right)$ be the central index of f, then

$\underset{r\to \mathrm{\infty }}{\overline{lim}}\frac{{log}_{p}{v}_{f}\left(r\right)}{{log}_{q}\phi \left(r\right)}={\sigma }_{2},\phantom{\rule{2em}{0ex}}\underset{r\to \mathrm{\infty }}{\underline{lim}}\frac{{log}_{p}{v}_{f}\left(r\right)}{{log}_{q}\phi \left(r\right)}={\mu }_{1}.$

Proof Let $f\left(z\right)={\sum }_{n=0}^{\mathrm{\infty }}{a}_{n}{z}^{n}$. Without loss of generality, we can assume that $|{a}_{0}|\ne 0$. From (3.2), for any $1<{\alpha }_{1}<\alpha$, we have

$log\mu \left({\alpha }_{1}r\right)=log|{a}_{0}|+{\int }_{0}^{{\alpha }_{1}r}\frac{{v}_{f}\left(t\right)}{t}\phantom{\rule{0.2em}{0ex}}dt\ge log|{a}_{0}|+{\int }_{r}^{{\alpha }_{1}r}\frac{{v}_{f}\left(t\right)}{t}\phantom{\rule{0.2em}{0ex}}dt\ge log|{a}_{0}|+{v}_{f}\left(r\right)log{\alpha }_{1}.$

By the Cauchy inequality, it is easy to see $\mu \left({\alpha }_{1}r\right)\le M\left({\alpha }_{1}r,f\right)$, hence

${v}_{f}\left(r\right)log{\alpha }_{1}\le logM\left({\alpha }_{1}r,f\right)+{c}_{3},$
(3.4)

where ${c}_{3}>0$ is a constant. By Proposition 1.2, (3.4) and ${lim}_{r\to \mathrm{\infty }}\frac{{log}_{q}\phi \left({\alpha }_{1}r\right)}{{log}_{q}\phi \left(r\right)}=1$ ($1<{\alpha }_{1}<\alpha$), we have

$\underset{r\to \mathrm{\infty }}{\overline{lim}}\frac{{log}_{p}{v}_{f}\left(r\right)}{{log}_{q}\phi \left(r\right)}\le \underset{r\to \mathrm{\infty }}{\overline{lim}}\frac{{log}_{p+1}M\left({\alpha }_{1}r,f\right)}{{log}_{q}\phi \left({\alpha }_{1}r\right)}\cdot \underset{r\to \mathrm{\infty }}{\overline{lim}}\frac{{log}_{q}\phi \left({\alpha }_{1}r\right)}{{log}_{q}\phi \left(r\right)}={\sigma }_{\left[p,q\right]}\left(f,\phi \right),$
(3.5)
$\underset{r\to \mathrm{\infty }}{\underline{lim}}\frac{{log}_{p}{v}_{f}\left(r\right)}{{log}_{q}\phi \left(r\right)}\le \underset{r\to \mathrm{\infty }}{\underline{lim}}\frac{{log}_{p+1}M\left({\alpha }_{1}r,f\right)}{{log}_{q}\phi \left({\alpha }_{1}r\right)}\cdot \underset{r\to \mathrm{\infty }}{lim}\frac{{log}_{q}\phi \left({\alpha }_{1}r\right)}{{log}_{q}\phi \left(r\right)}={\mu }_{\left[p,q\right]}\left(f,\phi \right).$
(3.6)

On the other hand, set $R={\alpha }_{1}r$, by (3.3), we have

$M\left(r,f\right)<\mu \left(r\right)\left({v}_{f}\left({\alpha }_{1}r\right)+\frac{{\alpha }_{1}}{{\alpha }_{1}-1}\right)=|{a}_{{v}_{f}\left({\alpha }_{1}r\right)}|{r}^{{v}_{f}\left({\alpha }_{1}r\right)}\left({v}_{f}\left({\alpha }_{1}r\right)+\frac{{\alpha }_{1}}{{\alpha }_{1}-1}\right).$
(3.7)

Since ${\left\{|{a}_{n}|\right\}}_{n=1}^{\mathrm{\infty }}$ is a bounded sequence, by (3.7), we have

${log}_{p+1}M\left(r,f\right)\le {log}_{p}{v}_{f}\left({\alpha }_{1}r\right)\left[1+\frac{{log}_{p+1}{v}_{f}\left({\alpha }_{1}r\right)}{{log}_{p}{v}_{f}\left({\alpha }_{1}r\right)}\right]+{log}_{p+1}r+{c}_{4},$
(3.8)

where ${c}_{4}>0$ is a constant. By Proposition 1.2, (3.8), ${lim}_{r\to \mathrm{\infty }}\frac{{log}_{q}\phi \left({\alpha }_{1}r\right)}{{log}_{q}\phi \left(r\right)}=1$ ($1<{\alpha }_{1}<\alpha$) and ${lim}_{r\to \mathrm{\infty }}\frac{{log}_{p+1}r}{{log}_{q}\phi \left(r\right)}=0$, we have

