\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2007(2007), No. 134, pp. 1--7.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2007 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2007/134\hfil Sectorial oscillation theory]
{Sectorial oscillation of linear differential equations and iterated
order}

\author[Z.-J. Wu, D.-C. Sun  \hfil EJDE-2007/134 \hfilneg]
{Zhao-Jun Wu, Dao-chun Sun}  

\address{Zhao-Jun Wu \newline
Department of Mathematics,  Xianning University, Hubei Xianning,
437100,  China}
\curraddr{School of Mathematics, South China Normal University,
 Guangzhou, 510631,  China}
\email{wuzj52@hotmail.com}

\address{Dao-chun Sun \newline
School of Mathematics, South China Normal University, Guangzhou,
510631,  China} \email{sundch@scnu.edu.cn}


\thanks{Submitted June 26 2007. Published October 12, 2007.}
\thanks{Supported by grants 10471048 from the NNSF of China,
 and KT0623,KZ0629 from the
NSF \hfill\break\indent of Xianning University}
\subjclass[2000]{34M10, 30D35}
\keywords{Iterated order; iterated convergence exponent}

\begin{abstract}
 In the present paper, we investigate higher order linear
 differential equations with entire coefficients of iterated order.
 Using value distribution theory of transcendental meromorphic
 functions and covering surface theory, we extend a result
 on the order of growth of solutions published by
 Bank and Langley \cite{bank2}.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}

\section{Introduction and main results}

In 1982, Bank and Laine \cite{bank1}  investigated
the exponent of convergence of zeros of the solutions for the
differential equation
\begin{equation}
f''+A(z)f=0,\label{e1.1}
\end{equation}
 where $A(z)$ is a transcendental entire
function and $E$ is the product of normalized linearly independent
solutions $f_{1},f_{2}$ for \eqref{e1.1}. They proved that
$$
\sigma(E)=\max\{\sigma(A), \lambda(E)\}.
$$
A considerable number of research results concerning
\eqref{e1.1} have been proved. We refer the reader to
the book by Laine \cite{laine} for a summary of those results.
We assume that the reader is  familiar with the basic
results and notation of the Nevanlinna's value distribution theory
of meromorphic functions (see \cite{yang},\cite{hayman}), such as
$\sigma(f), \lambda(f)$ to denote, respectively the order and
exponent of convergence of meromorphic function $f$.

For $k\geq 2$, we consider a linear differential equation
\begin{equation}
f^{(k)}+A_{k-2}f^{(k-2)}+\dots+A_{0}f=0, \label{e1.2}
\end{equation}
where $A_{0},\dots,A_{k-2}$ are entire functions with
$A_{0}\not\equiv 0$. It is well known that all solutions of
\eqref{e1.2} are entire functions, and if some of the coefficients
of \eqref{e1.2} are transcendental, then \eqref{e1.2} has at least
one solution with order $\sigma(f)=\infty$. Now there exists a
question: How to describe precisely the properties of growth of
solutions of infinite order of \eqref{e1.2}? It is to make use of
iterated order of entire functions, see Laine \cite{laine}. Let us
define inductively (see e.g. \cite{bernal}), for $r\in [0,+\infty),
\exp^{[1]}r=e^{r}$ and $\exp^{[n+1]}r=\exp(\exp^{[n]}r), n\in
\mathbb{N}$. For all $r$ sufficiently large, we define
$\log^{[1]}r=\log r$ and $\log^{[n+1]}r=$ $\log(\log^{[n]}r), n\in
\mathbb{N}$. We also denote $\exp^{[0]}r$$=r=$$\log^{[0]}r$,
$\log^{[-1]}r=\exp^{[1]}r$ and $\exp^{[-1]}r$$=\log^{[1]}r$. We
recall the following definitions and remarks (see
\cite{kinnunen,sato,cao}).

