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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 68, pp. 1--12.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2015/68\hfil 
 Oscillation of arbitrary-order derivatives of solutions]
{Oscillation of arbitrary-order derivatives of solutions to linear differential
equations taking small functions in the unit disc}

\author[P. Gong, L. P. Xiao \hfil EJDE-2015/68\hfilneg]
{Pan Gong, Li-Peng Xiao}

\address{Pan Gong \newline
Institute of Mathematics and Information Science,
Jiangxi Normal University, Nanchang 330022, China}
\email{gongpan12@163.com}

\address{Li Peng Xiao (corresponding author) \newline
Institute of Mathematics and Information Science,
Jiangxi Normal University,
Nanchang 330022, China}
\email{2992507211@qq.com}


\thanks{Submitted January 6, 2015. Published March 20, 2015.}
\subjclass[2000]{34M10, 30D35}
\keywords{Unit disc; iterated order; growth; exponent of convergence}

\begin{abstract}
 In this article, we study the relationship between solutions and
 their derivatives of the differential equation
 $$
 f''+A(z)f'+B(z)f=F(z),
 $$
 where $A(z), B(z), F(z)$ are meromorphic functions of finite iterated
 $p$-order in the unit disc.
 We obtain some oscillation theorems for $f^{(j)}(z)-\varphi(z)$,
 where $f$ is a solution and $\varphi(z)$ is a small function.
\end{abstract}

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


\section{Introduction and results}\label{intro}

Throughout this paper, we assume that the reader is familiar with
the fundamental results and the standard notations of the
Nevanlinna's value distribution theory on the complex plane and in
the unit disc $\Delta=\{z\in\mathbb{C}:|z|<1\}$ 
(see \cite{11,12,15,16,19}). 
In addition, we need to give some definitions and discussions. 
Firstly, let us give two definitions about the degree of small growth 
order of functions in $\Delta$ as polynomials on the complex plane
$\mathbb{C}$. There are many types of definitions of small growth
order of functions in $\Delta$ (see \cite{9,10}).

 \begin{definition}[\cite{9,10}] \label{def1.1} \rm
  Let $f$ be a meromorphic function in $\Delta$ , and
 $$
D(f)=\limsup_{r\to 1^-}\frac{T(r,f)}{\log \frac{1}{1-r}}=b.
$$ 
If $b<\infty$, then we say that  $f$ is
of finite $b$  degree (or is non-admissible). If $b=\infty$ , then
we say that $f$  is  of infinite degree (or is admissible), both defined by
characteristic function $T(r,f)$.
\end{definition}

\begin{definition}[\cite{9,10}] \label{def1.2} \rm
  Let $f$ be an analytic function in $\Delta$ , and
\[
 D_{M}(f)=\limsup_{r\to 1^-}\frac{\log^+M(r,f )}{\log \frac{1}{1-r}}=a 
\quad (\text{or }a=\infty).
\]
Then we say that  $f$ is a function of finite $a$ degree (or of infinite
degree) defined by maximum modulus function
$M(r,f)=\max_{| z|=r}| f(z)|$.

For $F\subset [0,1)$, the upper and lower densities of $F$
are defined by 
$$
\overline{\operatorname{dens}}_\triangle
F=\limsup_{r\to 1^-}\frac{m(F\cap[0,r))}{m([0,r))},\quad
\underline{\operatorname{dens}}_\triangle  F=\liminf_{r\to
1^-}\frac{m(F\cap[0,r))}{m([0,r))}
$$ 
respectively, where
$m(G)=\int_G\frac{dt}{1-t} $ for $G\subset[0,1)$.
\end{definition}

Now we give the definition of iterated order and growth index to
classify generally the functions of fast growth in $\Delta$
as those in $\mathbb{C}$, see \cite{3,14,15}. Let us define
inductively, for $r\in[0,1), \exp_1{r}=e^{r}$ and
$\exp_{p+1}{r}=\exp(\exp_p{r}),p\in \mathbb{N}$. We also define for
all $r$ sufficiently large in $(0,1)$, $\log_1{r}=\log{r}$ and
$\log_{p+1}{r}=\log(\log_p{r}), p\in \mathbb{N}$. Moreover, we
denote by $\exp_0{r}=r, \log_0{r}=r, \exp_{-1}{r}=\log_1{r},
\log_{-1}{r}=\exp_1{r}$.

\begin{definition}[\cite{4}] \label{def1.3} \rm
 The iterated $p$-order  of a
meromorphic function $f$  in $\Delta$  is defined by
\begin{equation*}
\rho_{p}(f)=\limsup_{r\to 1^-}\frac{\log_{p} ^{+}
T(r,f)}{\log \frac {1}{1-r}}\quad (p\geq 1).
\end{equation*}
For an analytic function $f$  in $\Delta$ , we also define
\begin{equation*}
\rho_{M,p}(f)=\limsup_{r\to 1^-}\frac{\log_{p+1} ^{+}
M(r,f)}{\log \frac {1}{1-r}}\quad (p\geq 1).
\end{equation*}
\end{definition}

\begin{remark} \label{rmk1.1} \rm
It follows by  Tsuji  \cite{19} that if
$f$ is an analytic function in $\Delta$, then
\begin{equation*}
\rho_1(f)\leq \rho_{M,1}(f)\leq \rho_1(f)+1.
\end{equation*}
However it follows by \cite[Proposition 2.2.2]{15} that
\begin{equation*}
\rho_{M,p}(f)= \rho_p(f)\quad(p\geq 2).
\end{equation*}
\end{remark}

\begin{definition}[\cite{4}] \label{def1.4} \rm
The growth index of the iterated order of a meromorphic function  $f$  
in  $\Delta$ is defined by 
\[
i(f)=\begin{cases}
   0, &  \text{if $f$ is non-admissible}; \\ 
\min\{p\in \mathbb{N},\rho_{p}(f)<\infty\}, &
   \text{if $f$ is admissible}; \\ 
\infty, &     \text{if $\rho_{p}(f)=\infty$  \ for all $p\in \mathbb{N}$}.
  \end{cases} 
\]
For an analytic function $f$ in $\Delta$, we also define
\[
i_{M}(f)= \begin{cases}
   0, &  \text{if $f$ is non-admissible}; \\
 \min\{p\in \mathbb{N},\rho_{M,p}(f)<\infty\}, &
   \text{if $f$ is admissible}; \\ 
\infty, & \text{if $\rho_{M,p}(f)=\infty$  for all $p\in \mathbb{N}$}.
  \end{cases} 
\]
\end{definition}