${\sigma }_{\left[p,q\right]}\left(f,\phi \right)=\underset{r\to \mathrm{\infty }}{\overline{lim}}\frac{{log}_{p+1}M\left(r,f\right)}{{log}_{q}\phi \left(r\right)}\le \underset{r\to \mathrm{\infty }}{\overline{lim}}\frac{{log}_{p}{v}_{f}\left({\alpha }_{1}r\right)}{{log}_{q}\phi \left({\alpha }_{1}r\right)}=\underset{r\to \mathrm{\infty }}{\overline{lim}}\frac{{log}_{p}{v}_{f}\left(r\right)}{{log}_{q}\phi \left(r\right)},$
(3.9)
${\mu }_{\left[p,q\right]}\left(f,\phi \right)=\underset{r\to \mathrm{\infty }}{\underline{lim}}\frac{{log}_{p+1}M\left(r,f\right)}{{log}_{q}\phi \left(r\right)}\le \underset{r\to \mathrm{\infty }}{\underline{lim}}\frac{{log}_{p}{v}_{f}\left({\alpha }_{1}r\right)}{{log}_{q}\phi \left({\alpha }_{1}r\right)}=\underset{r\to \mathrm{\infty }}{\underline{lim}}\frac{{log}_{p}{v}_{f}\left(r\right)}{{log}_{q}\phi \left(r\right)}.$
(3.10)

By (3.5), (3.6), (3.9) and (3.10), we obtain the conclusion of Lemma 3.4. □

Lemma 3.5 Let ${f}_{1}\left(z\right)$ and ${f}_{2}\left(z\right)$ be entire functions of $\left[p,q\right]-\phi$ order and denote $F={f}_{1}{f}_{2}$. Then

${\lambda }_{\left[p,q\right]}\left(F,\phi \right)=max\left\{{\lambda }_{\left[p,q\right]}\left({f}_{1},\phi \right),{\lambda }_{\left[p,q\right]}\left({f}_{2},\phi \right)\right\}.$

Proof Let $n\left(r,F\right)$, $n\left(r,{f}_{1}\right)$ and $n\left(r,{f}_{2}\right)$ be unintegrated counting functions for the number of zeros of $F\left(z\right)$, ${f}_{1}\left(z\right)$ and ${f}_{2}\left(z\right)$. For any $r>0$, it is easy to see

$n\left(r,F\right)\ge max\left\{n\left(r,{f}_{1}\right),n\left(r,{f}_{2}\right)\right\}.$
(3.11)

By Definition 1.7 and (3.11), we have

${\lambda }_{\left[p,q\right]}\left(F,\phi \right)\ge max\left\{{\lambda }_{\left[p,q\right]}\left({f}_{1},\phi \right),{\lambda }_{\left[p,q\right]}\left({f}_{2},\phi \right)\right\}.$
(3.12)

On the other hand, since the zeros of $F\left(z\right)$ must be the zeros of ${f}_{1}\left(z\right)$ or the zeros of ${f}_{2}\left(z\right)$, for any $r>0$, we have

$n\left(r,F\right)\le n\left(r,{f}_{1}\right)+n\left(r,{f}_{2}\right)\le 2max\left\{n\left(r,{f}_{1}\right),n\left(r,{f}_{2}\right)\right\}.$
(3.13)

By Definition 1.7 and (3.13), we have

${\lambda }_{\left[p,q\right]}\left(F,\phi \right)\le max\left\{{\lambda }_{\left[p,q\right]}\left({f}_{1},\phi \right),{\lambda }_{\left[p,q\right]}\left({f}_{2},\phi \right)\right\}.$
(3.14)

Therefore, by (3.12) and (3.14), we have ${\lambda }_{\left[p,q\right]}\left(F,\phi \right)=\left\{{\lambda }_{\left[p,q\right]}\left({f}_{1},\phi \right),{\lambda }_{\left[p,q\right]}\left({f}_{2},\phi \right)\right\}$. □

Lemma 3.6 Let $f\left(z\right)$ be a transcendental meromorphic function satisfying ${\sigma }_{\left[p,q\right]}\left(f,\phi \right)={\sigma }_{3}$, where $\phi \left(r\right)$ only satisfies $\frac{{log}_{p+1}r}{{log}_{q}\phi \left(r\right)}=0$, and let k be any positive integer. Then, for any $\epsilon >0$, there exists a set ${E}_{3}$ having finite linear measure such that for all $r\notin {E}_{3}$, we have

$m\left(r,\frac{{f}^{\left(k\right)}}{f}\right)=O\left\{{exp}_{p-1}\left\{\left({\sigma }_{3}+\epsilon \right){log}_{q}\phi \left(r\right)\right\}\right\}.$

Proof Set $k=1$, since ${\sigma }_{\left[p,q\right]}\left(f,\phi \right)={\sigma }_{3}<\mathrm{\infty }$, for sufficiently large r and for any given $\epsilon >0$, we have

$T\left(r,f\right)<{exp}_{p}\left\{\left({\sigma }_{3}+\epsilon \right){log}_{q}\phi \left(r\right)\right\}.$
(3.15)

By the lemma of logarithmic derivative, we have

$m\left(r,\frac{{f}^{\prime }}{f}\right)=O\left\{logT\left(r,f\right)+logr\right\}\phantom{\rule{1em}{0ex}}\left(r\notin {E}_{3}\right),$
(3.16)

where ${E}_{3}\subset \left[0,+\mathrm{\infty }\right)$ is a set of finite linear measure, not necessarily the same at each occurrence. By (3.15), (3.16) and $\frac{{log}_{p+1}r}{{log}_{q}\phi \left(r\right)}=0$, we have $m\left(r,\frac{{f}^{\prime }}{f}\right)=O\left\{{exp}_{p-1}\left\{\left(\sigma +\epsilon \right){log}_{q}\phi \left(r\right)\right\}\right\}$ ($r\notin {E}_{3}$).