\begin{definition} \label{def1} \rm
The iterated $p$-order $\sigma _{p}(f)$ of a meromorphic function
$f(z)$ is defined by
$$
\sigma_{p}(f)=\limsup_{r\to \infty} \frac{\log^{[p]}T(r,f)}{\log r }
\quad(p\in \mathbb{N}).
$$
\end{definition}

\begin{remark} \label{rmk1} \rm
(1) If $p=1$, then we denote $\sigma _{1}(f)=\sigma(f)$.
(2) If $p=2$, then we denote by $\sigma_{2}(f)$ the so-called hyper order
(see \cite{yi-yang}).
(3) If $f(z)$ is an entire function, then
$$
\sigma _{p}(f)=\limsup_{r\to\infty} \frac{\log^{[p+1]}M(r,f)}{\log r}.
$$
\end{remark}

\begin{definition} \label{def2} \rm
 The growth index of the
iterated order of a meromorphic function $f(z)$ is defined by
$$
i(f)=\begin{cases}
0 & \text{if $f$ is  rational},  \\
\min\{n\in \mathbb{N}:\sigma _{n}(f)<\infty\}
& \text{if $f$  is transcendental and}\\
& \text{$\sigma_{n}(f)<\infty $ for  some $n\in \mathbb{N}$}, \\
\infty & \text{if $\sigma _{n}(f)=\infty $ for all $n\in
\mathbb{N}$.}
\end{cases}
$$
\end{definition}

\begin{definition} \label{def2b} \rm
The iterated convergence exponent of the sequence of $a$-points
$(a\in \mathbb{C}\cup\{\infty\})$ is defined by
$$
\lambda_{n}(f-a)=\lambda_{n}(f,a)=\limsup_{r\to \infty}
\frac{\log^{[n]}N(r,\frac{1}{f-a})}{\log r}\quad(n\in \mathbb{N}),
$$
and $\overline{\lambda}_{n}(f-a)$, the  iterated convergence
exponent of the sequence of distinct $a$-points is defined by
$$
\overline{\lambda}_{n}(f-a)=\overline{\lambda}_{n}(f,a)=
\limsup_{r\to\infty}
\frac{\log^{[n]}\overline{N}(r,\frac{1}{f-a})}{\log r}\quad(n\in
\mathbb{N}).
$$
\end{definition}

\begin{remark} \label{rmk2} \rm
(1) $\lambda_{1}(f-a)=\lambda(f-a)$.
(2) $\overline{\lambda}_{1}(f-a)=\overline{\lambda}(f-a)$.
\end{remark}

For the sake of convenience, we also make the following definitions
and remarks.

\begin{definition} \label{def3} \rm
The iterated sectorial convergence exponent of the sequence of
$a$-points $(a\in \mathbb{C}\cup\{\infty\})$ is defined by
$$
\lambda_{n,\alpha,\beta}(f-a)=\lambda_{n,\alpha,\beta}(f,a)=\limsup_{r\to
\infty} \frac{\log^{[n]}n(r,X(\alpha,\beta),\frac{1}{f-a})}{\log
r}\quad(n\in \mathbb{N}),
$$
and $\overline{\lambda}_{n}(f-a)$, the  iterated sectorial
convergence exponent of the sequence of distinct $a$-points is
defined by
$$
\overline{\lambda}_{n}(f-a)=\overline{\lambda}_{n}(f,a)=
\limsup_{r\to\infty}
\frac{\log^{[n]}\overline{n}(r,X(\alpha,\beta),\frac{1}{f-a})}{\log
r}\quad(n\in \mathbb{N}).
$$
where $X(\alpha,\beta)=\{z|
  \alpha< \arg z<\beta\}, 0<\beta-\alpha\leq \pi$ and $n(r,X(\alpha,\beta),f=a)$
  is the roots of $f(z)-a=0$ in $\Omega(\alpha,\beta)\cap\{|z|<r\}$,
  counting multiplicities, and $\overline{n}(r,X(\alpha,\beta),f=a)$ is
  the corresponding notion ignoring multiplicities.
\end{definition}

