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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 226, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2016/226\hfil Growth of local solutions]
{Growth of local solutions to linear differential equations around an isolated
essential singularity}

\author[H. Fettouch, S. Hamouda \hfil EJDE-2016/226\hfilneg]
{Houari Fettouch, Saada Hamouda}

\address {Houari Fettouch \newline
 Laboratory of Pure and Applied Mathematics,
 University of Mostaganem, UMAB, Algeria}
\email{houari.fettouch@univ-mosta.dz}

\address {Saada Hamouda  \newline
 Laboratory of Pure and Applied Mathematics,
 University of Mostaganem, UMAB, Algeria}
\email{saada.hamouda@univ-mosta.dz}

\thanks{Submitted April 18, 2016. Published August 18, 2016.}
\subjclass[2010]{34M10, 30D35}
\keywords{Linear differential equations; local growth of solutions;
\hfill\break\indent  isolated essential singularity}

\begin{abstract}
 In this article we study the growth of solutions to a class of linear
 differential equations around an isolated essential singularity point.
 By using conformal mapping we apply some results from the complex
 plane to a neighborhood of a singular point.   We point out that  there
 are several similarities between the results for complex plane and
 results in this article.
\end{abstract}

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


\section{Introduction and statement of results}

Throughout this article, we assume that the reader is familiar with the
fundamental results and the standard notations of the Nevanlinna value
distribution theory of meromorphic function on the complex plane 
$\mathbb{C}$ and in the unit disc $D=\{ z\in \mathbb{C} :| z| <1\}$
(see \cite{haym, yang}). The
importance of this theory has inspired many authors to find modifications
and generalizations to different domains. Extensions of Nevanlinna Theory to
annuli have been made by \cite{bieb, khri, kond, korh, mark}. In this paper,
we concentrate our investigation near an isolated essential singular point.
We start to give the appropriate definitions. 
Set $\overline{\mathbb{C}}=\mathbb{C}\cup \{ \infty \} $ 
and suppose that $f( z) $ is meromorphic in $\overline{\mathbb{C}}-\{ z_0\} $,
where $z_0\in\mathbb{C}$. Define the counting function of $f$ by
\begin{equation}
N_{z_0}( r,f) =-\int_{\infty }^{r}\frac{n(
t,f) -n( \infty ,f) }{t}dt-n( \infty ,f) \log r,
\label{d1}
\end{equation}
where $n( t,f) $ counts the number of poles of $f( z) $
in the region $\{ z\in\mathbb{C}:t\leq | z-z_0| \} \cup \{ \infty \} $
each pole according to its multiplicity; and the proximity function by
\begin{equation}
m_{z_0}( r,f) =\frac{1}{2\pi }\int_0^{2\pi }\ln^{+}| f( z_0-re^{i\varphi }) 
| d\varphi . \label{d2}
\end{equation}
The characteristic function of $f$ is defined in the usual manner by
\begin{equation}
T_{z_0}( r,f) =m_{z_0}( r,f) +N_{z_0}(
r,f) . \label{d3}
\end{equation}

In addition, the order of meromorphic function $f( z) $ near $z_0$ is defined by
\begin{equation}
\sigma _{T}( f,z_0) =\limsup_{r\to 0} \frac{
\log ^{+}T_{z_0}( r,f) }{-\log r}. \label{d4}
\end{equation}
For an analytic function $f( z) $ in $\overline{\mathbb{C}}-\{ z_0\} $, 
we have also the definition
\begin{equation}
\sigma _{M}( f,z_0) =\limsup_{r\to 0} \frac{
\log ^{+}\log ^{+}M_{z_0}( r,f) }{-\log r}, \label{d5}
\end{equation}
where $M_{z_0}( r,f) =\max \{ | f( z)| :| z-z_0| =r\} $. 

For example, the function $f( z) =\exp \{ \frac{1}{(z_0-z) ^{n}}\} $, where 
$n\in\mathbb{N}\backslash \{ 0\} $, we have 
$M_{z_0}( r,f) =\exp\{ \frac{1}{r^{n}}\} $, and then 
$\sigma _{M}(f,z_0) =n$. We have also 
$T_{z_0}( r,f)=m_{z_0}( r,f) =\frac{1}{2\pi }\int_0^{2\pi }\ln
^{+}| f( z_0-re^{i\varphi }) | d\varphi =
\frac{1}{r^{n}}$, and so $\sigma _{T}( f,z_0) =n$.

For the function $f( z) =\exp \{ \frac{-1}{(1-z) }\} $, we have 
$\sigma ( f,1) =1$ while in the unit disc we have 
$\sigma _{T}( f) =\sigma _{M}( f) =0$.

We see that in the unit disc we have 
$\sigma _{T}( f) \leq \sigma_{M}( f) \leq \sigma _{T}( f) +1$ 
and in the complex plane we have $\sigma _{T}( f) =\sigma _{M}( f) $.
Now, how about the relation between $\sigma _{T}( f,z_0) $ and 
$\sigma _{M}( f,z_0) $? Below, in Lemma \ref{lem2}, we will
prove that if $f( z) $ is meromorphic function in 
$\overline{\mathbb{C}}-\{ z_0\} $ and 
$g( w) =f( z_0-\frac{1}{w}) $, then $g( w) $ is a meromorphic function in
$\mathbb{C} $ and we have $T( R,g) =T_{z_0}( r,f) $; where 
$R=\frac{1}{r}$; which implies that $\sigma _{T}( f,z_0) =\sigma_{M}( f,z_0) $. 
So, we can use the notation $\sigma (f,z_0) $ without any ambiguity.