 \begin{definition}[\cite{5,6}] \label{def1.5} \rm
Let $f$  be a meromorphic function in  $\Delta$. Then the iterated 
$p$-exponent of convergence of the sequence of zeros of $f(z)$  is defined by
\begin{equation*}
\lambda_{p}(f)=\limsup_{r\to 1^-}\frac{\log_{p} ^{+}
N(r,\frac{1}{f})}{\log \frac {1}{1-r}},
\end{equation*}
where $N(r,\frac{1}{f})$  is the integrated counting function of zeros of
$f(z)$ in $\{z\in \mathbb{C}: |z|  \leq r\}$.
Similarly, the iterated p-exponent of convergence of the sequence of distinct zeros
of $f(z)$ is defined by
\begin{equation*}
\overline{\lambda}_{p}(f)=\limsup_{r\to 1^-}\frac{\log_{p}
^{+} \overline{N}(r,\frac{1}{f})}{\log \frac {1}{1-r}},
\end{equation*}
where $\overline{N}(r,\frac{1}{f})$  is the integrated counting function of
distinct zeros of $f(z)$ in $\{z\in \mathbb{C}:|z| \leq r\}$.
\end{definition}

\begin{definition}[\cite{7}] \label{def1.6} \rm
 The growth index of the iterated convergence exponent of the sequence of 
zeros of  $f(z)$ in $\Delta$ is defined by
\[
i_{\lambda}(f)= \begin{cases}
   0, &  \text{if } N(r,\frac{1}{f})=O(\log \frac{1}{1-r}); \\ 
\min\{p\in \mathbb{N},\lambda_{p}(f)<\infty\}, &
   \text{if some $p\in \mathbb{N}$  with  } \lambda_p(f)<\infty; \\ 
\infty, &     \text{if $\lambda_{p}(f)=\infty$ for all $p\in \mathbb{N}$}.
  \end{cases} 
\]
Similarly, the growth index of the
iterated convergence exponent of the sequence of distinct zeros of  $f(z)$ in
$\Delta$ is defined by
\[
i_{\overline{\lambda}}(f)=\begin{cases}
 0, &  \text{if } \overline{N}(r,\frac{1}{f})=O(\log \frac{1}{1-r}); \\ 
\min\{p\in \mathbb{N},\overline{\lambda}_{p}(f)<\infty\}, &
   \text{if some $p\in \mathbb{N}$  with  }\overline{\lambda}_p(f)<\infty; \\ 
\infty, &    \text{if } 
\overline{\lambda}_{p}(f)=\infty \text{ for all }  p\in \mathbb{N}.
  \end{cases} 
\]
\end{definition}

\begin{definition}[\cite{11}] \label{def1.7} \rm 
For $a\in \mathbb{\overline{C}}=\mathbb{C}\cup \{\infty\}$, the deficiency of
$f$ is defined by
\begin{equation*}
\delta (a,f)=1-\limsup_{r\to 1^-}\frac{N(r,\frac{1}{f-a})}{T(r,f)},
\end{equation*}
provided $f$ has unbounded characteristic.
\end{definition}

The complex oscillation theory of solutions of linear differential
equations in the complex plane $\mathbb{C}$ was started by Bank and
Laine in 1982.  After their well known work, many important results
have been obtained on the growth and the complex oscillation theory
of solutions of linear differential equation in  $\mathbb{C}$. 
It arises naturally an interesting subject of complex oscillation theory 
of differential equations in the unit disc, which is more difficult 
to study than that in the complex plane, and there exist some results
 (see \cite{1,2,4,5,6,7,9, 10,12,13,16,18,21}).
 Recently,  Latreuch and Bela\"{\i}di studied 
the oscillation problem of solutions and their
derivatives of second-order non-homogeneous linear differential
equation
\begin{equation}\label{e1.1}
f''+A(z)f'+B(z)f=F(z),
\end{equation}
where $A(z), B(z)\not\equiv0$ and $F(z)\not\equiv0$ are meromorphic
functions of finite iterated $p$-order in $\Delta$. For some
related papers in the complex plane on the usual order see, \cite{20}.
 Before we state their results we need to define the following:
\begin{gather}\label{e1.2}
A_j(z)=A_{j-1}(z)-\frac{B'_{j-1}(z)}{B_{j-1}(z)}, \quad(j=1,2,3,\dots) \\
\label{e1.3}
B_j(z)=A'_{j-1}(z)-A_{j-1}(z)\frac{B'_{j-1}(z)}{B_{j-1}(z)}+B_{j-1}(z)
\quad(j=1,2,3,\dots) \\
\label{e1.4}
F_j(z)=F'_{j-1}(z)-F_{j-1}(z)\frac{B'_{j-1}(z)}{B_{j-1}(z)},
\quad(j=1,2,3,\dots)
\end{gather}
where $A_0(z)=A(z)$, $B_0(z)=B(z)$  and $F_0(z)=F(z)$.
 Latreuch and  Bela\"{\i}di obtained the following results.


 \begin{theorem}[\cite{17}] \label{thmA}  Let $A(z)$, $B(z)\not\equiv0$  and  
$F(z)\not\equiv0$  be meromorphic functions of finite iterated  $p$-order 
in $\Delta$  such that  $B_j(z)\not\equiv0$ and $F_j(z)\not\equiv0$  
$(j=1,2,3\dots)$.
 If $f$ is a meromorphic solution in $\Delta$ of \eqref{e1.1} with
$\rho_p(f)=\infty$ and $\rho_{p+1}(f)=\rho$, then $f$  satisfies
\begin{gather*}
\overline{\lambda}_p(f^{(j)})=\lambda_p(f^{(j)})=\rho_p(f)
=\infty \quad (j=0,1,2,\dots) \\
\overline{\lambda}_{p+1}(f^{(j)})=\lambda_{p+1}(f^{(j)})
=\rho_{p+1}(f)=\rho \quad (j=0,1,2,\dots).
\end{gather*}
\end{theorem}