\begin{remark}[\cite{wang}] \label{rmk3} \rm
(1) $\lambda_{1,\alpha,\beta}(f-a)=\lambda_{\alpha,\beta}(f-a)$.
(2) $\overline{\lambda}_{1,\alpha,\beta}(f-a)
 =\overline{\lambda}_{\alpha,\beta}(f-a)$.
\end{remark}

\begin{definition} \label{def4} \rm
The iterated radial convergence exponent of the sequence of
$a$-points $(a\in \mathbb{C}\cup\{\infty\})$ is defined by
$$
\lambda_{n,\theta}(f-a)=\lambda_{n,\theta}(f,a)=
\lim_{\varepsilon\to0^{+}}\lambda_{n,\theta-\varepsilon,\theta+\varepsilon}(f,a).\quad(n\in
\mathbb{N}),
$$
\end{definition}

\begin{remark}[\cite{wang}] \label{rmk4} \rm
(1) $\lambda_{1,\theta}(f-a)=\lambda_{\theta}(f-a)$.
(2) $\overline{\lambda}_{1,\theta}(f-a)=\overline{\lambda}_{\theta}(f-a)$.
\end{remark}


In 1991,  Bank and  Langley considered the higher order linear
differential equations and obtained the following result.

\begin{theorem}[\cite{bank2}] \label{thm1}
 Let $A_{0},\dots,A_{k-2}$ be entire functions of finite order,
 and assume that \eqref{e1.2} possesses a solution base
 $f_{1},f_{2},\dots,f_{n}$
 such that $\lambda(f_{i})<+\infty$ for $i=1,2,\dots,n$. Then the
 product $E=f_{1}\dots f_{n}$ is of finite order of growth, $\sigma(E)<\infty$.
\end{theorem}

In this paper, we  extend Theorem \ref{thm1} by using value
distribution theory of a transcendental meromorphic function due to
Nevanlinna \cite{nevanlinna}
 and the covering surface theory (see e.g. \cite{tsuji}). In fact,
 we shall prove the following theorem.

\begin{theorem} \label{thm2}
 Assume that some (or all) of
$A_{0},\dots,A_{k-2}$ are transcendental entire functions, and
$p=\max\{i(A_{j}), j=1,\dots,k-2\}<\infty$.  Suppose that
\eqref{e1.2} possesses a solution base $f_{1},f_{2},\dots,f_{n}$.
If $E:=f_{1}\dots f_{n}$ is of infinite iterated $p$-order growth,
i.e. $\sigma_{p}(E)=\infty$, then there at least exists a ray $L:
\arg z=\theta$ such that $\lambda_{p,\theta}(E)=\infty$.
\end{theorem}

From Theorem \ref{thm2}, we can deduce the following result.

\begin{corollary}\label{coro1}
Under the  conditions of Theorem \ref{thm2}, we
assume that \eqref{e1.2} possesses a solution base
$f_{1},f_{2},\dots,f_{n}$
 such that $\lambda_{p}(f_{i})<+\infty$ for $i=1,2,\dots,n$. Then the
 product $E=f_{1}\dots f_{n}$ is of finite iterated $p$-order growth, i.e.
 $\sigma_{p}(E)<+\infty$.
\end{corollary}

When $p=1$, Corollary \ref{coro1} becomes Theorem \ref{thm1}.