In the usual manner, we define the hyper order near $z_0$ as 
\begin{gather}
\sigma _{2,T}( f,z_0) =\limsup_{r\to 0}
\frac{\log ^{+}\log ^{+}T_{z_0}( r,f) }{-\log r}, \label{d6} \\
\sigma _{2,M}( f,z_0) =\limsup_{r\to 0}
\frac{\log ^{+}\log ^{+}\log ^{+}M_{z_0}( r,f) }{-\log r}.
\label{d7}
\end{gather}
The linear differential equation
\begin{equation}
f''+A( z) e^{az}f'+B( z)e^{bz}f=0, \label{1}
\end{equation}
where $A( z) $ and $B( z) $ are entire functions, is
investigated by many authors; see for example \cite{ozaw,chen1,chen2,gund2}.
In \cite{chen1}, Chen proved that if $ab\neq 0$ and $\arg a\neq \arg b$ or 
$a=cb$ ($0<c<1$ or $c>1$), then every solution 
$f(z) \not\equiv 0$ of \eqref{1} is of infinite order. Recently, the
second author proved results similar to \eqref{1} in the unit disc
for the differential equation
\begin{equation}
f''+A( z) e^{\frac{a}{( z_0-z) ^{\mu
}}}f'+B( z) e^{\frac{b}{( z_0-z) ^{\mu }}}f=0, \label{2}
\end{equation}
where $A( z) $ and $B( z) $ are analytic in the unit
disc, $\mu >0$ and $\arg a\neq \arg b$ or $a=cb$ $( 0<c<1) $, see
\cite{ham12}. However, the method of \cite{ham12} does not work in general
for the case $0<\mu \leq 1$: see the discussion in \cite[Remark 3.1]{ham12}.
The case $\mu =1$ will be investigated in the following theorem with certain
modifications on $A( z) $ and $B( z) $.

\begin{theorem}\label{thm1}
Let $z_0,a,b$ be complex constants such that $\arg a\neq \arg b$
or $a=cb$ $(0<c<1) $ and $n$ be a positive integer. 
Let $A(z) ,B( z) \not\equiv 0$ be analytic functions in 
$\overline{\mathbb{C}}-\{ z_0\} $ with 
$\max \{ \sigma ( A,z_0),\sigma ( B,z_0) \} <n$. Then, every solution 
$f(z) \not\equiv 0$ of the differential equation
\begin{equation}
f''+A( z) e^{\frac{a}{( z_0-z) ^{n}}
}f'+B( z) e^{\frac{b}{( z_0-z) ^{n}}}f=0.
\label{eq0}
\end{equation}
satisfies $\sigma ( f,z_0) =\infty $ with $\sigma _2(f,z_0) =n$.
\end{theorem}

In \cite{frei}, Frei proved the following result in the complex plane.

\begin{theorem}[\cite{frei}]\label{thma} 
 If the differential equation
\begin{equation}
g''+e^{-w}g'+cg=0 \label{eq1}
\end{equation}
where $c\neq 0$ is a complex constant, possesses a solution $g\not\equiv 0$
of finite order, then $c=-k^2$ where $k$ is a positive integer.
Conversely, for each positive integer $k$, the equation \eqref{eq1} with
$c=-k^2$ possesses a solution $g$ which is a polynomial in $e^{w}$ of
degree $k$.
\end{theorem}

The analogous of this result, near a singular point $z_0$, is as 
follows.

\begin{theorem} \label{thm2}
Let $c\neq 0, z_0$ be complex numbers. If the differential equation
\begin{equation}
f''+\Big( \frac{1}{( z_0-z) ^2}e^{\frac{-1}{
( z_0-z) }}-\frac{2}{( z_0-z) }\Big) f'+
\frac{c}{( z_0-z) ^{4}}f=0 \label{eq2}
\end{equation}
possesses a solution $f( z) \not\equiv 0$ of finite order 
$\sigma( f,z_0) <\infty $ then $c=-k^2$, where $k$ is an integer.
Conversely, for each positive integer $k$, the equation \eqref{eq2} with 
$c=-k^2$, possesses a solution $f$ which is a polynomial in
$e^{\frac{1}{( z_0-z) }}$ of degree $k$.
\end{theorem}

\begin{example} \label{examp1} \rm
$f_1( z) =1+e^{\frac{1}{( z_0-z) }}$ is a
solution of the differential equation
\begin{equation*}
f''+( \frac{1}{( z_0-z) ^2}e^{\frac{-1}{
( z_0-z) }}-\frac{2}{( z_0-z) }) f'-
\frac{1}{( z_0-z) ^{4}}f=0.
\end{equation*}
\end{example}

\begin{example} \label{examp2} \rm
$f_2( z) =1+4e^{\frac{1}{( z_0-z) }}+6e^{\frac{2
}{( z_0-z) }}$ is a solution of the differential equation
\begin{equation*}
f''+\Big( \frac{1}{( z_0-z) ^2}e^{\frac{-1}{
( z_0-z) }}-\frac{2}{( z_0-z) }\Big) f'-
\frac{4}{( z_0-z) ^{4}}f=0.
\end{equation*}
\end{example}

\begin{theorem}\label{thm3}
Let $A_0( z) \not\equiv 0,A_1( z),\dots ,A_{k-1}( z) $ be meromorphic 
functions in $\overline{\mathbb{C}}-\{ z_0\} $ satisfying
\begin{gather}
| A_0( z) | \geq \exp \{ \frac{\alpha }{r^{\mu }}\} , \label{c1} \\
| A_{j}( z) | \leq \exp \{ \frac{\beta }{
r^{\mu }}\} ,\ j\neq 0, \label{c2}
\end{gather}
where $\alpha >\beta \geq 0$, $\mu >0$, $\arg ( z_0-z) =\theta
\in ( \theta _1,\theta _2) \subset [ 0,2\pi ) $
and $| z_0-z| =r\to 0$. Then, every solution $f( z) \not\equiv 0$ 
of the differential equation
\begin{equation}
f^{( k) }+A_{k-1}( z) f^{( k-1)
}+\dots +A_1( z) f'+A_0( z) f=0, \label{eq3}
\end{equation}
satisfies $\sigma _2( f,z_0) \geq \mu $.
\end{theorem}