\begin{theorem}[\cite{17}] \label{thmB} 
Let $A(z)$, $B(z)\not\equiv0$  and $F(z)\not\equiv0$
 be meromorphic functions in $\Delta$ with finite iterated  $p$-order
 such that  $B_j(z)\not\equiv0$ and $F_j(z)\not\equiv0$ $(j=1,2,3\dots)$.
 If $f$ is a meromorphic solution in $\Delta$ of \eqref{e1.1} with
$$
\rho_p(f)>\max\{\rho_p(A),\rho_p(B),\rho_p(F)\},
$$
then
$$
\overline{\lambda}_{p}(f^{(j)})=\lambda_{p}(f^{(j)})
=\rho_{p}(f) \quad (j=0,1,2,\dots).
$$
\end{theorem}

\begin{theorem}[\cite{17}] \label{thmC}
 Let $A(z)$, $B(z)\not\equiv0$  and $F(z)\not\equiv0$
 be analytic functions in $\Delta$ with finite iterated  $p$-order
 such that $\beta=\rho_p(B)>\max\{\rho_p(A),\rho_p(F)\}$.
Then all nontrivial solutions of  \eqref{e1.1}  satisfy
$$
\rho_p(B)\leq\overline{\lambda}_{p+1}(f^{(j)})=\lambda_{p+1}(f^{(j)})
=\rho_{p+1}(f)\leq\rho_{M,p}(B) \quad (j=0,1,2,\dots)
$$
with at most one possible exceptional solution $f_0$  such that
$$
\rho_{p+1}(f_0)<\rho_p(B).
$$
\end{theorem}

\begin{theorem}[\cite{17}] \label{thmD} 
Let $A(z)$, $B(z)\not\equiv0$  and $F(z)\not\equiv0$
 be meromorphic  functions in $\Delta$ with finite iterated  $p$-order
 such that  $\sigma_p(B)>\max\{\sigma_p(A),\sigma_p(F)\}$.
 If $f$ is a meromorphic solution in $\Delta$ of \eqref{e1.1} with
$\rho_p(f)=\infty$ and $\rho_{p+1}(f)=\rho$, then $f$  satisfies
\begin{gather*}
\overline{\lambda}_p(f^{(j)})=\lambda_p(f^{(j)})=\rho_p(f)
=\infty \quad (j=0,1,2,\dots) \\
\overline{\lambda}_{p+1}(f^{(j)})
=\lambda_{p+1}(f^{(j)})=\rho_{p+1}(f)=\rho \quad (j=0,1,2,\dots),
\end{gather*}
where 
$$
\sigma_{p}(f)=\limsup_{r\to 1^-}\frac{\log_{p}  m(r,f)}{\log \frac {1}{1-r}}.
$$
\end{theorem}

In this article, we continue to study the oscillation problem of
solutions and their derivatives of second order non-homogeneous
linear differential equation of \eqref{e1.1}. Let  $\varphi(z)$ be a
meromorphic function in $\Delta$ with finite iterated  $p$-order
$\rho_p(\varphi)<\infty $. We need to define the notation
\begin{equation}\label{e1.5}
D_j=F_{j}-(\varphi''+A_j \varphi'+B_j
\varphi),\quad(j=0,1,2,\dots)
\end{equation}
where $A_j(z), B_j(z), F_j(z)$ are defined in \eqref{e1.2}--\eqref{e1.4}. 
We obtain the following results.

\begin{theorem} \label{thm1.1} 
Let $\varphi(z)$ be a meromorphic function in
 $\Delta$ with $\rho_p(\varphi)<\infty $. Let $A(z)$, $B(z)\not\equiv0$  and
$F(z)\not\equiv0$  be meromorphic functions of finite iterated  $p$-order in $\Delta$
 such that  $B_j(z)\not\equiv0$ and $D_j(z)\not\equiv0$
 $(j=0,1,2,\dots)$.

 (a) If $f$ is a meromorphic solution in $\Delta$ of \eqref{e1.1} with
$\rho_p(f)=\infty$ and $\rho_{p+1}(f)=\rho<\infty$ , then $f$  satisfies
\begin{gather*}
\overline{\lambda}_p(f^{(j)}-\varphi)
=\lambda_p(f^{(j)}-\varphi)=\rho_p(f)=\infty \quad (j=0,1,2,\dots),\\
\overline{\lambda}_{p+1}(f^{(j)}-\varphi)
=\lambda_{p+1}(f^{(j)}-\varphi)=\rho_{p+1}(f)=\rho \quad (j=0,1,2,\dots).
\end{gather*}

(b) If $f$ is a meromorphic solution in $\Delta$ of \eqref{e1.1} with
$$
\max\{\rho_p(A),\rho_p(B),\rho_p(F),\rho_p(\varphi)\}<\rho_p(f)<\infty,
$$
then
$$
\overline{\lambda}_{p}(f^{(j)}-\varphi)=\lambda_{p}(f^{(j)}-\varphi)=\rho_{p}(f) 
\quad (j=0,1,2,\dots).
$$
\end{theorem}

Next, we give some sufficient conditions on the coefficients which
guarantee $B_j(z)\not\equiv0$ and $D_j(z)\not\equiv0$
 $(j=1,2,\dots)$, and we obtain

 \begin{theorem} \label{thm1.2} 
Let $\varphi(z)$ be an analytic function in
 $\Delta$ with $\rho_p(\varphi)<\infty $ and be not a solution of \eqref{e1.1}.
 Let $A(z)$, $B(z)\not\equiv0$  and $F(z)\not\equiv0$
 be analytic functions in $\Delta$ with finite iterated  $p$-order such that 
$\beta=\rho_p(B)>\max\{\rho_p(A),\rho_p(F),\rho_p(\varphi)\}$ and 
$\rho_{M,p}(A)\leq\rho_{M,p}(B)$.
Then all nontrivial solutions of  \eqref{e1.1}  satisfy
\[
\rho_p(B)\leq\overline{\lambda}_{p+1}(f^{(j)}-\varphi)
=\lambda_{p+1}(f^{(j)}-\varphi)=\rho_{p+1}(f)\leq\rho_{M,p}(B) \quad 
(j=0,1,2,\dots)
\]
with at most one possible exceptional solution $f_0$  such that
$$
\rho_{p+1}(f_0)<\rho_p(B).
$$
\end{theorem}

\begin{theorem} \label{thm1.3} 
Let $\varphi(z)$ be a
meromorphic function in  $\Delta$ with $\rho_p(\varphi)<\infty $ and
 be not a solution of \eqref{e1.1}. Let $A(z)$, $B(z)\not\equiv0$
 and $F(z)\not\equiv0$
 be meromorphic functions in $\Delta$ with finite iterated  $p$-order
 such that $\rho_p(B)>\max\{\rho_p(A),\rho_p(F),\rho_p(\varphi)\}$ and 
$\delta(\infty,B)>0$. If $f$ is a meromorphic solution in $\Delta$ 
of \eqref{e1.1} with $\rho_p(f)=\infty$ and $\rho_{p+1}(f)=\rho$, then $f$  
satisfies
\begin{gather*}
\overline{\lambda}_p(f^{(j)}-\varphi)=\lambda_p(f^{(j)}-\varphi)=\rho_p(f)
=\infty \quad (j=0,1,2,\dots), \\
\overline{\lambda}_{p+1}(f^{(j)}-\varphi)
=\lambda_{p+1}(f^{(j)}-\varphi)=\rho_{p+1}(f)=\rho \quad (j=0,1,2,\dots).
\end{gather*}
\end{theorem}