We assume that $m\left(r,\frac{{f}^{\left(k\right)}}{f}\right)=O\left\{{exp}_{p-1}\left\{\left({\sigma }_{3}+\epsilon \right){log}_{q}\phi \left(r\right)\right\}\right\}$ ($r\notin {E}_{3}$) holds for any positive integer k. By $N\left(r,{f}^{\left(k\right)}\right)\le \left(k+1\right)N\left(r,f\right)$, for all $r\notin {E}_{3}$, we have

$\begin{array}{rl}T\left(r,{f}^{\left(k\right)}\right)& =m\left(r,{f}^{\left(k\right)}\right)+N\left(r,{f}^{\left(k\right)}\right)\le m\left(r,\frac{{f}^{\left(k\right)}}{f}\right)+m\left(r,f\right)+\left(k+1\right)N\left(r,f\right)\\ \le \left(k+1\right)T\left(r,f\right)+O\left\{{exp}_{p-1}\left\{\left({\sigma }_{3}+\epsilon \right){log}_{q}\phi \left(r\right)\right\}\right\}.\end{array}$
(3.17)

By (3.16) and (3.17), for $r\notin {E}_{3}$, we have

$m\left(r,\frac{{f}^{\left(k+1\right)}}{f}\right)\le m\left(r,\frac{{f}^{\left(k+1\right)}}{{f}^{\left(k\right)}}\right)+m\left(r,\frac{{f}^{\left(k\right)}}{f}\right)=O\left\{{exp}_{p-1}\left\{\left({\sigma }_{3}+\epsilon \right){log}_{q}\phi \left(r\right)\right\}\right\}.$

□

Lemma 3.7 ([19])

Let $f\left(z\right)$ be an entire function of $\left[p,q\right]$-order, and $f\left(z\right)$ can be represented by the form

$f\left(z\right)=U\left(z\right){e}^{V\left(z\right)},$

where $U\left(z\right)$ and $V\left(z\right)$ are entire functions such that

${\lambda }_{\left[p,q\right]}\left(f\right)={\lambda }_{\left[p,q\right]}\left(U\right)={\sigma }_{\left[p,q\right]}\left(U\right),\phantom{\rule{2em}{0ex}}{\sigma }_{\left[p,q\right]}\left(f\right)=max\left\{{\sigma }_{\left[p,q\right]}\left(U\right),{\sigma }_{\left[p,q\right]}\left({e}^{V}\right)\right\}.$

If $f\left(z\right)$ is an entire function of $\left[p,q\right]-\phi$ order, we have a similar result as follows.

Lemma 3.8 Let $f\left(z\right)$ be an entire function of $\left[p,q\right]-\phi$ order, and $f\left(z\right)$ can be represented by the form

$f\left(z\right)=U\left(z\right){e}^{V\left(z\right)},$

where $U\left(z\right)$ and $V\left(z\right)$ are entire functions of $\left[p,q\right]-\phi$ order such that

$\begin{array}{c}{\lambda }_{\left[p,q\right]}\left(f,\phi \right)={\lambda }_{\left[p,q\right]}\left(U,\phi \right)={\sigma }_{\left[p,q\right]}\left(U,\phi \right),\hfill \\ {\sigma }_{\left[p,q\right]}\left(f,\phi \right)=max\left\{{\sigma }_{\left[p,q\right]}\left(U,\phi \right),{\sigma }_{\left[p,q\right]}\left({e}^{V},\phi \right)\right\}.\hfill \end{array}$

## 4 Proofs of Theorems 2.1-2.4

Proof of Theorem 2.1 Set ${\sigma }_{\left[p,q\right]}\left(A,\phi \right)={\sigma }_{4}>0$. First, we prove that every solution of (1.1) satisfies ${\sigma }_{\left[p+1,q\right]}\left(f,\phi \right)\le {\sigma }_{4}$. If $f\left(z\right)$ is a polynomial solution of (1.1), it is easy to know that ${\sigma }_{\left[p+1,q\right]}\left(f,\phi \right)=0\le {\sigma }_{4}$ holds. If $f\left(z\right)$ is a transcendental solution of (1.1), by (1.1) and Lemma 3.1, there exists a set ${E}_{1}\subset \left(1,+\mathrm{\infty }\right)$ having finite logarithmic measure such that for all z satisfying $|z|=r\notin \left[0,1\right]\cup {E}_{1}$ and $|f\left(z\right)|=M\left(r,f\right)$, we have

${\left(\frac{{v}_{f}\left(r\right)}{r}\right)}^{2}\left(1+o\left(1\right)\right)\le {exp}_{p+1}\left\{\left({\sigma }_{4}+\frac{\epsilon }{2}\right){log}_{q}\phi \left(r\right)\right\}.$

And hence, we have

${v}_{f}\left(r\right)\le r{exp}_{p+1}\left\{\left({\sigma }_{4}+\epsilon \right){log}_{q}\phi \left(r\right)\right\}\phantom{\rule{1em}{0ex}}\left(r\notin {E}_{1}\right).$
(4.1)