\section{Auxiliary Lemmas}

Our proof requires the Nevanlinna's theory in an angular domain. Let
$f(z)$ be
  a meromorphic function and
  $X(\alpha,\beta)=\{z|
  \alpha\leq \arg z\leq\beta\}$ be an angular domain, where $0<\beta-\alpha\leq2\pi$.
Nevanlinna defined the following notation (\cite{nevanlinna}),
\begin{gather*}
A_{\alpha, \beta}(r,f)=\frac{k}{\pi}\int_{1}^{r}(\frac{1}{t^{k}}
 -\frac{t^{k}}{r^{2k}})
 \{\log^{+}|f(te^{i\alpha})|+\log^{+}|f(te^{i\beta})|\}\frac{dt}{t};
\\
B_{\alpha, \beta}(r,f)=\frac{2k}{\pi r^{k}}\int_{\alpha}^{\beta}
\log^{+}|f(re^{i\theta})|\sin k(\theta-\alpha)d\theta;
\\
C_{\alpha, \beta}(r,f)=2\sum_{b\in\Delta}(\frac{1}{|b_{v}|^{k}}
 -\frac{|b_{v}|^{k}}{r^{2k}})\sin k(\beta_{v}-\alpha),
\end{gather*}
where $k=\frac{\pi}{\beta-\alpha},1\leq r<\infty$ and the summation
$\sum_{b\in\Delta}$ is taken over all poles
$b=|b|e^{i\theta}$ of the function $f(z)$ in the sector
$\Delta: 1<|z|<r,  \; \alpha<\arg z <\beta$, counting multiplicity. The
corresponding notation $\overline{C}(r,f)$ then applies to distinct
poles. Furthermore, for $r>1$, we define
$$
D_{\alpha, \beta}(r,f)=A_{\alpha, \beta}(r,f)+B_{\alpha, \beta}(r,f),\quad
S_{\alpha, \beta}(r,f)=C_{\alpha, \beta}(r,f)+D_{\alpha,
\beta}(r,f).
$$

 For the sake of simplicity, we omit the subscript of all the notation and
 use the notation $A(r,f)$, $ B(r,f)$, $ C(r,f)$, $ D(r,f)$ and $S(r,f)$
instead of
 $A_{\alpha, \beta}(r,f)$, $ B_{\alpha, \beta}(r,f)$, $C_{\alpha, \beta}(r,f)$,
 $D_{\alpha, \beta}(r,f)$ and $S_{\alpha, \beta}(r,f)$.

\begin{lemma}[\cite{wusj}] \label{lem1}
 Suppose that $f(z)$ is a meromorphic function and
$\Omega (\alpha,\beta)$ be an angular
domain, where $0<\beta-\alpha\leq 2\pi$. Then,
\begin{itemize}
\item[(i)] for any value $a\in \mathbb{C}$, we have
$$
S(r,\frac{1}{f-a})=S(r,f)+O(1),
$$
holds for any $r>1$.
\item[(ii)] for any $r<R$,
\begin{gather*}
A(r,\frac{f'}{f})\leq k \{(\frac{R}{r})^{k}\int_{1}^{R}
 \frac{\log T(t,f)}{t^{1+k}}dt+\log \frac{r}{R-r}+\log\frac{R}{r}+1\},\\
B(r,\frac{f'}{f})\leq\frac{4k}{r^{k}}m(r,\frac{f' }{f}).
\end{gather*}
\end{itemize}
\end{lemma}

We also need the Ahlfors' theory in an angular domain. We
firstly recall some notation (see e.g. Tsuji \cite{tsuji}).

Let $f(z)$ be a meromorphic function in an angular domain
$\Delta(\theta,\alpha_{0})=\{z:|\arg z-\theta|\leq\alpha_{0}\}$
and $\Delta(\theta,\alpha)=\{z:|\arg z-\theta|\leq \alpha\}$ be
an angular domain which was contained in
$\Delta(\theta,\alpha_{0})$, where $\theta\in[0,2\pi)$ and
$\alpha\leq\alpha_{0}$. Let $\Delta_{0}(r)$, $\Delta(r)$ be
the part of $\Delta(\theta,\alpha_{0})$,
$\Delta(\theta,\alpha)$, which is contained in $|z|\leq r$,
respectively. We put
 \begin{gather*}
S_{0}(r,\Delta(\theta,\alpha))=\frac{1}{\pi}
\iint_{\Delta(r)} (\frac{|f^{'}(z)|}{(1+|f(z)|^2)})^2 r
d\theta dr,\quad z=re^{i\theta},\\
T_{0}(r,\Delta(\theta,\alpha))
 =\int_{0}^{r}\frac{S_{0}(t,\Delta(\theta,\alpha))}{t}dt,
\end{gather*}
which is called as Ahlfors-Shimizu characteristics. We denote the
above characteristic functions of $f(z)$ in the whole complex plane
by $S_{0}(r,f), T_{0}(r,f)$. From  \cite[Theorem 1.4]{hayman}, we
have
\begin{equation}
|T(r,f)-T_{0}(r,f)-\log|f(0)||\leq\frac{1}{2}\log 2.\label{e2.1}
\end{equation}