\section{Preliminary Lammas}\label{preps}

\begin{lemma}[\cite{2}] \label{lem2.1}
 Let $f(z)$ be a  meromorphic
function in the unit disc for which $i(f)=p\geq1$ and
$\rho_p(f)=\beta<\infty$ and let $k\in \mathbb{N}$. Then for any
$\varepsilon>0$,
$$
m\Big(r,\frac{f^{(k)}}{f}\Big)
=O\Big(\exp_{p-2}\big(\frac{1}{1-r}\big)^{\beta+\varepsilon}\Big)
$$
for all $r$  outside a set $E_1 \subset[0,1)$  with 
$\int_{E_1} \frac{dr}{1-r}<\infty$.
\end{lemma}


\begin{lemma}[\cite{6}] \label{lem2.2}
Let $A_0,A_1,\dots,A_{k-1},F\not\equiv0$  be meromorphic functions
in $\Delta$ , and let $f$ be a meromorphic solution of the
differential equation
\begin{equation} \label{e2.1}
f^{(k)}+A_{k-1}(z)f^{(k-1)}+\dots+A_0(z)f=F(z)
\end{equation}
such that $i(f)=p  (0<p<\infty)$. If either
$$
\max\{i(A_j)\quad (j=0,1,\dots,k-1),i(F)\}<p
$$
or
$$
\max\{\rho_p(A_j)\quad (j=0,1,\dots,k-1),\rho_p(F)\}<\rho_p(f),
$$
then
\begin{gather*}
i_{\overline{\lambda}}(f)=i_\lambda{(f)}=i(f)=p, \\
\overline{\lambda}_p(f)=\lambda_p(f)=\rho_p(f).
\end{gather*}
\end{lemma}

\begin{lemma}[\cite{17}]\label{lem2.3}
Let $A_0,A_1,\dots,A_{k-1},F\not\equiv0$   be finite iterated
$p$-order meromorphic functions in the unit disc $\Delta$. If
$f$ is a meromorphic solution with $\rho_p(f)=\infty$ and
$\rho_{p+1}(f)=\rho<\infty$ of equation \eqref{e2.1}, then
\begin{gather*}
\overline{\lambda}_p(f)=\lambda_p(f)=\rho_p(f)=\infty ,\\
\overline{\lambda}_{p+1}(f)=\lambda_{p+1}(f)=\rho_{p+1}(f)=\rho .
\end{gather*}
\end{lemma}


\begin{lemma}\label{lem2.4}
Let $\varphi,A_0,A_1,\dots,A_{k-1},F\not\equiv0$  be finite
iterated $p$-order meromorphic functions in the unit disc
$\Delta$ such that
\[
F-\varphi^{(k)}-A_{k-1}\varphi^{(k-1)}-\dots-A_1\varphi'-A_0\varphi
\not\equiv0.
\]
 If $f$ is a meromorphic solution with
$\rho_p(f)=\infty$ and $\rho_{p+1}(f)=\rho<\infty$ of equation
\eqref{e2.1}, then
\begin{gather*}
\overline{\lambda}_p(f-\varphi)=\lambda_p(f-\varphi)=\rho_p(f)=\infty, \\
\overline{\lambda}_{p+1}(f-\varphi)=\lambda_{p+1}(f-\varphi)=\rho_{p+1}(f)=\rho .
\end{gather*}
\end{lemma}

\begin{proof}
 Suppose that $g=f-\varphi$, we obtain $f=g+\varphi$, 
then from \eqref{e2.1} we have
$$
g^{(k)}+A_{k-1}g^{(k-1)}+\dots+A_1g'+A_{0}g
=F-\varphi^{(k)}-A_{k-1}\varphi^{(k-1)}-\dots-A_1\varphi'-A_0\varphi.
$$
Since $\rho_p(f-\varphi)=\infty$ and
$\rho_{p+1}(f-\varphi)=\rho<\infty$, then by using Lemma \ref{lem2.3} we
obtain
\begin{gather*}
\overline{\lambda}_p(f-\varphi)=\lambda_p(f-\varphi)=\rho_p(f)=\infty,\\
\overline{\lambda}_{p+1}(f-\varphi)=\lambda_{p+1}(f-\varphi)=\rho_{p+1}(f)=\rho .
\end{gather*}
\end{proof}

\begin{lemma}[\cite{6}]\label{lem2.5}
Let $p\in \mathbb{N}$, and assume that the coefficients $A_0,\dots,A_{k-1}$
and $F\not\equiv0$ are analytic in $\Delta$ and
$\rho_p(A_j)<\rho_p(A_0)$ for all $j=1,\dots,k-1$. Let
$\alpha_M=\max \{\rho_{M,p}(A_{j}):j=0,\dots,k-1\}$. If
$\rho_{M,p+1}(F)<\rho_p(A_0)$, then all solutions $f$ of \eqref{e2.1}
satisfy
$$
\rho_p(A_0)\leq\overline{\lambda}_{p+1}(f)=\lambda_{p+1}(f)
=\rho_{M,p+1}(f)\leq\alpha_M ,
$$
with at most one exception $f_0$ satisfying
$\rho_{M,p+1}(f_0)<\rho_p(A_0)$.
\end{lemma}

By a similar reasoning as Lemma \ref{lem2.4} and by using Lemma \ref{lem2.5}, we can
obtain the following lemma.