Similar results to Theorem \ref{thm3} in the complex plane are given in 
\cite{bel, gund3}.

\begin{theorem} \label{thm4}
Let $A_0( z) \not\equiv 0,A_1( z),\dots ,A_{k-1}( z) $ be analytic functions in 
$\overline{\mathbb{C}}-\{ z_0\} $ satisfying 
$\max \{ \sigma (A_{j},z_0) :j\neq 0\} <\sigma ( A_0,z_0) $.
Then, every solution $f( z) \not\equiv 0$ of \eqref{eq3}
satisfies $\sigma _2( f,z_0) =\sigma (A_0,z_0) $.
\end{theorem}

\section{Preliminaries lemmas}

Throughout this paper, we use the following symbols that do not have 
necessarily the same at each occurrence:
$r_0>0$, $\varepsilon >0$, $\gamma >1$, $\lambda >0$ are real constants.
The set $E_1^{\ast }\subset ( 0,r_0] $ has finite logarithmic
measure $\int_0^{r_0}\frac{\chi _{E_1^{\ast }}}{t}dt<\infty $.
The set $E_2^{\ast }\subset [ 0,2\pi ) $ has a linear measure zero,
$\int_0^{2\pi }\chi _{E_2^{\ast }}dt=0$.

\begin{lemma}[\cite{gund1}] \label{lem1}
 Let $g$ be a transcendental meromorphic function in
$\mathbb{C}$, and let $\gamma >1$, $\varepsilon >0$ be given real constants; 
then

(i) there exists a set $E_1\subset ( 1,\infty ) $ that has a
finite logarithmic measure and a constant $\lambda >0$ that depends only on 
$\gamma $ such that for all $R=| w| $ satisfying $R\notin E_1$, we have
\begin{equation}
\big| \frac{g^{( k) }( w) }{g( w) }\big| 
\leq \lambda [ T( \gamma R,g) \log T(
\gamma R,g) ] ^{k}; \label{l1}
\end{equation}

(ii) there exists a set $E_2\subset [ 0,2\pi ) $ that has a
linear measure zero and a constant $\lambda >0$ that depends only on 
$\gamma $ such that for all $\theta \in [ 0,2\pi ) \backslash E_2$
there exists a constant $R_0=R_0( \theta ) >0$ such that for
all $z$ satisfying $\arg z\in [ 0,2\pi ) \backslash E_2$ and 
$r=| z| >R_0$, we have
\begin{equation}
\big| \frac{g^{( k) }( w) }{g( w) }\big| 
\leq \lambda [ T( \gamma R,g) R^{\varepsilon}\log T( \gamma R,g) ] ^{k}. \label{l2}
\end{equation}
\end{lemma}

\begin{lemma} \label{lem2}
Let $f$ be a non constant meromorphic function in
 $\overline{\mathbb{C}}-\{ z_0\} $ and set 
$g( w) =f( z_0-\frac{1}{w}) $. Then, $g( w) $ is meromorphic in 
$\mathbb{C}$ and we have
\begin{equation*}
T( R,g) =T_{z_0}\big( \frac{1}{R},f\big) .
\end{equation*}
\end{lemma}

\begin{proof}
It is easy to prove the following statements:

(i) $w_0\neq 0$ is a pole of $g$ of order $n$ if and only if 
$\frac{1}{w_0}-z_0$ is a pole of $f$ of order $n$.

(ii) $0$ is a pole of $g$ of order $n$ if and only if $\infty $ is a pole of 
$f$ of order $n$.

(iii) The change of variable $w=\frac{1}{z_0-z}$ maps the region 
$\{z\in\mathbb{C}:t\leq | z-z_0| \} \cup \{ \infty \} $
on the region $\{ w\in \mathbb{C}:| w| \leq \frac{1}{t}\} $.

