\documentclass[reqno]{amsart}
\usepackage{hyperref}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2010(2010), No. 87, 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}
\thanks{\copyright 2010 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2010/87\hfil Growth and oscillation]
{Growth and oscillation of differential polynomials in the unit
disc}

\author[A. El Farissi, B. Bela\"idi,  Z. Latreuch\hfil EJDE-2010/87\hfilneg]
{Abdallah El Farissi, Benharrat Bela\"idi, Zinela\^abidine Latreuch}

\address{Abdallah El Farissi \newline
Department of Mathematics, Laboratory of Pure and Applied Mathematics,
University of Mostaganem, B. P. 227 Mostaganem, Algeria}
\email{elfarissi.abdallah@yahoo.fr}

\address{Benharrat Bela\"idi \newline
Department of Mathematics, Laboratory of Pure and Applied Mathematics,
University of Mostaganem, B. P. 227 Mostaganem, Algeria}
\email{belaidibenharrat@yahoo.fr}


\address{Zinela\^abidine Latreuch \newline
Department of Mathematics, Laboratory of Pure and Applied Mathematics,
University of Mostaganem, B. P. 227 Mostaganem, Algeria}
\email{z.latreuch@gmail.com}


\thanks{Submitted March 24, 2010. Published June 21, 2010.}
\subjclass[2000]{34M10, 30D35}
\keywords{Linear differential equations; analytic
solutions; hyper order; \hfill\break\indent
exponent of convergence; hyper exponent of convergence}

\begin{abstract}
 In this article, we give sufficiently conditions for  the solutions
 and the differential polynomials generated by second-order
 differential equations to have the same properties
 of growth and oscillation. Also answer to the question posed
 by Cao \cite{c4} for the second-order linear differential equations in
 the unit disc.

\end{abstract}

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

\section{Introduction and main results}

 The study on value distribution of differential polynomials
generated by solutions of a given complex differential equation
in the case of complex plane seems to have been started by  Bank
\cite{b1}. Since then a number of authors have been working on the subject.
Many authors have investigated the growth and oscillation of
the solutions of complex linear differential equations in
$\mathbb{C}$, see \cite{b2,c2,c5,e1,g1,k1,l1,l2,l4,w1,z1}.
In the unit disc, there already
exist many results \cite{c1,c3,c4,c6,c7,h2,h3,l3,s1,t1,z2},
but the study is more
difficult than that in the complex plane.
Recently, Fenton-Strumia \cite{f1} obtained some results of
Wiman-Valiron type for power series in the unit disc,
and Fenton-Rossi \cite{f2} obtained an asymptotic equality
of Wiman-Valiron type for the derivatives of analytic
functions in the unit disc and applied to ODEs with analytic
coefficients.

  In this article, we assume that the reader is familiar with
the fundamental results and the standard notation of the Nevanlinna's
theory on the complex plane and in the unit disc
$D=\{z:|z|<1\}$, see \cite{h1,l1,n1,t1,y1,y2}.
In addition, we will use $\lambda (f)(\lambda _{2}(f))$ and
$\overline{\lambda }(f)(\overline{\lambda }_{2}(f))$ to
denote respectively the exponents (hyper-exponents) of convergence
of the zero-sequence and the sequence of distinct zeros of a
meromorphic function
$f $, $\rho (f)$ to denote the order and $\rho _{2}(
f)$ to denote the hyper-order of $f$. See \cite{c7,h2,l3,t1}
for notation and definitions.


\begin{definition}  \label{defA} \rm
The type of a meromorphic function $f$ in
$D$ with order $0<\rho (f)<\infty $ is defined by
\[
\tau (f)=\limsup_{r\to 1^{-}}(1-r)^{\rho (f)}T(r,f).
\]
\end{definition}

 Consider the linear differential equation
\begin{equation}
f^{(k)}+A_{k-1}(z)f^{(k-1)
}+\dots +A_{1}(z)f'+A_{0}(z)f=0,
  \label{e1.1}
\end{equation}
where $A_{0},A_{1},\dots ,A_{k-1}$ are analytic functions in $D$,
and $k$ is an integer, $k\geq 1$.