\begin{lemma}\label{lem2.6}
Let $p\in \mathbb{N}$, $\varphi$ be finite iterated $p$-order analytic
functions in the unit disc $\Delta$ and assume that the
coefficients $A_0,\dots,A_{k-1}$ ,$F\not\equiv0$ and
$F-\varphi^{(k)}-A_{k-1}\varphi^{(k-1)}-\dots-A_1\varphi'-A_0\varphi
\not\equiv0$ are analytic in $\Delta$ and
$\rho_p(A_j)<\rho_p(A_0)$ for all $j=1,\dots,k-1$. Let
$\alpha_M=\max \{\rho_{M,p}(A_{j}):j=0,\dots,k-1)$. If
$\rho_{M,p+1}(F-\varphi^{(k)}-A_{k-1}\varphi^{(k-1)}-\dots-A_1\varphi'-A_0\varphi)
<\rho_p(A_0)$, then all solutions $f$ of \eqref{e2.1} satisfy
$$\rho_p(A_0)\leq\overline{\lambda}_{p+1}(f-\varphi)=\lambda_{p+1}(f-\varphi)=\rho_{M,p+1}(f)\leq\alpha_M ,$$
with at most one  exception  $f_0$  satisfying
$\rho_{M,p+1}(f_0)<\rho_p(A_0)$.
\end{lemma}

\begin{lemma}\label{lem2.7}
Let  $\varphi$, $A_0,\dots,A_{k-1}$, $F\not\equiv0$ be meromorphic
functions in the unit disc $\Delta$ such that
$F-\varphi^{(k)}-A_{k-1}\varphi^{(k-1)}-\dots-A_1\varphi'-A_0\varphi
\not\equiv0$ , and let $f$ be a meromorphic solution of the
differential equation of \eqref{e2.1}, such that $i(f)=p (0<p<\infty)$. If
either
$$\max\{i(A_j): (j=0,1,\dots,k-1),i(F),\;i(\varphi)\}<p$$
or
$$\max\{\rho_p(A_j): (j=0,1,\dots,k-1),\;\rho_p(F),\rho_p(\varphi)\}<\rho_p(f),$$
then
\begin{gather*}
i_{\overline{\lambda}}(f-\varphi)=i_\lambda{(f-\varphi)}=i(f)=p,\\
\overline{\lambda}_p(f-\varphi)=\lambda_p(f-\varphi)=\rho_p(f).
\end{gather*}
\end{lemma}

The proof of the above lemma follows a similar reasoning as in 
Lemmas \ref{lem2.4} and \ref{lem2.2}.


\section{Proofs of theorems}\label{proofs}

\begin{proof}[Proof of Theorem \ref{thm1.1}]
(a) For the proof, we use the principle of mathematical induction. Since
 $D_0=F-(\varphi''+A\varphi'+B\varphi)\not\equiv0$, then by using 
Lemma \ref{lem2.4}, we have
\begin{gather*}
\overline{\lambda}_p(f-\varphi)=\lambda_p(f-\varphi)=\rho_p(f)=\infty,\\
\overline{\lambda}_{p+1}(f-\varphi)=\lambda_{p+1}(f-\varphi)=\rho_{p+1}(f)=\rho .
\end{gather*}
Since $B(z)\not\equiv0$, dividing both sides of \eqref{e1.1} by $B$,
 we obtain
\begin{equation} \label{e3.1}
\frac{1}{B}f''+\frac{A}{B}f'+f=\frac{F}{B}.
\end{equation}
Differentiating both sides of  \eqref{e3.1}, we have
\begin{equation} \label{e3.2}
\frac{1}{B}f^{(3)}+\Big(\big(\frac{1}{B}\big)'+\frac{A}{B}\Big)f''
+\Big(\big(\frac{A}{B}\big)'+1\Big)f'=\big(\frac{F}{B}\big)'.
\end{equation}
Multiplying  \eqref{e3.2} by $B$, we obtain
\begin{equation} \label{e3.3}
f^{(3)}+A_1f''+B_1f'=F_1,
\end{equation}
where
$$
A_1=A-\frac{B'}{B},\quad B_1=A'-A\frac{B'}{B}+B,\quad
F_1=F'-F\frac{B'}{B}.
$$
 Since $A_1,B_1$  and $F_1$  are meromorphic
functions with finite iterated $p$-order, and
$D_1=F_1-(\varphi''+A_1 \varphi'+B_1\varphi)\not\equiv0$, then using 
Lemma \ref{lem2.4},
we obtain
\begin{gather*}
\overline{\lambda}_p(f'-\varphi)=\lambda_p(f'-\varphi)=\rho_p(f)=\infty, \\
\overline{\lambda}_{p+1}(f'-\varphi)=\lambda_{p+1}(f'-\varphi)=\rho_{p+1}(f)=\rho .
\end{gather*}
Since $B_1(z)\not\equiv0$, dividing now both sides of  \eqref{e3.3} by $B_1$,
we obtain
\begin{equation} \label{e3.4}
\frac{1}{B_1}f^{(3)}+\frac{A_1}{B_1}f''+f'=\frac{F_1}{B_1}.
\end{equation}
Differentiating both sides of equation \eqref{e3.4} and multiplying by
$B_1$, we obtain
\begin{equation} \label{e3.5}
f^{(4)}+A_2f^{(3)}+B_2f''=F_2,
\end{equation}
where $A_{2}$, $B_{2},F_2$ are
meromorphic functions defined in \eqref{e1.2}-\eqref{e1.4}.
 Since $D_2=F_2-(\varphi''+A_2\varphi'+B_2\varphi)\not\equiv0$, by using 
Lemma \ref{lem2.4} again, we
obtain
\begin{gather*}
\overline{\lambda}_p(f''-\varphi)=\lambda_p(f''-\varphi)=\rho_p(f)=\infty,\\
\overline{\lambda}_{p+1}(f''-\varphi)=\lambda_{p+1}(f''-\varphi)
=\rho_{p+1}(f)=\rho.
\end{gather*}
Suppose now that
\begin{gather} \label{e3.6}
\overline{\lambda}_p(f^{(k)}-\varphi)=\lambda_p(f^{(k)}-\varphi)
=\rho_p(f)=\infty, \\
\label{e3.7}
\overline{\lambda}_{p+1}(f^{(k)}-\varphi)=\lambda_{p+1}(f^{(k)}-\varphi)
=\rho_{p+1}(f)=\rho
\end{gather}
for all $k=0,1,2,\dots,j-1$, and we prove that \eqref{e3.6} and \eqref{e3.7} are
true for $k=j$. By the same procedure as before, we can obtain
\begin{equation} \label{e3.8}
f^{(j+2)}+A_{j}f^{(j+1)}+B_{j}f^{(j)}=F_{j},
\end{equation}
where $A_j$, $B_j$ and $F_j$  are meromorphic
functions defined in \eqref{e1.2}-\eqref{e1.4}. Since
$D_j=F_j-(\varphi''+A_j\varphi'+B_j\varphi)\not\equiv0$, by using Lemma \ref{lem2.4},
we obtain
\begin{gather*}
\overline{\lambda}_p(f^{(j)}-\varphi)=\lambda_p(f^{(j)}-\varphi)=\rho_p(f)=\infty,\\
\overline{\lambda}_{p+1}(f^{(j)}-\varphi)=\lambda_{p+1}(f^{(j)}-\varphi)
=\rho_{p+1}(f)=\rho .
\end{gather*}