Let $n(r,\theta,\alpha,a)$ be the number of zeros of $f(z)-a$
contained in $\Delta(r)$, counting multiplicities. We can assume
that $f(0)\neq a$ and put
 $$
N(r,\theta,\alpha,a)=\int_{0}^{r}\frac{n(t,\theta,\alpha,a)}{t}dt.
$$
If not, then the definition has to be modified, in a well known
manner. Now, we give the following lemmas.

\begin{lemma}[\cite{tsuji}] \label{lem2}
 Let $f(z)$ be meromorphic in the complex
plane, then
\begin{gather*}
S_{0}(r,\Delta(\theta,\alpha))\leq 3\sum_{i=1}^{3}n(2r,\theta,\alpha_{0},
a_{i})+O(\log r),\\
T_{0}(r,\Delta(\theta,\alpha))\leq 3\sum_{i=1}^{3}N(2r,\theta,\alpha_{0},
a_{i})+O(\log^{2} r).
\end{gather*}
where $a_{1},a_{2},a_{3}$ be any three distinct points in
$\mathbb{C}_{\infty}$.
\end{lemma}


\section{Proof of main results}

 \begin{proof}[Proof of Theorem \ref{thm2}]
The Wronskian determinant $W( f_{1},f_{2},\dots,f_{n} )$ of the
fundamental system of solutions $\{ f_{1},f_{2},\dots,f_{n}\} $ is
given by
$$
W=W( f_{1},f_{2},\dots,f_{n} )=
\det  \begin{bmatrix}
 1&1&\dots&1\\
\frac{f'_{1}}{f_{1}}&\frac{f'_{2}}{f_{2}}&\dots&\frac{f'_{n}}{f_{n}}\\
&\dots&\dots\\
\frac{f^{(n-1)}_{1}}{f_{1}}&\frac{f^{(n-1)}_{2}}{f_{2}}&\dots&\frac{f^{(n-1)}_{n}}{f_{n}}
  \end{bmatrix}
$$
Apply the  \cite[Proposition 1.4.8 pp.16]{laine}, we can derive that
$W$ is a positive constant  denoted by $K$. Hence
$$
\frac{1}{E}=\frac{1}{K}\frac{W}{E}=\frac{1}{K}\sum_{1\leq i_{l}
\neq i_{t}\leq n}(-1)^{\tau}\Pi_{l=1}^{n-1}
\frac{f_{i_{l}}^{(l)}}{f_{i_{l}}}.
$$
Let $f\not\equiv 0$ be a solution of  \eqref{e1.2}. It follows from
\cite[Theorem 4 (i)]{bernal} that the iterated $p$-order of
$\log T(r,f)$ is at most $\sigma$, where $\sigma<\infty$ is a constant.