\begin{theorem}[\cite{c3}] \label{thmA}
Let $A_{0}(z), \dots,A_{k-1}(z)$, the coefficients of
\eqref{e1.1}, be analytic in $D$. If
$\max \{\rho (A_{j}):j=1,\dots ,k-1\}<\rho (A_{0})$, then
$\rho (A_{0})\leqslant \rho _{2}(f)\leqslant
\alpha _{M}$ for all solutions $f\not\equiv 0$ of
\eqref{e1.1}, where
$\alpha _{M}=\max \{\rho _{M}(A_{j}):j=0,\dots ,k-1\}$.
\end{theorem}

 Recall that the order of an
analytic function $f$ in $D$ is defined by
\[
\rho _{M}(f)=\limsup_{r\to 1^{-}}
\frac{\log ^{+}\log ^{+}M(r,f)}{\log \frac{1}{1-r}},
\]
where $M(r,f)=\max_{|z|=r}|f(z)|$. The following two
statements hold \cite[p. 205]{t1}.
\begin{itemize}
\item[(a)]  If $f$ is an analytic function in $D$,
then $\rho (f)\leqslant \rho _{M}(f)\leqslant \rho (f)+1$.

\item[(b)]  There exist analytic functions $f$ in $D$
which satisfy $\rho _{M}(f)\neq \rho (f)$. For
example, let $\mu >1$ be a constant, and set
\[
\psi (z)=\exp \{ (1-z)^{-\mu }\},
\]
where we choose the principal branch of the logarithm.
Then $\rho (\psi )=\mu -1$ and $\rho _{M}(\psi )=\mu $,
see \cite{c7}.
\end{itemize}

 In contrast, the possibility that occurs in (b) cannot occur
in the whole plane $\mathbb{C}$,
 because if $\rho (f)$ and $\rho _{M}(f)$ denote
the order of an entire function $f$ in the plane
$\mathbb{C}$ (defined by the Nevanlinna characteristic and the
maximum modulus, respectively), then it is well know that
$\rho (f)=\rho_{M}(f)$.


 \begin{theorem}[\cite{c3}] \label{thmB}
Under the hypotheses of Theorem \ref{thmA},
if $\rho _{2}(A_{j})<\infty $, $(j=0,\dots , k-1)$,
then every solution $f\not\equiv 0$ of \eqref{e1.1}
satisfies $\overline{\lambda }_{2}(f-z)=\rho_{2}(f)$.
\end{theorem}

 Consider a linear differential equation of the form
\begin{equation}
f''+A_{1}(z)f'+A_{0}(z)f=F,  \label{e1.2}
\end{equation}
where $A_{1}(z)$, $A_{0}(z)\not\equiv 0$,
$F(z)$ are analytic functions in the unit disc
$D=\{z:|z|<1\}$. It is
well-known that all solutions of equation \eqref{e1.2} are analytic
functions in $D$ and that there are exactly two linearly independent
solutions of \eqref{e1.2}; see \cite{h2}.


  Many important results have been obtained on the fixed
points of general transcendental meromorphic functions for almost four
decades, see \cite{z1}. However, there are few studies on the fixed points of
solutions of differential equations, specially in the unit disc.
Chen \cite{c5}  studied the problem on the fixed points and
hyper-order of solutions of second order linear differential
equations with entire coefficients. After that, there were some
results which improve those of  Chen, see \cite{b2,e1,l2,l4,w1}.
It is natural to ask what can be said
about similar situations in the unit disc $D$. Recently,
Cao \cite{c4} investigated the fixed points of solutions of
linear complex differential equations in the unit disc.

