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\title [Growth of solutions of linear difference equations of $\varphi $-order]{Growth properties of solutions of linear difference equations with coefficients having $\varphi $-order}
\author{Nityagopal Biswas}
\address{Department of Mathematics, Chakdaha College, \\ Chakdaha, Nadia,
Pin: 741222, West Bengal, \\
India.}
\email{nityamaths@gmail.com}
%
\author{Pulak Sahoo}
\address{Department of Mathematics, Midnapore College (Autonomous), \\ Midnapore, Paschim Medinipur,
 Pin: 721101, West Bengal,\\
 India.}
\email{pulak.pmath19@gmail.com}
%
\subjclass{30D35, 39A10, 39A12.}
\keywords{Nevanlinna's Theory; Linear difference equation; Meromorphic solution; $\varphi $-order.}
\begin{abstract}
In this paper, we investigate the relations between the growth of entire
coefficients and that of solutions of complex homogeneous and
non-homogeneous linear difference equations with entire coefficients of $%
\varphi $-order by using a slow growth scale, the $\varphi $-order, where $%
\varphi $ is a non-decreasing unbounded function. We extend some precedent
results due to Zheng and Tu (2011) \cite{ZhengTu} and others.
\end{abstract}
\maketitle

\section{Introduction and Preliminaries}

We assume that the readers are familiar with the fundamental results and
standard notations of the Nevanlinna's value distribution theory of entire
and meromorphic functions. In addition, let us recall some notations such as $%
m\left( r,f\right) $ and $N\left( r,f\right) $ (see \cite{Hayman1964, Laine}%
). Let $n\left( r,f\right) $ be the number of poles of a function $f$
(counting multiplicities) in $\left\vert z\right\vert \leq r$. Then we
define the integrated counting function $N\left( r,f\right) $ by
\begin{equation*}
N\left( r,f\right) =\int_{0}^{r}\frac{n\left( t,f\right) -n\left( 0,f\right)
}{t}dt+n\left( 0,f\right) \log r,
\end{equation*}%
and we define the proximity function $m\left( r,f\right) $ by
\begin{equation*}
m\left( r,f\right) =\frac{1}{2\pi }\int_{0}^{2\pi }\log ^{+}\left\vert
f\left( re^{i\phi }\right) \right\vert d\phi ,\text{\ }
\end{equation*}%
where $\log ^{+}x=\max \left\{ 0,\log x\right\} .$ We should think of $%
m\left( r,f\right) $ as a measure of how close $f$ is to infinity on $%
\left\vert z\right\vert =r.$ Nevertheless, within that context, we recall
that $T\left( r,f\right) $ stands for the Nevanlinna characteristic function
of the meromorphic function $f$ defined on each positive real value $r$ by
\begin{equation*}
T\left( r,f\right) =m\left( r,f\right) +N\left( r,f\right) .
\end{equation*}%
And $M\left( r,f\right) $ stands for the so called maximum modulus function
defined for each non-negative real value $r$ by
\begin{equation*}
M\left( r,f\right) =\max_{\left\vert z\right\vert =r}\left\vert f\left(
z\right) \right\vert .
\end{equation*}%
The applications of Nevanlinna's value distribution theory has been
developed since 1960's. Recently, the properties of meromorphic solutions of
complex linear difference equations have become a subject of great interest
from the viewpoint of Nevanlinna's theory and its difference analogues.
Since then, many authors investigated the linear difference equations for
example, \cite{ChaingFeng, LaineYang, LiuMao}. Moreover, we use notations $%
\sigma \left( f\right) $ for the order of a meromorphic function $f\left(
z\right) $ and defined as%
\begin{equation*}
\sigma \left( f\right) =\underset{r\rightarrow \infty }{\limsup }\frac{\log
T\left( r,f\right) }{\log r}.
\end{equation*}%
We denote the linear measure for a set $E\subset \lbrack 0,\infty ),$ by $%
m\left( E\right) =\int_{E}dt$ and logarithmic measure for a set $E\subset
\left( 1,\infty \right) ,$ by $m_{l}\left( E\right) =\int_{E}\frac{dt}{t}$.
The upper density of a set $E\subset \lbrack 0,\infty )$ is defined as
\begin{equation*}
\overline{dens}\text{ }E=\underset{r\rightarrow \infty }{\limsup }\frac{%
m\left( E\cap \left[ 0,r\right] \right) }{r},
\end{equation*}%
and the upper logarithmic density of a set $E\subset \left( 1,\infty \right)
$ is defined as%
\begin{equation*}
\overline{\log dens}\text{ }\left( E\right) =\underset{r\rightarrow \infty }{%
\limsup }\frac{m_{l}\left( E\cap \left[ 1,r\right] \right) }{\log r}.
\end{equation*}

\begin{proposition}
\label{Proposition1.1}\cite{Belaidi20127(83)}For all $H\subset \left[
1,\infty \right) $ the following statements hold:

(i) If $m_{l}\left( H\right) =\infty ,$ then $m\left( H\right) =\infty ;$

(ii) If $\overline{dens}H>0,$ then $m\left( H\right) =\infty ;$

(iii) If $\overline{\log dens}H>0,$ then $m_{l}\left( H\right) =\infty .$
\end{proposition}