For any $\theta\in\mathbb{R}$, if $\varepsilon>0$ is sufficiently
small, we deduce from Lemma \ref{lem1} (ii) in which $R=2r $ that
\begin{equation}
A_{\theta-\varepsilon,\theta+\varepsilon}(r,\frac{f'_{i}}{f_{i}})
=\begin{cases}
O(1) & \text{if }p=1,  \\
O(\int_{1}^{2r}\frac{\log^{+}T(t,f_{i})}{t^{1+\frac{\pi}{2\varepsilon}}}dt)\\
=O(\int_{1}^{2r}\frac {e^{[p-1]}t^{\sigma +
1}}{t^{1+\frac{\pi}{2\varepsilon}}}dt)
=O(e^{[p-1]}r^{\sigma + 1}). & \text{if }p\geq2.
\end{cases}\label{e3.1}
\end{equation}
Since
$$
m(r,\frac{f'_{i}}{f_{i}})=O(\log rT(r,f_{i}))=O(e^{[p-1]}r^{\sigma+1}),
\quad  r\not\in F,
$$
where $F$ is a set of finite linear measure, we can deduce from
lemma \ref{lem1} (ii) that
\begin{equation}
B_{\theta-\varepsilon,\theta+\varepsilon}(r,\frac{f'_{i}}{f_{i}})
=\begin{cases}
O(1) & \text{if } p=1,  \\
O(e^{[p-1]}r^{\sigma+1}). & \text{if } p\geq2.
\end{cases}\label{e3.2}
\end{equation}
holds for any $r\not\in F$.
Since
$$
D_{\theta-\varepsilon,\theta+\varepsilon}(r,\frac{f_{i}^{(h)}}{f_{i}})
 \leq \sum_{i=1}^{h}
D_{\theta-\varepsilon,\theta+\varepsilon}
(r,\frac{f_{i}^{(l)}}{f_{i}^{(l-1)}})+O(1),
$$
where $i=1,2,\dots,n$, $h=2,3,\dots,n-1$. Therefore we have
\begin{equation*}
D_{\theta-\varepsilon,\theta+\varepsilon}(r,\frac{f'_{i}}{f_{i}})
=\begin{cases}
O(1) & \text{if } p=1,  \\
O(e^{[p-1]}r^{\sigma+1}). & \text{if } p\geq2.
\end{cases}
\end{equation*}
By the definition and Lemma \ref{lem1} (i), we can deduce that for any
$\theta\in\mathbb{R}$ and any sufficiently small $\varepsilon>0$,
\begin{equation}
S(r,E)\leq C(r,\frac{1}{E})+O(e^{[p-1]}r^{\sigma+1}), \quad r\not\in
F\label{e3.3}
\end{equation}
holds in the angular domain $\{z|\theta-\varepsilon<\arg
z<\theta+\varepsilon\}$.