  The main purpose of this article is to give sufficiently
conditions for the solutions and the differential polynomials
generated by the second order linear differential equation
\eqref{e1.2} to have the same properties of the growth and oscillation.
Also, we answer to the following question posed by  Cao \cite{c4}:
\begin{quote}
How about the fixed points and iterated order of
differential polynomial generated by solutions of linear differential
equations in the unit disc?
\end{quote}

Before we state our results, we denote
\begin{gather}
\alpha _{0}=d_{0}-d_{2}A_{0},\quad
\beta _{0}=d_{2}A_{0}A_{1}-(d_{2}A_{0})'-d_{1}A_{0}+d_{0}',  \label{e1.3}
\\
\alpha _{1}=d_{1}-d_{2}A_{1},\quad
\beta _{1}=d_{2}A_{1}^{2}-(d_{2}A_{1})'-d_{1}A_{1}-d_{2}A_{0}
+d_{0}+d_{1}',  \label{e1.4}
\\
h=\alpha _{1}\beta _{0}-\alpha _{0}\beta _{1},  \label{e1.5}
\\
\psi (z)=\frac{\alpha _{1}\big(\varphi '-(
d_{2}F)'-\alpha _{1}F\big)-\beta _{1}(\varphi
-d_{2}F)}{h},  \label{e1.6}
\end{gather}
where $A_{0},A_{1},d_{0},d_{1},d_{2},\varphi $ and $F$ are analytic
functions in the unit disc $D=\{z:|z|<1\}$ with finite
order.

 \begin{theorem} \label{thm1.1}
Let $A_{1}(z),A_{0}(z) \not\equiv 0$ and
 $F$ be analytic functions in $D$, of finite order.
Let $d_{0},d_{1},d_{2}$ be analytic functions in $D$
that are not all equal to zero with $\rho (d_{j})<\infty $
$(j=0,1,2)$ such that $h\not\equiv 0$, where $h$
is defined by \eqref{e1.5}. If $f$ is an infinite order solution of
\eqref{e1.2} with $\rho _{2}(f)=\rho $,
then the differential polynomial $g_{f}=d_{2}f''+d_{1}f'+d_{0}f$
 satisfies
\begin{equation}
\rho (g_{f})=\rho (f)=\infty ,\quad
\rho_{2}(g_{f})=\rho _{2}(f)=\rho .  \label{e1.7}
\end{equation}
\end{theorem}

 \begin{theorem} \label{thm1.2}
Let $A_{1}(z)$, $A_{0}(z)\not\equiv 0$  and $F$  be
analytic functions in $D$ of finite order. Let
$d_{0}(z),d_{1}(z),d_{2}(z)$ be analytic
functions in $D$ which are not all equal to zero with
$\rho (d_{j})<\infty $ $(j=0,1,2)$
such that $h\not\equiv 0$, and let $\varphi (z)$
 be an analytic function in $D$ with finite order such
that $\psi (z)$ is not a solution of \eqref{e1.2}.
If $f$ is an infinite order solution of
\eqref{e1.2} with $\rho _{2}(f)=\rho$, then the differential
polynomial $g_{f}=d_{2}f''+d_{1}f'+d_{0}f$  satisfies
\begin{gather}
\overline{\lambda }(g_{f}-\varphi )=\lambda (
g_{f}-\varphi )=\rho (g_{f})=\rho (f)=\infty,
\label{e1.8} \\
\overline{\lambda }_{2}(g_{f}-\varphi )=\lambda _{2}(
g_{f}-\varphi )=\rho _{2}(g_{f})=\rho _{2}(
f)=\rho .  \label{e1.9}
\end{gather}
\end{theorem}