By (4.1) and Lemma 3.2, there exists some ${\alpha }_{1}$ ($1<{\alpha }_{1}<\alpha$) such that for all $r\ge {r}_{0}$, we have

${v}_{f}\left(r\right)\le {\alpha }_{1}r{exp}_{p+1}\left\{\left({\sigma }_{4}+\epsilon \right){log}_{q}\phi \left({\alpha }_{1}r\right)\right\}.$
(4.2)

By Lemma 3.4, (4.2) and the two conditions on $\phi \left(r\right)$, we have

${\sigma }_{\left[p+1,q\right]}\left(f,\phi \right)=\underset{r\to \mathrm{\infty }}{\overline{lim}}\frac{{log}_{p+1}{v}_{f}\left(r\right)}{{log}_{q}\phi \left(r\right)}\le {\sigma }_{4}.$
(4.3)

On the other hand, by (1.1), we have

$m\left(r,A\right)=m\left(r,-\frac{{f}^{″}}{f}\right)=O\left\{logrT\left(r,f\right)\right\}.$
(4.4)

By (4.4), we have ${\sigma }_{\left[p,q\right]}\left(A,\phi \right)\le {\sigma }_{\left[p+1,q\right]}\left(f,\phi \right)$. Therefore, we have that ${\sigma }_{\left[p+1,q\right]}\left(f,\phi \right)={\sigma }_{\left[p,q\right]}\left(A,\phi \right)$ holds for all non-trivial solutions of (1.1). □

Proof of Theorem 2.2 Set ${\sigma }_{\left[p,q\right]}\left(A,\phi \right)={\sigma }_{5}>0$, by Theorem 2.1, we have ${\sigma }_{\left[p+1,q\right]}\left({f}_{1},\phi \right)={\sigma }_{\left[p+1,q\right]}\left({f}_{2},\phi \right)={\sigma }_{\left[p,q\right]}\left(A,\phi \right)={\sigma }_{5}$. Hence, we have

${\lambda }_{\left[p+1,q\right]}\left(F,\phi \right)\le {\sigma }_{\left[p+1,q\right]}\left(F,\phi \right)\le max\left\{{\sigma }_{\left[p+1,q\right]}\left({f}_{1},\phi \right),{\sigma }_{\left[p+1,q\right]}\left({f}_{2},\phi \right)\right\}={\sigma }_{\left[p,q\right]}\left(A,\phi \right).$
(4.5)

By Lemma 3.5 and (4.5), we have

$max\left\{{\lambda }_{\left[p+1,q\right]}\left({f}_{1},\phi \right),{\lambda }_{\left[p+1,q\right]}\left({f}_{2},\phi \right)\right\}={\lambda }_{\left[p+1,q\right]}\left(F,\phi \right)\le {\sigma }_{\left[p+1,q\right]}\left(F,\phi \right)\le {\sigma }_{\left[p,q\right]}\left(A,\phi \right).$
(4.6)

It remains to show that ${\lambda }_{\left[p+1,q\right]}\left(F,\phi \right)={\sigma }_{\left[p+1,q\right]}\left(F,\phi \right)$. By (1.1), we have (see [[13], pp.76-77]) that all zeros of $F\left(z\right)$ are simple and that

${F}^{2}={C}^{2}{\left({\left(\frac{{F}^{\prime }}{F}\right)}^{2}-2\left(\frac{{F}^{″}}{F}\right)-4A\right)}^{-1},$
(4.7)

where $C\ne 0$ is a constant. Hence,

$\begin{array}{rl}2T\left(r,F\right)& =T\left(r,{\left(\frac{{F}^{\prime }}{F}\right)}^{2}-2\left(\frac{{F}^{″}}{F}\right)-4A\right)+O\left(1\right)\\ \le O\left(\overline{N}\left(r,\frac{1}{F}\right)+m\left(r,\frac{{F}^{\prime }}{F}\right)+m\left(r,\frac{{F}^{″}}{F}\right)+m\left(r,A\right)\right).\end{array}$
(4.8)

By Lemma 3.6, for all $r\notin {E}_{3}$, we have $m\left(r,A\right)=m\left(r,\frac{{f}^{″}}{f}\right)=O\left\{{exp}_{p}\left\{\left({\sigma }_{5}+\epsilon \right){log}_{q}\phi \left(r\right)\right\}\right\}$, $m\left(r,\frac{{F}^{\prime }}{F}\right)=O\left\{{exp}_{p}\left\{\left({\sigma }_{5}+\epsilon \right){log}_{q}\phi \left(r\right)\right\}\right\}$ and $m\left(r,\frac{{F}^{″}}{F}\right)=O\left\{{exp}_{p}\left\{\left({\sigma }_{5}+\epsilon \right){log}_{q}\phi \left(r\right)\right\}\right\}$. By (4.8), for all $r\notin {E}_{3}$, we have

$T\left(r,F\right)=O\left\{\overline{N}\left(r,\frac{1}{F}\right)+{exp}_{p}\left\{\left({\sigma }_{5}+\epsilon \right){log}_{q}\phi \left(r\right)\right\}\right\}.$
(4.9)