In the following, we shall prove that there exists a ray
$L: \arg z=\theta$ such that for any $0<\varepsilon<\frac{\pi}{2}$,
we have
\begin{equation}
\limsup_{r\to \infty}  \frac{\log^{[p]} S(r,E)}{\log
r}=\infty\label{e3.4}
\end{equation}
holds  in the angular domain $\{z|\theta-\varepsilon<\arg
z<\theta+\varepsilon\}$. Otherwise, for any $\theta\in[0,2\pi)$, we
have a $\varepsilon_{\theta}\in(0,\frac{\pi}{2})$, such that
\begin{equation}
\limsup_{r\to \infty}  \frac{\log^{[p]} S(r,E)}{\log
r}<\infty.\label{e3.5}
\end{equation}
holds in the angular domain $\{z|\theta-\varepsilon_{\theta}<\arg
z<\theta+\varepsilon_{\theta}\}$. We deduce from Lemma \ref{lem1} (i) that
for any finite value $a$, we have $S(r,\frac{1}{E-a})=S(r,E)+O(1)$.
Since $C(r,a)\leq S(r,\frac{1}{E-a})$, then
\begin{equation}
C(r,\frac{1}{E-a})\leq S(r,\frac{1}{E-a})=S(r,E)+O(1).\label{e3.6}
\end{equation}
On the other hand, it follows from
$\theta-\frac{\varepsilon_{\theta}}{2}<\beta_{v}
<\theta+\frac{\varepsilon_{\theta}}{2}$
that $\sin k(\beta_{v}-\theta+\frac{\varepsilon_{\theta}}{2})\geq
\sin\frac{\pi}{4}=\frac{\sqrt{2}}{2}$, where
$k=\frac{\pi}{2\varepsilon_{\theta}}$. Hence
\begin{align*}
C(2r,\frac{1}{E-a})
&\geq C_{\theta-\frac{\varepsilon_{\theta}}{2},\theta
 +\frac{\varepsilon_{\theta}}{2}}(2r,\frac{1}{E-a})\\
&\geq 2\sum_{1<|b_{v}|<r,\theta-\frac{\varepsilon_{\theta}}{2}
 <\beta_{v}<\theta+\frac{\varepsilon_{\theta}}{2}}
 (\frac{1}{|b_{v}|^{k}}-\frac{|b_{v}|^{k}}{(2r)^{2k}})\sin
 k(\beta_{v}-\theta+\frac{\varepsilon_{\theta}}{2})\\
&\geq \sqrt{2}\sum_{1<|b_{v}|<r,\theta-\frac{\varepsilon_{\theta}}{2}
 <\beta_{v}<\theta+\frac{\varepsilon_{\theta}}{2}}
 (\frac{1}{|b_{v}|^{k}}-\frac{|b_{v}|^{k}}{(2r)^{2k}})\\
&\geq \sqrt{2}[\int_{1}^{r}\frac {1}{t^{k}}dn(t)+
 \frac{1}{(2r)^{2k}}\int_{1}^{r}t^{k}dn(t)]\\
&\geq \sqrt{2}[ k\int_{1}^{r}\frac {1}{t^{k+1}}n(t)dt+
 \frac{n(r)}{r^{k}}-
 \frac{r^{k}n(r)}{r^{2k}}+\frac{k}{(2r)^{2k}}\int_{1}^{r}t^{k-1}n(t)dt]\\
&\geq   \sqrt{2}[ \frac{n(r)}{r^{k}}-
 \frac{r^{k}n(r)}{(2r)^{2k}}]\\
&\geq \sqrt{2}(1- \frac{1}{2^{2k}})\frac{n(r)}{r^{k}},
\end{align*}
where $n(t)=n(t,\theta,\frac{\varepsilon_{\theta}}{2},a)$.
 From \eqref{e3.5}, \eqref{e3.6} and the above equation,
\begin{equation}
\limsup_{r\to \infty}  \frac{\log^{[p]}
n(r,\theta,\frac{\varepsilon_{\theta}}{2},a)}{\log
r}<\infty.\label{e3.7}
\end{equation}
Because $[0,2\pi]$ is compact and $[0,2\pi]\subset
\cup\{(\theta-\frac{\varepsilon_{\theta}}{4},\theta-\frac{\varepsilon_{\theta}}{4}),\theta\in[0,2\pi)\}$,
then we can choose finitely many
$(\theta_{i}-\frac{\varepsilon_{\theta_{i}}}{4},
\theta_{i}-\frac{\varepsilon_{\theta_{i}}}{4})(i=1,2,\dots,T)$,
such that $[0,2\pi]\subset
\cup\{(\theta_{i}-\frac{\varepsilon_{\theta_{i}}}{4},
\theta_{i}-\frac{\varepsilon_{\theta_{i}}}{4}),i=1,2,\dots,T\}$.

By using Lemma \ref{lem2}, for any three distinct complex numbers
$a_{j}$, $j=1,2,3$, we have
\begin{align*}
S_{0}(r,f)
&\leq \sum_{i=1}^{T}S_{0}(r,\Delta(\theta_{i},
\frac{\varepsilon_{\theta_{i}}}{4}))\\
& \leq \sum_{i=1}^{T}\{3\sum_{i=j}^{3}n(2r,\theta_{i},
\frac{\varepsilon_{\theta_{i}}}{2},a_{j})\}+O(\log r)
\end{align*}
 From \eqref{e2.1}, \eqref{e3.7} and the definition of $T_{0}(r,f)$
and the above equation, we can get that $E$ is of finite
$p$-iterated order. This contradicts with the hypothesis and so
\eqref{e3.4} follows.