\begin{remark} \label{rmk1.1} \rm
In Theorem \ref{thm1.2}, if we do not have the
condition $\psi (z)$ is not a solution of \eqref{e1.2},
then the conclusions of Theorem \ref{thm1.2} does not hold. For
example, the functions $f_{1}(z)=1-z$ and
$f_{2}(z) =(1-z)\exp (\exp \frac{1}{1-z})$ are linearly
independent solutions of the equation
\begin{equation}
f''+A_{1}(z)f'+A_{0}(
z)f=0,  \label{e1.10}
\end{equation}
where
\[
A_{0}(z)=-\frac{\exp \frac{1}{1-z}}{(1-z)^{3}}-
\frac{1}{(1-z)^{3}}, \quad
A_{1}(z)=-\frac{ \exp \frac{1}{1-z}}{(1-z)^{2}}-\frac{1}{(1-z)^{2}}.
\]
Clearly  $f=f_{1}+f_{2}$ is a solution of  \eqref{e1.10}.
Set $d_{2}=d_{1}\equiv 0$ and $d_{0}=\frac{1}{1-z}$.
Then $g_{f}=d_{0}f$,
$h=-d_{0}^{2}$ and $\psi (z)=\frac{\varphi }{d_{0}}$. If we take
$\varphi =d_{0}f_{1}$, then $\psi (z)=f_{1}$ is a solution of
\eqref{e1.10} and we have
\[
\lambda (g_{f}-\varphi )=\lambda (
d_{0}f-d_{0}f_{1})=\lambda (d_{0}f_{2})=\lambda (
\exp(\exp \frac{1}{1-z}))=0.
\]
On the other hand,
\[
\rho (g_{f})=\rho (d_{0}f)=\rho (
d_{0}f_{1}+d_{0}f_{2})=\rho (1+\exp (\exp \frac{1}{1-z}
))=\infty .
\]
\end{remark}

\begin{theorem} \label{thm1.3} Let $A_{1}(z)$,
$A_{0}(z)\not\equiv 0$ and $F$ be finite
order analytic functions in $D$ such that all solutions of
\eqref{e1.2} are of infinite order. Let
$d_{0}(z) ,d_{1}(z),d_{2}(z)$ be analytic functions
in $D$ which are not all equal to zero with
$\rho (d_{j})<\infty $ $(j=0,1,2)$  such that
$h\not\equiv 0$, and let $\varphi (z)$ be an
analytic function in $D$  with finite order. If
$f$ is a solution of \eqref{e1.2} with
$\rho _{2}(f)=\rho $,  then the differential polynomial
$g_{f}=d_{2}f''+d_{1}f'+d_{0}f$ satisfies \eqref{e1.8}
 and \eqref{e1.9}.
\end{theorem}

\begin{remark} \label{rmk1.2} \rm
In Theorems \ref{thm1.1}, \ref{thm1.2}, \ref{thm1.3}, if we do not have
the condition $h\not\equiv 0$, then the differential polynomial
can be of finite order. For example, if $d_{2}(z)\not\equiv 0$, is a
finite order analytic function in $D$ and
$d_{0}(z)=A_{0}(z)d_{2}(z)$,
$d_{1}(z)=A_{1}(z)d_{2}(z)$, then $h\equiv 0$ and
$g_{f}=F(z) d_{2}(z)$ is of finite order.
\end{remark}

In the following we give an application of the above
results.

\begin{corollary} \label{coro1.1}
Let $A_{0}(z)$, $A_{1}(z),d_{0},d_{1},d_{2}$
 be analytic functions in $D$ such that
$\max \{\rho (A_{1}),\rho (d_{j})$
$(j=0,1,2)\}<\rho (A_{0})=\rho $ $(0<\rho <\infty )$,
$\tau(A_{0})=\tau $ $(0<\tau <\infty )$,  and let
 $\varphi \not\equiv 0$ be an analytic function in $D$
with $\rho (\varphi )<\infty $. If $f\not\equiv 0$ is a
solution of equation \eqref{e1.10},  then the differential
polynomial $g_{f}=d_{2}f''+d_{1}f'+d_{0}f$
satisfies
\begin{gather}
\overline{\lambda }(g_{f}-\varphi )=\lambda (
g_{f}-\varphi )=\rho (g_{f})=\rho (f)
=\infty ,  \label{e1.11}
\\
\alpha _{m}\geqslant \overline{\lambda }_{2}(g_{f}-\varphi )
=\lambda _{2}(g_{f}-\varphi )=\rho _{2}(g_{f})
=\rho _{2}(f)\geqslant \rho (A_{0}),  \label{e1.12}
\end{gather}
where $\alpha _{M}=\max \{ \rho _{M}(A_{j}):j=0,1\} $.
\end{corollary}

\begin{remark} \label{rmk1.3}\rm
The special case $\varphi (z)=z$ in the above theorems reduces
to the fixed points of the differential
polynomial $g_{f}$.
\end{remark}