From these statements,  $g( w) $ is meromorphic in $\mathbb{C}$ and by using
the change of variable $T=\frac{1}{t}$, we obtain 
\begin{align*}
N( R,g) 
&=\int_0^{R}\frac{n( T,g) -n(0,g) }{T}dT+n( 0,g) \ln R \\
&=-\int_{\infty }^{\frac{1}{R}}\frac{n( t,f) -n(\infty ,f) }{t}dt
+n( \infty ,f) \ln R \\
&=-\int_{\infty }^{\frac{1}{R}}\frac{n( t,f) -n(
\infty ,f) }{t}dt-n( \infty ,f) \ln \frac{1}{R}
=N_{z_0}( \frac{1}{R},f) .
\end{align*}
Which means that $N( R,g) =N_{z_0}( \frac{1}{R},f) $.
We have
\begin{equation}
\begin{aligned}
m( R,g)
&=\frac{1}{2\pi }\int_0^{2\pi }\ln ^{+}| g( Re^{i\varphi }) | d\varphi \\
&=\frac{1}{2\pi }\int_0^{2\pi }\ln ^{+}| f( z_0-\frac{1}{R}
e^{-i\varphi }) | d\varphi \label{l2b} \\
&= \frac{-1}{2\pi }\int_0^{-2\pi }\ln ^{+}| f( z_0-
\frac{1}{R}e^{i\varphi }) | d\varphi   \\
&= \frac{1}{2\pi }\int_{-2\pi }^{0}\ln ^{+}| f( z_0-
\frac{1}{R}e^{i\varphi }) | d\varphi   \\
&=\frac{1}{2\pi }\int_0^{2\pi }\ln ^{+}| f( z_0-
\frac{1}{R}e^{i\varphi }) | d\varphi =m_{z_0}( \frac{1
}{R},f) .
\end{aligned}
\end{equation}
So, we conclude that $T( R,g) =T_{z_0}( \frac{1}{R},f) $.
\end{proof}

\begin{remark} \label{rmk1} \rm 
By Lemma \ref{lem2}, if $f$ is a non constant meromorphic
function in $\overline{\mathbb{C}}-\{ z_0\} $ and 
$g( w) =f( z_0-\frac{1}{w}) $ then $\sigma ( f,z_0) =\sigma ( g) $.
\end{remark}

\begin{lemma} \label{lem3}
Let $f$ be a non constant meromorphic function in $\overline{\mathbb{C}}-\{ z_0\} $ 
and let $\gamma >1$, $\varepsilon >0$ be given
constants; then 

(i) there exists a set $E_1^{\ast }\subset ( 0,r_0] $ that has
finite logarithmic measure $\int_0^{r_0}\frac{\chi _{E_1^{\ast
}}}{t}dt<\infty $ and a constant $\lambda >0$ that depends only on $\gamma $
such that for all $r=| z-z_0| $ satisfying
 $r\in ( 0,r_0] \backslash E_1^{\ast }$, we have
\begin{equation}
| \frac{f^{( k) }( z) }{f( z) }
| \leq \lambda [ \frac{1}{r^2}T_{z_0}( \frac{r}{
\gamma },f) \log T_{z_0}( \frac{r}{\gamma },f) ]^{k}\quad
 ( k\in\mathbb{N}) ; \label{l3}
\end{equation}

(ii) there exists a set $E_2^{\ast }\subset [ 0,2\pi ) $ that
has a linear measure zero and a constant $\lambda >0$ that depends only on 
$\gamma $ such that for all $\theta \in [ 0,2\pi ) \backslash
E_2^{\ast }$ there exists a constant $r_0=r_0( \theta ) >0$
such that for all $z$ satisfying $\arg ( z-z_0) =\theta $ and 
$r=| z-z_0| <r_0$, we have
\begin{equation}
| \frac{f^{( k) }( z) }{f( z) }| 
\leq \lambda [ \frac{1}{r^{2+\varepsilon }}T_{z_0}(
\frac{r}{\gamma },f) \log T_{z_0}( \frac{r}{\gamma },f)
] ^{k}\ ( k\in \mathbb{N}) . \label{l4}
\end{equation}
\end{lemma}

\begin{proof}
Set $g( w) =f( z_0-\frac{1}{w}) $. $g( w)$ is meromorphic in 
$\mathbb{C}$ and by Lemma \ref{lem1},\ we have \eqref{l1} and \eqref{l2}.
 We have $f( z) =g( w) $ such that $w=\frac{1}{z_0-z}$;
 then $f'( z) =\frac{1}{( z_0-z) ^2}g'( w) $ and then
\begin{equation}
\frac{f'( z) }{f( z) }=\frac{1}{(
z_0-z) ^2}\frac{g'( w) }{g( w) }. \label{l5}
\end{equation}
By Lemma \ref{lem1}, we have
\begin{equation*}
| \frac{g'( w) }{g( w) }|
\leq \lambda [ T( \gamma R,g) \log T( \gamma R,g)
] ,\quad  R\notin E_1;
\end{equation*}
and by Lemma \ref{lem2} and \eqref{l5}, we obtain
\begin{align*}
| \frac{f'( z) }{f( z) }|
&\leq \lambda [ \frac{1}{r^2}T_{z_0}( \frac{1}{\gamma R}
,f) \log T_{z_0}( \frac{1}{\gamma R},f) ] \\
&\leq \lambda [ \frac{1}{r^2}T_{z_0}( \frac{r}{\gamma }
,f) \log T_{z_0}( \frac{r}{\gamma },f) ] ,\ r\notin
E_1^{\ast }.
\end{align*}
where $\frac{1}{r}=R\notin E_1 \Leftrightarrow r\notin E_1^{\ast }$ and 
$\int_0^{r_0}\frac{\chi _{E_1^{\ast }}}{t}dt
=\int _{1/r_0}^{\infty }\frac{\chi _{E_1}}{T}dT<\infty $.