Let us assume ${\lambda }_{\left[p+1,q\right]}\left(F,\phi \right)<\beta <{\sigma }_{\left[p+1,q\right]}\left(F,\phi \right)$. Since all zeros of $F\left(z\right)$ are simple, we have

$\overline{N}\left(r,\frac{1}{F}\right)=N\left(r,\frac{1}{F}\right)=O\left\{{exp}_{p+1}\left\{\beta {log}_{q}\phi \left(r\right)\right\}\right\}.$
(4.10)

By (4.9) and (4.10), for all $r\notin {E}_{3}$, we have

$T\left(r,F\right)=O\left\{{exp}_{p+1}\left\{\beta {log}_{q}\phi \left(r\right)\right\}\right\}.$

By Definition 1.6 and Lemma 3.2, we have ${\sigma }_{\left[p+1,q\right]}\left(F,\phi \right)\le \beta <{\sigma }_{\left[p+1,q\right]}\left(F,\phi \right)$, this is a contradiction. Therefore, the first assertion is proved.

If ${\sigma }_{\left[p+1,q\right]}\left(F,\phi \right)<{\sigma }_{\left[p,q\right]}\left(A,\phi \right)$, let us assume that ${\lambda }_{\left[p+1,q\right]}\left(f,\phi \right)<{\sigma }_{\left[p,q\right]}\left(A,\phi \right)$ holds for any solution of type $f={c}_{1}{f}_{1}+{c}_{2}{f}_{2}$ (${c}_{1}{c}_{2}\ne 0$). We denote $F={f}_{1}{f}_{2}$ and ${F}_{1}=f{f}_{1}$, then we have ${\lambda }_{\left[p+1,q\right]}\left(F,\phi \right)<{\sigma }_{\left[p,q\right]}\left(A,\phi \right)$ and ${\lambda }_{\left[p+1,q\right]}\left({F}_{1},\phi \right)<{\sigma }_{\left[p,q\right]}\left(A,\phi \right)$. Since (4.9) holds for $F\left(z\right)$ and ${F}_{1}\left(z\right)$ and ${F}_{1}=f{f}_{1}=\left({c}_{1}{f}_{1}+{c}_{2}{f}_{2}\right){f}_{1}={c}_{1}{f}_{1}^{2}+{c}_{2}F$, we have

$\begin{array}{rl}T\left(r,{f}_{1}\right)& =O\left(T\left(r,{F}_{1}\right)+T\left(r,F\right)\right)\\ =O\left\{\overline{N}\left(r,\frac{1}{{F}_{1}}\right)+\overline{N}\left(r,\frac{1}{F}\right)+{exp}_{p}\left\{\left({\sigma }_{5}+\epsilon \right){log}_{q}\phi \left(r\right)\right\}\right\}.\end{array}$
(4.11)

By ${\lambda }_{\left[p+1,q\right]}\left(F,\phi \right)<{\sigma }_{\left[p,q\right]}\left(A,\phi \right)$, ${\lambda }_{\left[p+1,q\right]}\left({F}_{1},\phi \right)<{\sigma }_{\left[p,q\right]}\left(A,\phi \right)$ and (4.10), for some $\beta <{\sigma }_{\left[p,q\right]}\left(A,\phi \right)$, we have

$T\left(r,{f}_{1}\right)=O\left\{{exp}_{p+1}\left\{\beta {log}_{q}\phi \left(r\right)\right\}\right\}.$
(4.12)

By Definition 1.6 and (4.12), we have ${\sigma }_{\left[p+1,q\right]}\left({f}_{1},\phi \right)\le \beta <{\sigma }_{\left[p,q\right]}\left(A,\phi \right)$, this is a contradiction with Theorem 2.1. Therefore, we have that ${\lambda }_{\left[p+1,q\right]}\left(f,\phi \right)={\sigma }_{\left[p,q\right]}\left(A,\phi \right)$ holds for all solutions of type $f={c}_{1}{f}_{1}+{c}_{2}{f}_{2}$, where ${c}_{1}{c}_{2}\ne 0$. □

Proof of Theorem 2.3 By Theorem 2.1 and ${\lambda }_{\left[p+1,q\right]}\left(f,\phi \right)\le {\sigma }_{\left[p+1,q\right]}\left(f,\phi \right)$, it is easy to know that ${\lambda }_{\left[p+1,q\right]}\left(f,\phi \right)\le {\sigma }_{\left[p,q\right]}\left(A,\phi \right)$ holds. It remains to show that ${\sigma }_{\left[p,q\right]}\left(A,\phi \right)\le {\lambda }_{\left[p,q\right]}\left(f,\phi \right)$. Let us assume ${\sigma }_{\left[p,q\right]}\left(A,\phi \right)>{\lambda }_{\left[p,q\right]}\left(f,\phi \right)$. By (1.1) and a similar proof of Theorem 5.6 in [[13], p.82], we have

$T\left(r,\frac{f}{{f}^{\prime }}\right)=O\left\{\overline{N}\left(r,\frac{1}{f}\right)+\overline{N}\left(r,\frac{1}{A}\right)\right\}\phantom{\rule{1em}{0ex}}\left(r\notin {E}_{3}\right).$
(4.13)