(b) Since $D_{0}=F-(\varphi''+A\varphi'+B\varphi)\not\equiv0$, and 
$\max\{\rho_p(A),\rho_p(B),\rho_p(F),\rho_p(\varphi)\}<\rho_p(f)<\infty$,
then by using Lemma \ref{lem2.7} we have
$$
\overline{\lambda}_p(f-\varphi)=\lambda_p(f-\varphi)=\rho_p(f).
$$
By (a), we have \eqref{e3.3} and
 $\max\{\rho_p(A_1),\rho_p(B_1),\rho_p(F_1),\rho_p(\varphi)\}<\rho_p(f)<\infty$.
 Since $D_1\not\equiv0$, then by using Lemma \ref{lem2.7} we obtain
$$
\overline{\lambda}_p(f'-\varphi)=\lambda_p(f'-\varphi)=\rho_p(f).
$$
By (a), we have \eqref{e3.5} and
$\max\{\rho_p(A_2),\rho_p(B_2),\rho_p(F_2),\rho_p(\varphi)\}<\rho_p(f)<\infty$.
 Since  $D_2\not\equiv0$, then by using Lemma \ref{lem2.7} we obtain
$$
\overline{\lambda}_p(f''-\varphi)=\lambda_p(f''-\varphi)=\rho_p(f).
$$
Suppose now that
\begin{equation} \label{e3.9}
\overline{\lambda}_p(f^{(k)}-\varphi)=\lambda_p(f^{(k)}-\varphi)=\rho_p(f)
\end{equation}
for all $k=0,1,2,\dots,j-1$, and we prove that \eqref{e3.9} is true for
$k=j$. By (a) we have \eqref{e3.8} and
$\max\{\rho_p(A_j),\rho_p(B_j),\rho_p(F_j),\rho_p(\varphi)\}<\rho_p(f)<\infty$.
 Since $D_j\not\equiv0$, then by using Lemma \ref{lem2.7} we obtain
$$
\overline{\lambda}_{p}(f^{(j)}-\varphi)=\lambda_{p}(f^{(j)}-\varphi)=\rho_{p}(f) ,
$$
The proof is complete.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm1.2}]
Since $F-(\varphi''+A \varphi'+B\varphi)\not\equiv0$,
$\rho_{M,p+1}(F-(\varphi''+A \varphi'+B\varphi))<\rho_p(B)$.
By Lemma \ref{lem2.6}, all nontrivial solutions of \eqref{e1.1} satisfy
$$
\rho_p(B)\leq\overline{\lambda}_{p+1}(f-\varphi)
=\lambda_{p+1}(f-\varphi)=\rho_{p+1}(f)\leq\rho_{M,p}(B) 
$$
with at most one possible exceptional solution $f_0$  such that
$\rho_{p+1}(f_0)<\rho_p(B)$. By using \eqref{e1.2} and Lemma \ref{lem2.1} 
we have for any $\varepsilon>0$,
$$
m(r,A_j)\leq m(r,A_{j-1})+O\Big(\exp_{p-2}
\big(\frac{1}{1-r}\big)^{\beta+\varepsilon}\Big) 
\quad (\beta=\rho_p(B_{j-1}))
$$
outside a set $E_1 \subset[0,1)$  with 
$\int_{E_1} \frac{dr}{1-r}<\infty$, for all $j=1,2,3,\dots$, which we 
can write as
\begin{equation} \label{e3.10}
m(r,A_j)\leq
m(r,A)+O\Big(\exp_{p-2}\big(\frac{1}{1-r}\big)^{\beta+\varepsilon}\Big).
\end{equation}
On the other hand,  from \eqref{e1.3}, we have
\begin{equation} \label{e3.11}
\begin{split}
B_j&= A_{j-1}
\Big(\frac{A'_{j-1}}{A_{j-1}}-\frac{B'_{j-1}}{B_{j-1}}\Big)+B_{j-1}\\
&=A_{j-1}\Big(\frac{A'_{j-1}}{A_{j-1}}-\frac{B'_{j-1}}{B_{j-1}}\Big)
+A_{j-2}\Big(\frac{A'_{j-2}}{A_{j-2}}-\frac{B'_{j-2}}{B_{j-2}}\Big)+B_{j-2}\\
&=\sum_{k=0}^{j-1}A_k\Big(\frac{A'_k}{A_k}-\frac{B'_k}{B_k}\Big)+B.
\end{split}
\end{equation}
Now we prove that  $B_j\not\equiv0$ for all $j=1,2,3,\dots$. For
that we suppose there exists $j\in \mathbb{N}$ such that $B_j=0$.
By \eqref{e3.10} and \eqref{e3.11} we have
\begin{equation} \label{e3.12}\begin{split}
T(r,B)=m(r,B)
&\leq \sum_{k=0}^{j-1}m(r,A_k)
+O\Big(\exp_{p-2}\big(\frac{1}{1-r}\big)^{\beta+\varepsilon}\Big)\\
&\leq j m(r,A)+O\Big(\exp_{p-2}\big(\frac{1}{1-r}\big)^{\beta+\varepsilon}\Big)\\
&=j T(r,A)+O\Big(\exp_{p-2}\big(\frac{1}{1-r}\big)^{\beta+\varepsilon}\Big),
\end{split}
\end{equation}
which implies the contradiction $\rho_p(B)\leq\rho_p(A)$. Hence
$B_j\not\equiv0$ for all $j=1,2,3,\dots$. We prove that
$D_j\not\equiv0$ for all $j=1,2,3,\dots$. For that we suppose there
exists $j\in \mathbb{N}$ such that $D_j=0$. We have $F_{j}-(\varphi''+A_j
\varphi'+B_j\varphi)=0$ from \eqref{e1.5}, which implies
$$
F_{j}=\varphi\Big(\frac{\varphi''}{\varphi}+A_j
\frac{\varphi'}{\varphi}+B_j\Big)
=\varphi\Big[\frac{\varphi''}{\varphi}+A_j\frac{\varphi'}{\varphi}
+\sum_{k=0}^{j-1}A_{k}\Big(\frac{A'_k}{A_k}-\frac{B'_k}{B_k}\Big)+B\Big].
$$
Here we suppose that $\varphi(z)\not\equiv0$, otherwise by Theorem \ref{thmC}
there is nothing to prove. Therefore,
\begin{equation} \label{e3.13}
B=\frac{F_j}{\varphi}-\Big[\frac{\varphi''}{\varphi}+A_j\frac{\varphi'}{\varphi}
+\sum_{k=0}^{j-1}A_{k}\Big(\frac{A'_k}{A_k}-\frac{B'_k}{B_k}\Big)\Big].
\end{equation}
On the other hand, from \eqref{e1.4},