\section{Auxiliary Lemmas}

\begin{lemma}[\cite{c3}] \label{lem2.1}
 Let $f(z)$ be a meromorphic solution of the equation
\begin{equation}
L(f)=f^{(k)}+A_{k-1}(z)f^{(k-1)}+\dots +A_{0}(z)f=F(z),  \label{e2.1}
\end{equation}
where $k$ is an positive integer,
$A_{0},\dots $, $A_{k-1}$, $F\not\equiv 0$ are
meromorphic functions in $D$  such that
$\max \{\rho _{i}(F),\rho _{i}(A_{j})$ $(j=0,\dots ,k-1)\}
<\rho _{i}(f)$, $(i=1,2)$.
 Then,
\begin{equation}
\overline{\lambda }_{i}(f)=\lambda _{i}(f)=\rho _{i}(f)\quad
(i=1,2).  \label{e2.2}
\end{equation}
\end{lemma}

Using the properties of the order of growth see
\cite[Proposition 1.1]{c1} and the
definition of the type, we easily obtain the following result
which we omit the proof.

 \begin{lemma} \label{lem2.2}
Let $f$ and $g$ be meromorphic functions in
$D$ such that $0<\rho(f)$, $\rho (g)<\infty $ and
$0<\tau (f)$, $\tau (g)<\infty $.
 Then the following two statements hold:
\begin{itemize}
\item[(i)] If $\rho (f)>\rho(g)$, then
\begin{equation}
\tau (f+g)=\tau (fg)=\tau (f). \label{e2.3}
\end{equation}

\item[(ii)]  If $\rho (f)=\rho(g)$  and $\tau (f)>\tau (g)$,
then
\begin{equation}
\rho (f+g)=\rho (fg)=\rho (f)=\rho (g).  \label{e2.4}
\end{equation}
\end{itemize}
\end{lemma}

\begin{lemma} \label{lem2.3}
Let $A_{0}(z)$, $A_{1}(z)$, $d_{0}$, $d_{1}$, $d_{2}$
 be analytic functions in $D$  such that
$\max \{\rho (A_{1}),\rho (d_{j}), (j=0,1,2)\}
<\rho (A_{0})=\rho$
$(0<\rho <\infty )$, $\tau (A_{0})=\tau $
$(0<\tau <\infty )$. Then $h\not\equiv0 $, where
$h$ is given by \eqref{e1.5}.
\end{lemma}

 \begin{proof}
First we suppose that $d_{2}(z)\not\equiv 0$. Set
\begin{equation}
\begin{aligned}
h&=\alpha _{1}\beta _{0}-\alpha _{0}\beta _{1}=(d_{1}-d_{2}A_{1})
(d_{2}A_{0}A_{1}-(d_{2}A_{0})'-d_{1}A_{0}+d_{0}')\\
&\quad -(d_{0}-d_{2}A_{0})(d_{2}A_{1}^{2}-(
d_{2}A_{1})'-d_{1}A_{1}-d_{2}A_{0}+d_{0}+d_{1}').
\end{aligned} \label{e2.5}
\end{equation}
Now check all the terms of $h$. Since the term $d_{2}^{2}A_{1}^{2}A_{0}$
is eliminated, by \eqref{e2.5} we can write
\begin{equation}
\begin{aligned}
h&=-d_{2}^{2}A_{0}^{2}-d_{0}d_{2}A_{1}^{2}+(d_{1}'d_{2}
 +2d_{0}d_{2}-d_{2}'d_{1}-d_{1}^{2})A_{0}\\
&\quad +(d_{2}'d_{0}-d_{2}d_{0}'+d_{0}d_{1})
A_{1}+d_{1}d_{2}A_{0}A_{1}
-d_{1}d_{2}A_{0}'\\
&\quad +d_{0}d_{2}A_{1}' +d_{2}^{2}A_{0}'A_{1}
-d_{2}^{2}A_{0}A_{1}'+d_{0}'d_{1}-d_{0}d_{1}'-d_{0}^{2}.
\end{aligned}  \label{e2.6}
\end{equation}
By $d_{2}\not\equiv 0$, $A_{0}\not\equiv 0$ and Lemma \ref{lem2.2}
 we get from \eqref{e2.6} that $\rho (h)=\rho (A_{0})=\rho >0$,
then $h\not\equiv 0$.