We have $f''( z) =\frac{1}{( z_0-z)
^{4}}g''( w) +\frac{2}{( z_0-z) ^{3}}g'( w) $; 
and so
\begin{equation*}
\frac{f''( z) }{f( z) }
=\frac{1}{( z_0-z) ^{4}}\frac{g''( w) }{g(
w) }+\frac{2}{( z_0-z) ^{3}}\frac{g'(w) }{g( w) }.
\end{equation*}
and by Lemma \ref{lem1} and Lemma \ref{lem2}, we obtain
\begin{equation*}
| \frac{f''( z) }{f( z) }| 
\leq \lambda \big[ \frac{1}{r^2}T_{z_0}( \frac{r}{
\gamma },f) \log T_{z_0}( \frac{r}{\gamma },f) \big]^2\quad
 r\notin E_1^{\ast }.
\end{equation*}
In general, we can obtain 
\begin{equation*}
f^{( k) }( z) =\frac{1}{( z_0-z) ^{2k}}
g^{( k) }( w) +\frac{a_{k-1}}{( z_0-z)
^{2k-1}}g^{( k-1) }( w) +\dots +\frac{a_1}{(
z_0-z) ^{k+1}}g'( w) ,
\end{equation*}
where $a_{j}$  $(j=1,2,\dots ,k-1)$ are integers; and thus
\begin{equation}
\begin{aligned}
\frac{f^{( k) }( z) }{f( z) }
&=\frac{1}{( z_0-z) ^{2k}}\frac{g^{( k) }( w) }{
g( w) }+\frac{a_{k-1}}{( z_0-z) ^{2k-1}}\frac{
g^{( k-1) }( w) }{g( w) }+\dots\\
&\quad  +\frac{a_1}{( z_0-z) ^{k+1}}\frac{g'( w) }{g(w) }. 
\end{aligned} \label{l6}
\end{equation}
Also by suing Lemma \ref{lem1} and Lemma \ref{lem2} with \eqref{l6},
for $r=| z-z_0| <r_0$, we obtain
\begin{equation*}
\big| \frac{f^{( k) }( z) }{f( z) }\big| 
\leq \lambda \big[ \frac{1}{r^2}T_{z_0}( \frac{r}{
\gamma },f) \log T_{z_0}( \frac{r}{\gamma },f) \big]^{k}\quad r\notin E_1^{\ast }.
\end{equation*}
Now for \eqref{l4} we can use the same method as above and by using
Lemma \ref{lem1} and Lemma \ref{lem2} with \eqref{l6}, we obtain,
 for $ r=| z-z_0| <r_0$ and $\arg ( z-z_0) \in
[ 0,2\pi ) \backslash E_2^{\ast }$,
\begin{equation*}
| \frac{f^{( k) }( z) }{f( z) }
| \leq \lambda \big[ \frac{1}{r^{2+\varepsilon }}T_{z_0}(
\frac{r}{\gamma },f) \log T_{z_0}( \frac{r}{\gamma },f)
\big] ^{k},
\end{equation*}
where $\theta \in E_2\Leftrightarrow 2\pi -\theta \in E_2^{\ast }\
(E_2^{\ast }\subset [ 0,2\pi ) $ has a linear measure zero).
\end{proof}

The following lemma is a particular case of Lemma \ref{lem3}.

\begin{lemma}\label{lem4}
Let $f$ be a non constant meromorphic function in 
$\overline{\mathbb{C}}-\{ z_0\} $ of finite order 
$\sigma ( f,z_0)<\infty $; let $\varepsilon >0$ be a given constant. 
Then the following two statements hold.

(i) There exists a set $E_1^{\ast }\subset ( 0,r_0] $ that has
finite logarithmic measure 
$\int_0^{r_0}\frac{\chi _{E_1^{\ast}}}{t}dt<\infty $ such that for all 
$r=| z-z_0| \in( 0,r_0] \backslash E_1^{\ast }$, we have
\begin{equation}
| \frac{f^{( k) }( z) }{f( z) }
| \leq \frac{1}{r^{k( \sigma +2+\varepsilon ) }},\quad 
( k\in\mathbb{N}) . \label{l7}
\end{equation}

(ii) There exists a set $E_2^{\ast }\subset [ 0,2\pi ) $ that
has a linear measure zero such that for all 
$\theta \in [ 0,2\pi) \backslash E_2^{\ast }$ there exists a constant 
$r_0=r_0(\theta ) >0$ such that for all $z$ satisfying 
$\arg (z-z_0) =\theta $ and $r=| z-z_0| <r_0$, the
inequality \eqref{l7} holds.
\end{lemma}

The question which arises here is the following: can we get similar
estimations on $| \frac{f^{( k) }( z) }{f( z) }| $ in \eqref{l3}, 
\eqref{l4} and \eqref{l7} for a non constant function that is meromorphic 
only on a bounded region of the form 
$\{ z\in\mathbb{C}:0<| z-z_0| \leq r_0\}$?

\begin{lemma} \label{lem5}
Let $h$ be a non constant analytic function in $\overline{\mathbb{C}}-\{ z_0\} $ 
of order $\sigma ( f,z_0) >\alpha >0$. Then, there exists a 
set $F\subset ( 0,r_0] $ of infinite
logarirhmic measure $\int_0^{r_0}\frac{\chi _{F}}{t}dt=\infty $
such that for all $r\in F$ and
 $| h( z) |=M_{z_0}( r,h) $, we have
\begin{equation*}
\log | h( z) | >\frac{1}{r^{\alpha }}.
\end{equation*}
\end{lemma}