\begin{equation} \label{e3.14}
 m(r,F_j)\leq
m(r,F)+O\Big(\exp_{p-2}\big(\frac{1}{1-r}\big)^{\beta+\varepsilon}\Big). \quad
(j=1,2,3,\dots)
\end{equation}
By \eqref{e3.10}, \eqref{e3.13}, \eqref{e3.14} and Lemma \ref{lem2.1} we have
\begin{equation} \label{e3.15}
\begin{aligned}
T(r,B)&=m(r,B)\leq m(r,\frac{1}{\varphi})+m(r,F)+(j+1)m(r,A)\\
&\quad +O\Big(\exp_{p-2}\big(\frac{1}{1-r}\big)^{\beta_1+\varepsilon}\Big),
\end{aligned}
\end{equation}
where $\beta_1$ is some non-negative constant, which implies the
contradiction
$\rho_p(B)\leq \max\{\rho_p(A),\rho_p(F),\rho_p(\varphi)\}$.
Hence $D_j\not\equiv0$ for all $j=1,2,3,\dots$.
 Since $B_j\not\equiv0$, $D_j\not\equiv0$
$(j=1,2,3,\dots)$, then by  Theorem \ref{thm1.1} and Lemma \ref{lem2.6} we
have
$$
\rho_p(B)\leq\overline{\lambda}_{p+1}(f^{(j)}-\varphi)
=\lambda_{p+1}(f^{(j)}-\varphi)=\rho_{p+1}(f)\leq\rho_{M,p}(B)
\quad (j=0,1,2,\dots)
$$
with at most one possible exceptional solution $f_0$  such that
$\rho_{p+1}(f_0)<\rho_p(B)$.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm1.3}]
We need only to prove that $B_{j}\not\equiv0$ and $D_{j}\not\equiv0$
for all $j=1,2,3,\dots$. Then by Theorem \ref{thm1.1} we can obtain 
Theorem \ref{thm1.3}. Consider the assumption $\delta(\infty,B)=\delta >0$. 
Then for $r\to 1^-$ we have
\begin{equation} \label{e3.16}
 T(r,B)\leq \frac{2}{\delta}m(r,B).
\end{equation}
 Now we prove that $B_j\not\equiv0$ for
all $j=1,2,3,\dots$. For that we suppose there exists $j\in \mathbb{N} $
such that $B_j=0$. By \eqref{e3.10}, \eqref{e3.11} and \eqref{e3.16} we obtain
\begin{equation} \label{e3.17}
\begin{split}
T(r,B)\leq \frac{2}{\delta}m(r,B)
&\leq \frac{2}{\delta} \sum_{k=0}^{j-1}m(r,A_k)+\frac{2}{\delta}
O\Big(\exp_{p-2}\big(\frac{1}{1-r}\big)^{\beta+\varepsilon}\Big)\\
&\leq \frac{2}{\delta} j m(r,A)+\frac{2}{\delta}
O\Big(\exp_{p-2}\big(\frac{1}{1-r}\big)^{\beta+\varepsilon}\Big)\\
&\leq\frac{2}{\delta} j T(r,A)+\frac{2}{\delta}
O\Big(\exp_{p-2}\big(\frac{1}{1-r}\big)^{\beta+\varepsilon}\Big),
\end{split}
\end{equation}
which implies the contradiction $\rho_p(B)\leq \rho_p(A)$. Hence
$B_j\not\equiv0$ for all $j=1,2,3,\dots $. We prove that
$D_j\not\equiv0$ for all $j=1,2,3,\dots$. For that we suppose there
exists $j\in \mathbb{N} $ such that $D_j=0$.
If $\varphi(z)\not\equiv0$, then by \eqref{e3.10}, \eqref{e3.13}, \eqref{e3.14},
\eqref{e3.16} and Lemma \ref{lem2.1} we have
\begin{equation} \label{e3.18}
\begin{split}
T(r,B)&\leq \frac{2}{\delta}m(r,B)\\
&\leq \frac{2}{\delta} \Big[m(r,\frac{1}{\varphi})+m(r,F)+(j+1)
m(r,A)+O\Big(\exp_{p-2}\big(\frac{1}{1-r}\big)^{\beta+\varepsilon}\Big)\Big],
\end{split}
\end{equation}
which implies the contradiction
$\rho_p(B)\leq \max \{\rho_p(A),\rho_p(F),\rho_p(\varphi)\}$.
If $\varphi(z)\equiv0$, Then from \eqref{e1.4}, \eqref{e1.5}, we have
\begin{equation} \label{e3.19}
 F_{j-1}'-F_{j-1}\frac{B_{j-1}'(z)}{B_{j-1}(z)}=0,
\end{equation}
which implies $F_{j-1}(z)=cB_{j-1}(z)$, where $c$ is some constant.
By \eqref{e3.11} and \eqref{e3.19}, we have
\begin{equation} \label{e3.20}
\frac{1}{c}F_{j-1}=\sum_{k=0}^{j-2}A_k
\Big(\frac{A_k'}{A_k}-\frac{B_k'}{B_k}\Big)+B.
\end{equation}
On the other hand, from \eqref{e1.4},
\begin{equation} \label{e3.21}
m(r,F_{j-1})\leq m(r,F)
+O\Big(\exp_{p-2}\big(\frac{1}{1-r}\big)^{\beta+\varepsilon}\Big).
\end{equation}
By \eqref{e3.16}, \eqref{e3.20}, \eqref{e3.21} and Lemma \ref{lem2.1}, we have
\begin{equation} \label{e3.22}
\begin{split}
T(r,B)&\leq \frac{2}{\delta}m(r,B)\\
&\leq \frac{2}{\delta} \sum_{k=0}^{j-2}m(r,A_k)+\frac{2}{\delta}m(r,F_{j-1})
+ O\Big(\exp_{p-2}\big(\frac{1}{1-r}\big)^{\beta+\varepsilon}\Big)\\
&\leq \frac{2}{\delta} (j-1) T(r,A)+\frac{2}{\delta}T(r,F)+
O\Big(\exp_{p-2}\big(\frac{1}{1-r}\big)^{\beta+\varepsilon}\Big),
\end{split}
\end{equation}
which implies the contradiction $\rho_p(B)\leq \max \{\rho_p(A),\rho_p(F)\}$.
Hence $D_j\not\equiv0$ for all $j=1,2,3,\dots $. By Theorem \ref{thm1.1},
we obtain Theorem \ref{thm1.3}.
\end{proof}