 Now suppose $d_{2}\equiv 0$, $d_{1}\not\equiv 0$. Using a
similar reasoning as above we get $h\not\equiv 0$.
 Finally, if $d_{2}\equiv 0$, $d_{1}\equiv 0$,
$d_{0}\not\equiv 0$, then we have $h=-d_{0}^{2}\not\equiv 0$.
\end{proof}

\section{Proof of main results}

\begin{proof}[Proof of Theorem \ref{thm1.1}]
 Suppose that $f$ is a solution of \eqref{e1.2}
with $\rho (f)=\infty $ and $\rho _{2}(f)=\rho $.
Substituting $f''=F-A_{1}f'-A_{0}f$ into
$g_{f} $, we have
\begin{equation}
g_{f}-d_{2}F=(d_{1}-d_{2}A_{1})f'+(
d_{0}-d_{2}A_{0})f.  \label{e3.1}
\end{equation}
Differentiating both sides of  \eqref{e3.1} and using that
 $f''=F-A_{1}f'-A_{0}f$, we obtain
\begin{equation}
\begin{aligned}
g_{f}'-(d_{2}F)'-(d_{1}-d_{2}A_{1})F
&=[d_{2}A_{1}^{2}-(d_{2}A_{1})
'-d_{1}A_{1}-d_{2}A_{0}+d_{0}+d_{1}']f' \\
&\quad +[d_{2}A_{0}A_{1}-(d_{2}A_{0})'-d_{1}A_{0}+d_{0}']f.
\end{aligned}  \label{e3.2}
\end{equation}
Then, by \eqref{e1.3}, \eqref{e1.4}, \eqref{e3.1}
and \eqref{e3.2}, we have
\begin{gather}
\alpha _{1}f'+\alpha _{0}f=g_{f}-d_{2}F,  \label{e3.3}\\
\beta _{1}f'+\beta _{0}f=g_{f}'-(d_{2}F)
'-(d_{1}-d_{2}A_{1})F.  \label{e3.4}
\end{gather}
Set
\begin{equation}
\begin{aligned}
h&=\alpha _{1}\beta _{0}-\alpha _{0}\beta _{1}\\
&=(d_{1}-d_{2}A_{1}) (d_{2}A_{1}^{2}-(d_{2}A_{1})'-d_{1}A_{1}
-d_{2}A_{0}+d_{0}+d_{1}') \\
&\quad -(d_{0}-d_{2}A_{0})(d_{2}A_{0}A_{1}-(
d_{2}A_{0})'-d_{1}A_{0}+d_{0}').
\end{aligned}\label{e3.5}
\end{equation}
By the condition $h\not\equiv 0$ and \eqref{e3.3}-\eqref{e3.5},
we obtain
\begin{equation}
f=\frac{\alpha _{1}\big(g_{f}'-(d_{2}F)'-\alpha _{1}F\big)-\beta _{1}(g_{f}-d_{2}F)}{h}.  \label{e3.6}
\end{equation}
If $\rho (g_{f})<\infty $, then by \eqref{e3.6} we obtain
$\rho (f)<\infty $ and this is a contradiction. Hence $\rho
(g_{f})=\infty $.