 From \eqref{e3.3}, \eqref{e3.4} and definition \ref{def1}, we know that there
exists a ray $L: \arg z=\theta$ such that for any
$0<\varepsilon<\frac{\pi}{2}$, we have
\begin{equation}
\limsup_{r\to \infty}  \frac{\log^{[p]}
C(r,\frac{1}{E})}{\log r}=\infty\label{e3.9}
\end{equation}
holds in the angular domain $\{z|\theta-\varepsilon<\arg
z<\theta+\varepsilon\}$. Since $C(r,\frac{1}{E})\leq
2n(r,\theta,\varepsilon,E=0)$, then $
\lambda_{p,\theta-\varepsilon,\theta+\varepsilon}(E)=\infty$. Since
$\varepsilon$ is arbitrary, we have
 $\lambda_{p,\theta}(E)=\infty$. Therefore, we can deduce that Theorem
\ref{thm2}.
\end{proof}


\begin{thebibliography}{00}

\bibitem{bank1} S. Bank and I. Laine,
 \emph{On the oscillation theory of $f''+Af=0$ where
$A$ is entire}, Trans. Amer. Math. Soc. 273(1982), 351-363.

\bibitem{bank2}  S. Bank and J. Langley,
\emph{Oscillation theory for higher order linear differential
equations with entire coefficients}, Complex Variables Theory Appl.,
16(1991), 163-175.

\bibitem{bernal} Luis G. Bernal,
 \emph{ On growth $k$-order of solutions of a
complex homogenous linear differential equation,}  Proc. Amer. Math.
Soc., 101(1987), 317-322.

\bibitem{cao} T.-B. Cao,
\emph{Complex oscillation of entire solutions of higher order linear
differential equations}, Electronic Journal of Diff. Eqs.
2006(2006), no. 81, 1-8.

\bibitem{hayman} W. Hayman,
\emph{Meromorphic Functions}, Clarendon Press, Oxford, 1964.

\bibitem{kinnunen} L. Kinnunen, \emph{Linear differential equations
with solutions of finite iterated order}, Southeast Asian Bull.
Math.  22(1998), no. 4, 385-405.

\bibitem{laine} I. Laine, \emph{ Nevanlinna Theory and Complex
Differential Equations}, Walter de Gruyter, Berlin-New York, 1993.

\bibitem{nevanlinna} R. Nevanlinna,
\emph{\"Uber die Eigenschaften meromorpher Funktionen in
einem Winkelraum}, Acta Soc. Sci. Fenn., 50(1925), 12:1-45.

\bibitem{sato} D. Sato,
\emph{On the rate of growth of entire functions of fast growth},
Bull. Amer. Math. Soc. 69(1963), 411-414.

\bibitem{tsuji} M. Tsuji.,
\emph{Potential theory in modern function theory,} Maruzen
Co. Ltd, Tokyo,1959.

\bibitem{wang} S.-P. Wang.,
\emph{On the sectorial oscillation theory of
$f''+A(z)f=0$,} Ann. Acad. Sci. Fenn. A I Math. Diss, {\bf 92},
(1994).

\bibitem{wusj} S.-J. Wu,
\emph{ On the location of zeros of solution of $f^{"}+Af=0$ where
$A(z)$ is entire}, Math. Scand., 74(1994), 293-312.

\bibitem{yang} L. Yang,
\emph{Value Distribution Theory and New Research},
 Science Press, Beijing, 1982.

\bibitem{yi-yang} C.-C. Yang and H.-X. Yi,
\emph{Uniqueness Theory of Meromorphic Functions},
Science Press Kluwer, Beijing, 2003.

\end{thebibliography}

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