\begin{proof}
By the definition of $\sigma ( f,z_0) $, there exists a
decreasing sequence $\{ r_{m}\} \to 0$ satisfying 
$\frac{m}{m+1}r_{m}>r_{m+1}$ and
\begin{equation*}
\lim_{m\to \infty} \frac{\log \log M_{z_0}( r_{m},f) }{-\log r_{m}}>\alpha .
\end{equation*}
Then, there exists $m_0$ such that for all $m>m_0$ and for a given 
$\varepsilon >0$ small enough, we have
\begin{equation}
\log M_{z_0}( r_{m},f) >\frac{1}{r_{m}^{\alpha +\varepsilon }}.
\label{l8}
\end{equation}
There exists $m_1$ such that for all $m>m_1$, and for any 
$r\in [\frac{m}{m+1}r_{m},r_{m}] $ and for a given 
$\varepsilon >0$, we have
\begin{equation}
\big( \frac{m}{m+1}\big) ^{\alpha +\varepsilon }>r^{\varepsilon }.
\label{l9}
\end{equation}
By \eqref{l8} and \eqref{l9}, for all $m>m_2=\max \{m_0,m_1\} $ and for any
 $r\in [ \frac{m}{m+1}r_{m},r_{m}] $, we have
\begin{equation*}
\log M_{z_0}( r,f) >\log M_{z_0}( r_{m},f) >\frac{1
}{r_{m}^{\alpha +\varepsilon }}>\frac{1}{r^{\alpha +\varepsilon }}(
\frac{m}{m+1}) ^{\alpha +\varepsilon }>\frac{1}{r^{\alpha }}.
\end{equation*}
Set $F=\cup_{m=m_2}^{\infty }[ \frac{m}{m+1}r_{m},r_{m}]$; then we have
\begin{equation*}
\sum_{m=m_2}^{\infty }\int_{\frac{m}{m+1}r_{m}}^{r_{m}}
\frac{dt}{t}=\sum_{m>m_2}\log \frac{m+1}{m}=\infty .
\end{equation*}
\end{proof}

We recall a particular case of an important result due to Chiang and Hayman
in \cite{chian}.

\begin{lemma}[\cite{chian}] \label{lem7} 
Let $A_{j}$ be meromorphic functions in $\mathbb{C}$ and $f$ be a meromorphic 
solution of \eqref{eq3}, assuming that not all
coefficients $A_{j}$ are constants. Given a real constant $\gamma >1$, and
denoting $T( R) :=\sum_{j=0}^{k-1}T( R,A_{j}) $, we have
\begin{equation*}
\log m( R,f) <T( R) \{ \log R\log T(R) \} ^{\gamma }.
\end{equation*}
\end{lemma}

We can transform this result near a singular point as  follows.

\begin{lemma} \label{lem8}
Let $A_{j}$ be meromorphic functions in 
$\overline{\mathbb{C}}-\{ z_0\} $ and $f$ be a meromorphic 
solution of \eqref{eq3}
in $\overline{\mathbb{C}}-\{ z_0\} $, assuming that not all coefficients 
$A_{j}$ are constants. Given a real constant $\gamma >1$, and denoting 
$T_{z_0}(r) :=T_{z_0}( r,A_0)
+\sum_{j=1}^{k-1}\sum_{i=j}^{k-1}T_{z_0}(
r,A_{i}) +O( \log \frac{1}{r}) $, we have
\begin{equation*}
\log m_{z_0}( r,f) <T_{z_0}( r) \{ \log \frac{1}{r}\log ( T_{z_0}( r) ) \} ^{\gamma }.
\end{equation*}
\end{lemma}

\begin{proof}
Set $g( w) =f( z_0-\frac{1}{w}) $; $g( w)$ is meromorphic in $\mathbb{C}$. 
We have $f( z) =g( w) $ such that $w=\frac{1}{z_0-z}$;
then $f'( z) =\frac{1}{( z_0-z)^2}g'( w) =w^2g'( w)$,
$ f''( z) =w^{4}g''( w)+2w^{3}g'( w)$. In general, we can obtain that
\begin{equation}
f^{( k) }( z) =w^{2k}g^{( k) }(w) +a_{k-1}w^{2k-1}g^{( k-1) }( w)
+\dots +a_1w^{k+1}g'( w) , \label{l9b}
\end{equation}
where $a_{j}\ ( j=1,2,\dots ,k-1) $ are integers. Substituting 
\eqref{l9b} in \eqref{eq3}, we obtain
\begin{equation*}
g^{( k) }( w) +B_{k-1}( w) g^{(k-1) }( w) +\dots +B_1( w) g'(w) +B_0( w) g( w) =0,
\end{equation*}
such that $B_0( w) =\frac{1}{w^{2k}}A_0( z_0-\frac{1}{w}) $ and for 
$j\neq 0$, $B_{j}( w) =\sum_{i=j}^{k-1} \frac{c_{ij}}{w^{n_{ij}}}A_{i}
( z_0-\frac{1}{w}) $ where 
$c_{ij}$ and $n_{ij}$ are integers with $0<n_{ij}\leq 2k$. Since 
$A_{j}( z_0-\frac{1}{w}) $ $( j=0,1,\dots ,k-1) $ are meroporphic
functions in $\mathbb{C}$, then $B_{j}( w) $\ are meromorphic functions in 
$\mathbb{C}$ and by Lemma \ref{lem7}, we have
\begin{equation}
\log m( R,g) <T( R,B) \{ \log R\log T(R,B) \} ^{\gamma }, \label{l10}
\end{equation}
where $T( R,B) =\sum_{j=0}^{k-1}T( r,B_{j}) $.
By \eqref{l2b}, we have
\begin{equation}
m( R,g) =m( \frac{1}{R},f) =m_{z_0}( r,f) \label{l11}
\end{equation}
and by Lemma \ref{lem2}, we obtain that
\begin{equation}
\begin{aligned}
T( R,B) 
&\leq T\big( R,A_0( z_0+\frac{1}{w})
\big) +\sum_{j=1}^{k-1}\sum_{i=j}^{k-1}T\big(
R,A_{i}( z_0+\frac{1}{w}) \big) +O( \log R) \\
&\leq T_{z_0}( r,A_0)
+\sum_{j=1}^{k-1}\sum_{i=j}^{k-1}T_{z_0}(
r,A_{i}) +O( \log \frac{1}{r}) .
\end{aligned}  \label{l12}
\end{equation}
From \eqref{l10}--\eqref{l12}, we conclude that
\begin{equation*}
\log m_{z_0}( r,f) <T_{z_0}( r) \big\{ \log \frac{
1}{r}\log ( T_{z_0}( r) ) \big\} ^{\gamma }.
\end{equation*}
\end{proof}