By (4.13), the assumption ${\sigma }_{\left[p,q\right]}\left(A,\phi \right)>{\lambda }_{\left[p,q\right]}\left(f,\phi \right)$ and ${\overline{\lambda }}_{\left[p,q\right]}\left(A,\phi \right)\le {\sigma }_{\left[p,q\right]}\left(A,\phi \right)$, for some $\beta <{\sigma }_{\left[p,q\right]}\left(A,\phi \right)$, we have

$T\left(r,\frac{f}{{f}^{\prime }}\right)=O\left\{{exp}_{p}\left\{\beta {log}_{q}\phi \left(r\right)\right\}\right\}.$
(4.14)

By Definition 1.6 and (4.14), we have ${\sigma }_{\left[p,q\right]}\left(\frac{f}{{f}^{\prime }},\phi \right)={\sigma }_{\left[p,q\right]}\left(\frac{{f}^{\prime }}{f},\phi \right)\le \beta <{\sigma }_{\left[p,q\right]}\left(A,\phi \right)$. By

$-A\left(z\right)={\left(\frac{{f}^{\prime }}{f}\right)}^{\prime }+{\left(\frac{{f}^{\prime }}{f}\right)}^{2},$

we have ${\sigma }_{\left[p,q\right]}\left(A,\phi \right)\le {\sigma }_{\left[p,q\right]}\left(\frac{{f}^{\prime }}{f},\phi \right)<{\sigma }_{\left[p,q\right]}\left(A,\phi \right)$, this is a contradiction. Therefore, we have that ${\lambda }_{\left[p+1,q\right]}\left(f,\phi \right)\le {\sigma }_{\left[p,q\right]}\left(A,\phi \right)\le {\lambda }_{\left[p,q\right]}\left(f,\phi \right)$ holds for all non-trivial solutions of (1.1). □

Proof of Theorem 2.4 As a similar proof of Theorem 3.1 in [6], we denote $F={f}_{1}{f}_{2}$ and ${F}_{2}={g}_{1}{g}_{2}$. Let us assume

${\lambda }_{\left[p,q\right]}\left({F}_{2},\phi \right)=max\left\{{\lambda }_{\left[p,q\right]}\left({g}_{1},\phi \right),{\lambda }_{\left[p,q\right]}\left({g}_{2},\phi \right)\right\}<{\sigma }_{1}.$

By Theorem 2.1, we have ${\sigma }_{\left[p+1,q\right]}\left(F,\phi \right)\le max\left\{{\sigma }_{\left[p+1,q\right]}\left({f}_{1},\phi \right),{\sigma }_{\left[p+1,q\right]}\left({f}_{2},\phi \right)\right\}={\sigma }_{1}$, and hence, by Lemma 3.6, for any integer $k\ge 1$ and for any $\epsilon >0$, we have

$m\left(r,\frac{{F}^{\left(k\right)}}{F}\right)=O\left\{{exp}_{p}\left\{\left({\sigma }_{1}+\epsilon \right){log}_{q}\phi \left(r\right)\right\}\right\}\phantom{\rule{1em}{0ex}}\left(r\notin {E}_{3}\right).$

Furthermore, by Theorem 2.1, we have ${\lambda }_{\left[p,q\right]}\left(F,\phi \right)=max\left\{{\lambda }_{\left[p,q\right]}\left({f}_{1},\phi \right),{\lambda }_{\left[p,q\right]}\left({f}_{2},\phi \right)\right\}<{\sigma }_{1}$, and hence we have $\overline{N}\left(r,\frac{1}{F}\right)=O\left\{{exp}_{p}\left\{\beta {log}_{q}\phi \left(r\right)\right\}\right\}$ for some $\beta <{\sigma }_{1}$. And the $\left[p,q\right]-\phi$ order of the function $A\left(z\right)$ implies that

$T\left(r,A\right)=O\left\{{exp}_{p}\left\{\left({\sigma }_{1}+\epsilon \right){log}_{q}\phi \left(r\right)\right\}\right\}\phantom{\rule{1em}{0ex}}\left(r\to \mathrm{\infty }\right).$

By (4.9), we obtain

$T\left(r,F\right)=O\left\{\overline{N}\left(r,\frac{1}{F}\right)+{exp}_{p}\left\{\left({\sigma }_{1}+\epsilon \right){log}_{q}\phi \left(r\right)\right\}\right\}=O\left\{{exp}_{p}\left\{\left(\beta {log}_{q}\phi \left(r\right)\right\}\right\}.$
(4.15)

By Definition 1.6 and (4.15), we have ${\sigma }_{\left[p,q\right]}\left(F,\phi \right)\le {\sigma }_{1}$. On the other hand, by

$4A={\left(\frac{{F}^{\prime }}{F}\right)}^{2}-2\frac{{F}^{″}}{F}-\frac{1}{{F}^{2}},$
(4.16)

we have ${\sigma }_{\left[p,q\right]}\left(A,\phi \right)={\sigma }_{1}\le {\sigma }_{\left[p,q\right]}\left(F,\phi \right)$, hence ${\sigma }_{\left[p,q\right]}\left(F,\phi \right)={\sigma }_{1}$. The same reasoning is valid for the function ${F}_{2}$, we have