\subsection*{Acknowledgments}
The authors would like to thank the anonymous referee for making valuable 
suggestions and comments to improve this article.

 This research was supported by the National Natural Science Foundation of
China (11301232, 11171119), by the Natural Science
Foundation of Jiangxi province (20132BAB211009), and by the Youth
Science Foundation of Education Burean of Jiangxi province (GJJ12207).


\begin{thebibliography}{99}

\bibitem{1} B. Bela\"{\i}di, A. EI Farissi;
 \emph{Fixed points and iterated order of differential polynomial
generated by solutions of linear differential equations in the unit disc},
 J. Adv. Res. Pure Math. 3 (2011), no. 1, 161-172.

 \bibitem{2} B. Bela\"{\i}di;
 \emph{Oscillation of fast growing solutions of linear differential equations 
in the unit disc}, Acta Univ. Sapientiae Math. 2(2010), no. 1, 25-38.

\bibitem{3}  L. G. Bernal;
 \emph{On growth $k$-order of solutions of a complex homogeneous linear 
differential equation}, Proc Amer. Math. Soc. 101(1987), no. 2, 317-322.

\bibitem{4} T. B. Cao, H. Y. Yi;
 \emph{The growth of solutions of linear differential equations with coefficients 
of iterated order in the unit disc},
 J. Math. Anal. Appl. 319 (2006), no. 1, 278-294.

\bibitem{5} T. B. Cao;
\emph{The growth, oscillation and fixed points of solutions of complex linear 
differential equations in the unit disc},
 J. Math. Anal. Appl. 352 (2009), no. 2, 739-748.

\bibitem{6}  T. B. Cao, Z. S. Deng;
\emph{Solutions of non-homogeneous linear differential equations in the unit disc},
Ann. Polo. Math. 97 (2010), no. 1, 51-61.

\bibitem{7}  T. B. Cao, C. X. Zhu, K. Liu;
\emph{On the complex oscillation of meromorphic solutions of
second order linear differential equations in the unit disc }, 
J. Math. Anal. Appl. 374 (2011), no. 1, 272-281.

\bibitem{8} T. B. Cao, C. X. Zhu, H. Y. Xu;
\emph{On the complex oscillation of differential polynomials generated by 
meromorphic solutions of differential equations in the unit disc}, 
Proc. Indian Acad. Sci. 2010, 120 (4): 481-493.

\bibitem{9}  Z. X. Chen, K. H. Shon;
\emph{The growth of solutions of differential equations with coefficients 
of small growth in the disc},   J. Math. Anal. Appl. 297 (2004), no. 1, 285-304.

\bibitem{10}  I. E. Chyzhykov, G. G. Gundersen, J. Heittokangas;
\emph{Linear differential equations and logarithmic derivative estimates}, 
Proc. London Math. Soc. (3) 86 (2003), no. 3,735-754.

\bibitem{11}  W. K. Hayman;
\emph{Meromorphic Functions}, Oxford Mathematical Monographs Clarendon Press, 
Oxford, 1964.

\bibitem{12} J. Heittokangas;
 \emph{On complex differential equations in the unit disc}, Ann. Acad. Sci. 
Fenn. Math. Diss. 122 (2000), 1-54.

\bibitem{13}  J. Heittokangas, R. Korhonen, J. R\"{a}tty\"{a};
\emph{Fast growing solutions of linear differential equations in the unit
disc}, Results Math. 49 (2006), no. 3-4, 265-278.

\bibitem{14} L. Kinnunen;
\emph{Linear differential equations with solutions of finite iterated order},
 Southeast Asian Bull. Math., 22 (1998), no. 4, 385-405.

\bibitem{15}  I. Laine;
\emph{Nevanlinna theory and complex differential equations},
Walter de Gruyter, Berlin, 1993.

 \bibitem{16} I. Laine;
\emph{ Complex differential equations, Handbook of differential equations: 
ordinary differential equations},
 Vol. IV, 269-363, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2008.

\bibitem{17} Z. Latreuch, B. Bela\"{\i}di;
\emph{Complex oscillation of solutions and their derivatives of non-homogeneous 
linear differential equations in the unit disc}, 
Intern. J. Anal. Appl., 2 (2013), no.2, 111-123.

\bibitem{18}  D. Shea, L. Sons;
 \emph{Value distribution theory for meromorphic functions of slow growth in the disc},
 Houston J. Math. 12 (2) (1986), 249-266.

\bibitem{19} M. Tsuji;
\emph{Potential Theory in Modern Function Theory}, Chelsea, New York,(1975),
 reprint of the 1959 edition.

\bibitem{20} J. Tu, H. Y. Xu, C. Y. Zhang;
\emph{On the zeros of solutions of any order of derivative of second
order linear differential equations taking small functions},
Electron. J. Qual. Theory Differ. Equ. 2011, No. 23, 1-27.

\bibitem{21}  G. Zhang, A. Chen;
\emph{Fixed points of the derivative and  $k$-th power of solutions of complex 
linear differential equations in the unit disc}, 
Electron J. Qual. Theory Differ. Equ., 2009, No 48, 1-9.

\end{thebibliography}

\end{document}