\begin{lemma}\label{lem9}
Let $A( z) $ be analytic function in $\overline{\mathbb{C}}-\{ z_0\} $ with 
$\sigma ( A,z_0) <n$. Set $g( z) =A( z) \exp \{ \frac{a}{(z_0-z) ^{n}}\}$
($n\geq 1$ is an integer), $a=\alpha+i\beta \neq 0$,
$z_0-z=re^{i\varphi }$, $\delta _{a}( \varphi )
=\alpha \cos ( n\varphi ) +\beta \sin ( n\varphi )$,
and $H=\{ \varphi \in [ 0,2\pi ) :\delta _{a}( \varphi) =0\}$
(obviously, $H$ is of linear measure zero). 

Then for any given $\varepsilon >0$ and any 
$\varphi \in [0,2\pi ) \backslash H$, there exists 
$r_0>0$ such that for $0<r<r_0$,  we have

(i) if $\delta _{a}( \varphi ) >0$, then
\begin{equation}
\exp \{ ( 1-\varepsilon ) \delta _{a}( \varphi )
\frac{1}{r^{n}}\} \leq | g( z) | \leq
\exp \{ ( 1+\varepsilon ) \delta _{a}( \varphi )
\frac{1}{r^{n}}\} , \label{e1}
\end{equation}

(ii)  if $\delta _{a}( \varphi ) <0$,  then
\begin{equation}
\exp \{ ( 1+\varepsilon ) \delta _{a}( \varphi )
\frac{1}{r^{n}}\} \leq | g( z) | 
\leq \exp \{ ( 1-\varepsilon ) \delta _{a}( \varphi )
\frac{1}{r^{n}}\} . \label{e2}
\end{equation}
\end{lemma}

\begin{proof}
Set $h( w) :=g( z_0-\frac{1}{w}) =A( z_0-\frac{1}{w}) \exp \{ aw^{n}\} $, 
where  $A( z_0-\frac{ 1}{w}) $ is a analytic function in 
$\mathbb{C}$ of order $\sigma =\sigma ( A,z_0) <n$. We have
\begin{equation*}
| \exp \{ aw^{n}\} | =| \exp \{\frac{a}{( z_0-z) ^{n}}\} | 
=\exp \{\frac{\delta _{a}( \varphi ) }{r^{n}}\} .
\end{equation*}
Using the analogous lemma in $\mathbb{C}$ (see \cite{chen2, mark}), 
we get \eqref{e1} and \eqref{e2}.
\end{proof}

\section{Proof of main results}

\begin{proof}[Proof of Theorem \ref{thm1}]
From \eqref{eq0}, we can write
\begin{equation}
| B( z) | | e^{\frac{b}{(z_0-z) ^{n}}}| 
\leq | \frac{f''}{f}| +| A( z) | | e^{\frac{a}{( z_0-z) ^{n}}}|\,
 | \frac{f'}{f}| . \label{p1}
\end{equation}

\noindent\textbf{Case 1.} 
$\arg a\neq \arg b$: then there exist 
$( \varphi_1,\varphi _2) \subset [ 0,2\pi ) $ such that for 
$\arg( z_0-z) =\varphi \in ( \varphi _1,\varphi _2) $
we have $\delta _{b}( \varphi ) >0$ and $\delta _{a}(\varphi ) <0$. 
Since $\max \{ \sigma ( A,z_0),\sigma ( B,z_0) \} <n$, 
then by Lemma \ref{lem9}, \eqref{l4} and \eqref{p1}, we obtain
\begin{equation}
\exp \{ ( 1-\varepsilon ) \delta _{b}( \varphi )\frac{1}{r^{n}}\} 
\leq \frac{\lambda }{r^{2( 2+\varepsilon) }}[ T_{z_0}( \frac{r}{\gamma },f) 
] ^{4}\exp \{ ( 1-\varepsilon ) \delta _{a}( \varphi ) 
\frac{1}{r^{n}}\}. \label{p3}
\end{equation}
From \eqref{p3}, it is easy to obtain that 
$\sigma _2( f,z_0)\geq n$. In the other side, by Lemma \ref{lem8}, 
we can get $\sigma_2( f,z_0) \leq n$. Thus we conclude that 
$\sigma _2(f,z_0) =n$.
\smallskip

\noindent\textbf{Case 2.} 
$a=cb$ $(0<c<1)$: then there exist $(\varphi _1,\varphi _2) \subset [ 0,2\pi ) $ 
such that for $\arg ( z_0-z) =\varphi \in ( \varphi _1,\varphi_2) $ we have 
$\delta _{a}( \varphi ) =c\delta _{b}(\varphi ) >0$. 