$4\left(A+\mathrm{\Pi }\right)={\left(\frac{{F}_{2}^{\prime }}{{F}_{2}}\right)}^{2}-2\frac{{F}_{2}^{″}}{{F}_{2}}-\frac{1}{{F}_{2}^{2}},$
(4.17)

and ${\sigma }_{\left[p,q\right]}\left({F}_{2},\phi \right)={\sigma }_{1}$. Since ${\lambda }_{\left[p,q\right]}\left(F,\phi \right)<{\sigma }_{1}$ and ${\lambda }_{\left[p,q\right]}\left({F}_{2},\phi \right)<{\sigma }_{1}$, by Lemma 3.8, we may write

$F=Q{e}^{P},\phantom{\rule{2em}{0ex}}{F}_{2}=R{e}^{S},$
(4.18)

where P, Q, R, S are entire functions satisfying ${\sigma }_{\left[p,q\right]}\left(Q,\phi \right)={\lambda }_{\left[p,q\right]}\left(F,\phi \right)<{\sigma }_{1}$, ${\sigma }_{\left[p,q\right]}\left(R,\phi \right)={\lambda }_{\left[p,q\right]}\left({F}_{2},\phi \right)<{\sigma }_{1}$ and ${\sigma }_{\left[p,q\right]}\left({e}^{P},\phi \right)={\sigma }_{\left[p,q\right]}\left({e}^{S},\phi \right)={\sigma }_{1}$. Substituting (4.18) into (4.16) and (4.17), we have

$4A=-\frac{1}{{Q}^{2}{e}^{2P}}+{G}_{1}\left(z\right),$
(4.19)
$4\left(A+\pi \right)=-\frac{1}{{R}^{2}{e}^{2S}}+{G}_{2}\left(z\right),$
(4.20)

where ${G}_{1}\left(z\right)$ and ${G}_{2}\left(z\right)$ are meromorphic functions satisfying ${\sigma }_{\left[p,q\right]}\left({G}_{j},\phi \right)<{\sigma }_{1}$ ($j=1,2$). Equation (4.19) subtracting (4.20), we have

$\frac{1}{{R}^{2}{e}^{2S}}-\frac{1}{{Q}^{2}{e}^{2P}}={G}_{3}\left(z\right),$
(4.21)

where ${G}_{3}\left(z\right)$ is a meromorphic function satisfying ${\sigma }_{\left[p,q\right]}\left({G}_{3},\phi \right)<{\sigma }_{1}$. From (4.21), we have

${e}^{-2S}+{H}_{1}{e}^{-2P}={H}_{2},$
(4.22)

where ${H}_{1}\left(z\right)$ and ${H}_{2}\left(z\right)$ are meromorphic functions satisfying ${\sigma }_{\left[p,q\right]}\left({H}_{j},\phi \right)<{\sigma }_{1}$ ($j=1,2$), and ${H}_{1}=-\frac{{R}^{2}}{{Q}^{2}}$. Deriving (4.22), we have

$-2{S}^{\prime }{e}^{-2S}+\left({H}_{1}^{\prime }-2{P}^{\prime }{H}_{1}\right){e}^{-2P}={H}_{3},$
(4.23)

where ${H}_{3}\left(z\right)$ is a meromorphic function satisfying ${\sigma }_{\left[p,q\right]}\left({H}_{3},\phi \right)<{\sigma }_{1}$. Eliminating ${e}^{-2S}$ by (4.22) and (4.23), we have

$\left({H}_{1}^{\prime }-2\left({P}^{\prime }-{S}^{\prime }\right){H}_{1}\right){e}^{-2P}={H}_{4},$
(4.24)

where ${H}_{4}\left(z\right)$ is a meromorphic function satisfying ${\sigma }_{\left[p,q\right]}\left({H}_{4},\phi \right)<{\sigma }_{1}$. Since ${\sigma }_{\left[p,q\right]}\left({e}^{P},\phi \right)={\sigma }_{1}$, therefore by (4.24), we have ${H}_{1}^{\prime }-2\left({P}^{\prime }-{S}^{\prime }\right){H}_{1}\equiv 0$, thus we have ${H}_{1}=c{e}^{2\left(P-S\right)}$, $c\ne 0$. Hence

$\frac{{F}^{2}}{{F}_{2}^{2}}=\frac{{Q}^{2}}{{R}^{2}}{e}^{2\left(P-S\right)}=-\frac{1}{c}.$
(4.25)

From (4.16), (4.17) and (4.25), we have

$4\left(A+\mathrm{\Pi }+\frac{1}{c}A\right)={\left(\frac{{F}_{2}^{\prime }}{{F}_{2}}\right)}^{2}-2\frac{{F}_{2}^{″}}{{F}_{2}}+\frac{1}{c}{\left(\frac{{F}^{\prime }}{F}\right)}^{2}-\frac{2}{c}\frac{{F}^{″}}{F}.$

By Lemma 3.6, we obtain

$\begin{array}{rl}T\left(r,\left(1+\frac{1}{c}\right)A+\mathrm{\Pi }\right)& =m\left(r,\left(1+\frac{1}{c}\right)A+\mathrm{\Pi }\right)\\ =O\left\{{exp}_{p-1}\left\{\left({\sigma }_{1}+\epsilon \right){log}_{q}\phi \left(r\right)\right\}\right\}\phantom{\rule{1em}{0ex}}\left(r\to \mathrm{\infty }\right).\end{array}$

This implies

${\sigma }_{\left[p,q\right]}\left(\left(1+\frac{1}{c}\right)A+\mathrm{\Pi },\phi \right)=0.$