 Now, we prove that $\rho _{2}(g_{f})=\rho _{2}(f)=\rho $.
By $g_{f}=d_{2}f''+d_{1}f'+d_{0}f,$\ we obtain
$\rho _{2}(g_{f})\leqslant \rho_{2}(f)$ and by \eqref{e3.6},
we have $\rho _{2}(f)\leqslant \rho _{2}(g_{f})$. Hence
$\rho _{2}(g_{f})=\rho _{2}(f)=\rho $.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm1.2}]
 Suppose that $f$ is a solution of  \eqref{e1.2}
with $\rho (f)=\infty $ and $\rho _{2}(f)=\rho $.
Set $w(z)=g_{f}-\varphi $. Since $\rho (\varphi )<\infty $,
then by Theorem \ref{thm1.1}, we have
$\rho (w)=\rho (g_{f})=\rho (f)=\infty $ and
$\rho _{2}(w)=\rho _{2}(g_{f})=\rho _{2}(f)=\rho $. To
prove $\overline{\lambda }(g_{f}-\varphi )=\lambda (
g_{f}-\varphi )=\infty $ and $\overline{\lambda }_{2}(
g_{f}-\varphi )=\lambda _{2}(g_{f}-\varphi )=\rho $, we
need to prove only $\overline{\lambda }(w)=\lambda (
w)=\infty $ and $\overline{\lambda }_{2}(w)=\lambda
_{2}(w)=\rho $. By $g_{f}=w+\varphi $, and using
\eqref{e3.6}, we have
\begin{equation}
f=\frac{\alpha _{1}w'-\beta _{1}w}{h}+\psi (z), \label{e4.1}
\end{equation}
where $\alpha _{1}$, $\beta _{1}$, $h$, $\psi (z)$ are defined
in \eqref{e1.3}-\eqref{e1.6}. Substituting \eqref{e4.1}
into equation \eqref{e1.2}, we obtain
\begin{equation}
\frac{\alpha _{1}}{h}w'''+\phi _{2}w''+\phi _{1}w'+\phi _{0}w
=F-\big(\psi ''+A_{1}(z)\psi '+A_{0}(z)\psi \big)
=A,  \label{e4.2}
\end{equation}
where $\phi _{j}$ $(j=0,1,2)$ are meromorphic functions in $D$
with $\rho (\phi _{j})<\infty $ $(j=0,1,2)$. Since
$\psi (z)$ is not a solution of \eqref{e1.2}, it
follows that $A\not\equiv 0$. Then, by Lemma \ref{lem2.1}, we obtain
 $\overline{ \lambda }(w)=\lambda (w)=\rho (w)=\infty $ and
$\overline{\lambda }_{2}(w)=\lambda _{2}(
w)=\rho _{2}(w)=\rho $; i.e.,
$\overline{\lambda } (g_{f}-\varphi )=\lambda (g_{f}-\varphi )=\infty $
and $\overline{\lambda }_{2}(g_{f}-\varphi )=\lambda _{2}(g_{f}-\varphi )
=\rho $.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm1.3}]
By the hypotheses of Theorem \ref{thm1.3}, all solutions of
\eqref{e1.2} are of infinite order. From \eqref{e1.6}, we see
that $\psi (z)$ is of finite order, then $\psi (z)$
is not a solution of equation \eqref{e1.2}. By Theorem \ref{thm1.2}, we
obtain Theorem \ref{thm1.3}.
\end{proof}

\begin{proof}[Proof of Corollary \ref{coro1.1}]
 By Theorem \ref{thmA}, all solutions $f\not\equiv 0$ of
\eqref{e1.10} are of infinite order and satisfy
\[
\rho (A_{0})\leqslant \rho _{2}(f)\leqslant \max
\{\rho _{M}(A_{0}),\rho _{M}(A_{1})\}.
\]
Also, by Lemma \ref{lem2.3}, we have $h\not\equiv 0$. Then, by using
Theorem \ref{thm1.3} we obtain Corollary \ref{coro1.1}.
\end{proof}

 \subsection*{Acknowledgements}
The authors would like to thank the anonymous
referee for his/her helpful remarks and suggestions to
improve this article.

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