Since $\max \{ \sigma ( A,z_0),\sigma ( B,z_0) \} <n$,  by Lemma \ref{lem9}, 
\eqref{l4} and \eqref{p1}, we obtain
\begin{equation}
\exp \{ ( 1-\varepsilon ) \delta _{b}( \varphi )\frac{1}{r^{n}}\} 
\leq \frac{\lambda }{r^{2( 2+\varepsilon) }}
 \big[ T_{z_0}( \frac{r}{\gamma },f) \big] ^{4}\exp
\{ ( 1+\varepsilon ) c\delta _{b}( \varphi )
\frac{1}{r^{n}}\} . \label{p3b}
\end{equation}
From \eqref{p3b} and by taking $0<\varepsilon <\frac{1-c}{1+c}$, 
we obtain that $\sigma _2( f,z_0) \geq n$. 
By Lemma \ref{lem8}, we have $\sigma _2( f,z_0) \leq n$. Thus we conclude
 that $\sigma_2( f,z_0) =n$.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm2}]
Using the change of variable $w=\frac{1}{z_0-z}$ and setting 
$g(w) =f( z) $, we get $f'( z) =\frac{1}{( z_0-z) ^2}g'( w) 
=w^2g'( w) $, 
$f''( z) =w^{4}g''( w) +2w^{3}g'( w) $. Then the
differential equation \eqref{eq2} becomes 
\begin{equation}
g''( w) +e^{-w}g'( w)+cg( w) =0. \label{p4}
\end{equation}
By Theorem \ref{thma}, if \eqref{p4} possesses a solution $g\not\equiv 0$ of
finite order, then $c=-k^2$ where $k$ is a positive integer. Conversely,
for each positive integer $k$, the equation \eqref{eq1} with $c=-k^2$,
possesses a solution $g$ which is a polynomial in $e^{w}$ of degree $k$. 
By Remark \ref{rmk1}, we have $\sigma ( f,z_0) =\sigma (g) $. 
So, if the differential equation \eqref{eq2}\ possesses a solution 
$f( z) \not\equiv 0$ of finite order $\sigma (f,z_0) <\infty $ then 
$c=-k^2$. Conversely, for each positive integer $k$, the equation 
\eqref{eq2} with $c=-k^2$, possesses a solution $f$ which is a polynomial 
in $e^{\frac{1}{( z_0-z) }}$ of degree $k$.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm3}]
From \eqref{eq3}, we can write
\begin{equation}
| A_0( z) | \leq | \frac{f^{(
k) }}{f}| +| A_{k-1}( z) |
| \frac{f^{( k-1) }}{f}| +\dots +|
A_1( z) | | \frac{f'}{f}|
. \label{p5}
\end{equation}
Using \eqref{c1}, \eqref{c2} and \eqref{l4} in \eqref{p5}, for 
$\arg (z_0-z) =\theta \in ( \theta _1,\theta _2) \subset[ 0,2\pi ) $ and 
$| z_0-z| =r$ near enough to $0$, we obtain
\begin{equation}
\exp \{ \frac{\alpha }{r^{\mu }}\} 
\leq \frac{\lambda }{ r^{k( 2+\varepsilon ) }}
\big[ T_{z_0}( \frac{r}{\gamma }
,f) \big] ^{2k}\exp \{ \frac{\beta }{r^{\mu }}\} .
\label{p6}
\end{equation}
From \eqref{p6}, we obtain that $\sigma ( f,z_0) \geq \mu $.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm4}]
From \eqref{eq3}, we can write
\begin{equation}
| A_0( z) | \leq | \frac{f^{(k) }}{f}| 
+| A_{k-1}( z) || \frac{f^{( k-1) }}{f}| +\dots 
+|A_1( z) | | \frac{f'}{f}|. \label{p7}
\end{equation}
Set $\max \{ \sigma ( A_{j},z_0) :j\neq 0\} <\beta<\alpha <\sigma ( A_0,z_0) $. 
For any given $\varepsilon >0$, there exists $r_0>0$ such that for all 
$r$ satisfying $r_0\geq r>0$, we have
\begin{equation}
| A_{j}( z) | \leq \exp \{ \frac{1}{r^{\beta +\varepsilon }}\} ,\quad
 j=1,2,\dots ,k-1. \label{p8}
\end{equation}
By taking $\beta +\varepsilon <\alpha <\sigma ( A_0,z_0) $,
and by Lemma \ref{lem5}, there exists a set $F\subset ( 0,r_0] $
of infinite logarirhmic measure such that for all $r\in F$ and 
$|A_0( z) | =M_{z_0}( r,A_0) $, we have
\begin{equation}
| A_0( z) | >\exp \{ \frac{1}{r^{\alpha}}\} . \label{p9}
\end{equation}
Using \eqref{p8}--\eqref{p9} with \eqref{l3} in \eqref{p7}, we obtain
\begin{equation}
\exp \{ \frac{1}{r^{\alpha }}\} \leq \frac{\lambda }{r^{2k}}
\big[T_{z_0}( \frac{r}{\gamma },f) \big] ^{2k}\exp \{ \frac{1
}{r^{\beta +\varepsilon }}\} . \label{p10}
\end{equation}
From \eqref{p10}, we obtain that $\sigma ( f,z_0) \geq \alpha $.

On the other hand, applying Lemma \ref{lem8} with \eqref{eq3}, we obtain
that $\sigma ( f,z_0) \leq \sigma ( A_0,z_0) $.
Since $\alpha \leq \sigma ( f,z_0) \leq \sigma (A_0,z_0) $ holds for every 
$\alpha <\sigma (A_0,z_0) $, then we conclude that 
$\sigma ( f,z_0)=\sigma ( A_0,z_0) $.
\end{proof}

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\end{document}