Hence, by Proposition 1.1, we have $c=-1$. Since ${F}^{2}={F}_{2}^{2}$, we have

$\frac{{F}^{\prime }}{F}=\frac{{F}_{2}^{\prime }}{{F}_{2}},\phantom{\rule{2em}{0ex}}\frac{{F}^{″}}{F}=\frac{{F}_{2}^{″}}{{F}_{2}}.$

From (4.13) and (4.17), we have $\mathrm{\Pi }\equiv 0$, this is a contradiction. Therefore, we obtain the conclusion of Theorem 2.4. □

## References

1. Hayman W: Meromorphic Functions. Clarendon, Oxford; 1964.

2. Yang L: Vaule Distribution Theory and Its New Research. Science Press, Beijing; 1988. (in Chinese)

3. Yang CC, Yi HX Mathematics and Its Applications 557. In The Uniqueness Theory of Meromorphic Function. Kluwer Academic, Dordrecht; 2003.

4. Bank S, Laine I: On the oscillation theory of ${f}^{″}+Af=0$ where A is entire. Trans. Am. Math. Soc. 1982, 273: 352-363.

5. Bank S, Laine I: On the zeros of meromorphic solutions of second-order linear differential equations. Comment. Math. Helv. 1983, 58: 656-677. 10.1007/BF02564659

6. Bank S, Laine I, Langley J: Oscillation results for solutions of linear differential equations in the complex domain. Results Math. 1989, 16: 3-15. 10.1007/BF03322641

7. Gundersen GG: Finite order solutions of second order linear differential equations. Trans. Am. Math. Soc. 1988, 305: 415-429. 10.1090/S0002-9947-1988-0920167-5

8. Juneja OP, Kapoor GP, Bajpai SK:On the $\left(p,q\right)$-order and lower $\left(p,q\right)$-order of an entire function. J. Reine Angew. Math. 1976, 282: 53-67.

9. Juneja OP, Kapoor GP, Bajpai SK:On the $\left(p,q\right)$-type and lower $\left(p,q\right)$-type of an entire function. J. Reine Angew. Math. 1977, 290: 180-190.

10. Li LM, Cao TB:Solutions for linear differential equations with meromorphic coefficients of $\left(p,q\right)$-order in the plane. Electron. J. Differ. Equ. 2012, 2012: 1-15. 10.1186/1687-1847-2012-1

11. Liu J, Tu J, Shi LZ:Linear differential equations with entire coefficients of $\left[p,q\right]$-order in the complex plane. J. Math. Anal. Appl. 2010, 372: 55-67. 10.1016/j.jmaa.2010.05.014

12. Schönhage A: Über das wachstum zusammengesetzter funktionen. Math. Z. 1960, 73: 22-44. 10.1007/BF01163267

13. Kinnunen L: Linear differential equations with solutions of finite iterated order. Southeast Asian Bull. Math. 1998, 22(4):385-405.

14. Cao TB, Li LM: Oscillation results on meromorphic solutions of second order differential equations in the complex plane. Electron. J. Qual. Theory Differ. Equ. 2010, 68: 1-13.

15. Xu HY, Tu J:Oscillation of meromorphic solutions to linear differential equations with coefficients of $\left[p,q\right]$-order. Electron. J. Differ. Equ. 2014., 2014: Article ID 73

16. Chyzhykov I, Heittokangas J, Rättyä J: Finiteness of φ -order of solutions of linear differential equations in the unit disc. J. Anal. Math. 2009, 109: 163-198. 10.1007/s11854-009-0030-3

17. Hayman W: The local growth of power series: a survey of the Wiman-Valiron method. Can. Math. Bull. 1974, 17: 317-358. 10.4153/CMB-1974-064-0

18. He YZ, Xiao XZ: Algebroid Functions and Ordinary Differential Equations. Science Press, Beijing; 1988. (in Chinese)

19. Laine I: Nevanlinna Theory and Complex Differential Equations. de Gruyter, Berlin; 1993.

20. Gao SA, Chen ZX, Chen TW: The Complex Oscillation Theory of Linear Differential Equations. Huazhong Univ. Sci. Tech. Press, Wuhan; 1998. (in Chinese)

21. Jank G, Volkmann L: Einführung in die Theorie der ganzen und meromorphen Funktionen mit Anwendungen auf Differentialgleichungen. Birkhäuser, Basel; 1985.

## Acknowledgements

The authors thank the referee for his/her valuable suggestions to improve the present article. This work was supported by the NSFC (11301233), the Natural Science Foundation of Jiang-Xi Province in China (20132BAB211001, 20132BAB211002), and the Foundation of Education Department of Jiangxi (GJJ14272, GJJ14644) of China.

## Author information

Authors

### Corresponding author

Correspondence to Hong Yan Xu.

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

XS, JT and HYX completed the main part of this article, JT and HYX corrected the main theorems. All authors read and approved the final manuscript.

## Rights and permissions

The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.

Reprints and permissions

Shen, X., Tu, J. & Xu, H.Y. Complex oscillation of a second-order linear differential equation with entire coefficients of $\left[p,q\right]-\phi$ order. Adv Differ Equ 2014, 200 (2014). https://doi.org/10.1186/1687-1847-2014-200