In 2008, Chiang and Feng \cite{ChaingFeng} investigated the proximity
function and point wise estimates of $\frac{f\left( z+\eta \right) }{f\left(
z\right) }$, which are discrete versions of the classical logarithmic
derivative estimates of $f\left( z\right) $. They also applied their results
to obtain growth estimates of meromorphic solutions to higher order
homogeneous and non-homogeneous linear difference equations
\begin{equation}
A_{n}(z)f(z+n)+\dots +A_{1}(z)f(z+1)+A_{0}(z)f(z)=0  \tag{1.1}  \label{Equa1}
\end{equation}%
and%
\begin{equation}
A_{n}(z)f(z+n)+\dots +A_{1}(z)f(z+1)+A_{0}(z)f(z)=F\left( z\right) ,
\tag{1.2}  \label{Equa1.1}
\end{equation}%
where the coefficients $A_{0}\left( z\right) ,...,A_{n}\left( z\right) $ and
$F\left( z\right) \left( \not\equiv 0\right) $ are entire functions and they
obtained the following result.

\begin{theorem}
\label{Theorem1.1}\cite{ChaingFeng} Let $A_{0}\left( z\right)
,...,A_{n}\left( z\right) $ be entire functions such that there exists an
integer $l$ $\left( 0\leq l\leq n\right) $ such that%
\begin{equation*}
\underset{0\leq j\leq n}{\max }\left\{ \sigma \left( A_{j}\right)
;j\not=l\right\} <\sigma \left( A_{l}\right) ,
\end{equation*}%
then every meromorphic solution of Equation $\left( \ref{Equa1}\right) $
satisfies $\sigma \left( f\right) \geq \sigma \left( A_{l}\right) +1.$
\end{theorem}

Above results occur when there exists only one dominant coefficient. In the
case that there are more than one dominant coefficients, Laine and Yang \cite%
{LaineYang} obtained the following result.

\begin{theorem}
\label{Theorem1.2}\cite{LaineYang} Let $A_{0}\left( z\right)
,...,A_{n}\left( z\right) $ be entire functions of finite order such that
among those having the maximal order $\sigma =\underset{0\leq j\leq n}{\max }%
\sigma \left( A_{j}\right) $, exactly one has its type strictly greater than
the others. Then for any meromorphic solution $f\left( \not\equiv 0\right) $
of Equation $\left( \ref{Equa1}\right) $, we have $\sigma \left( f\right)
\geq \sigma +1.$
\end{theorem}

Recently, In 2011, Zheng and Tu \cite{ZhengTu}, studied the growth of
meromorphic solutions of homogeneous or non-homogeneous linear difference
equations and improved the previous results due to Chiang and Feng \cite%
{ChaingFeng} and Laine and Yang \cite{LaineYang}. In the case there are more
than one coefficients of Equation $\left( \ref{Equa1}\right) $ which have
the maximal orders Zheng and Tu \cite{ZhengTu} obtained the following
results.

\begin{theorem}
\label{Theorem1.3}\cite{ZhengTu} Let $H$ be a set of complex numbers
satisfying $\overline{\log dens}\left\{ \left\vert z\right\vert :z\in
H\right\} >0$ and let $A_{j}\left( z\right) $ $\left( j=0,1,...,n\right) $
be entire functions satisfying $\max \left\{ \sigma \left( A_{j}\right)
,j=0,1,...,n\right\} \leq \alpha _{1}.$ If there exists an integer $l$ $%
\left( 0\leq l\leq n\right) $ and a positive constant $\alpha _{2}~(\alpha
_{2}<\alpha _{1})~$\ such that for any given $\varepsilon \left(
0<\varepsilon <\alpha _{2}-\alpha _{1}\right) $, we have%
\begin{equation*}
\left\vert A_{l}\left( z\right) \right\vert \geq \exp \left\{ r^{\alpha
_{1}-\varepsilon }\right\}
\end{equation*}%
and%
\begin{equation*}
\left\vert A_{j}\left( z\right) \right\vert \leq \exp \left\{ r^{\alpha
_{2}}\right\} ,\text{ \ \ }\left( j\not=l\right) ,
\end{equation*}%
as $\left\vert z\right\vert =r\rightarrow +\infty $ for $z\in H,$ then every
meromorphic solution $f\left( \not\equiv 0\right) $ of Equation $\left( \ref%
{Equa1}\right) $ satisfies $\sigma \left( f\right) \geq \sigma \left(
A_{l}\right) +1.$\qquad
\end{theorem}

Recently, Chyzhykov et al. \cite{ChyzhykovHeittokangasRattya} introduced the
definition of $\varphi $-order of $f(z)$ in a unit disc, where $\varphi :%
\left[ 0,1\right) \rightarrow \left( 0,\infty \right) $ is a non-decreasing
unbounded function and $f(z)$ is a meromorphic function in the unit disc and
Shen et al. \cite{ShenTuXu}, introduced $\left[ p,q\right] -\varphi $ order
of entire and meromorphic functions in the complex plane $%
%TCIMACRO{\U{2102} }%
%BeginExpansion
\mathbb{C}
%EndExpansion
$ where $\varphi :\left[ 0,\infty \right) \rightarrow \left( 0,\infty
\right) $ is a non-decreasing unbounded function. Since then many
researchers investigated the growth oscillation of solutions of linear
differential equations and linear difference equations \{cf. \cite%
{BiswasDattaTamang, DattaBiswas, DattaBiswas2020, LiuTuZhang}\}. Revisiting
their ideas of $\varphi $-order we would like to prove some results using
the concepts of slow growth scale, the $\varphi $-order in the complex
plane. To investigate the growth of meromorphic solutions of Equations $%
\left( \ref{Equa1}\right) $ and $\left( \text{\ref{Equa1.1}}\right) $\ more
precisely, we recall the following definitions.

\begin{definition}
(\cite{ShenTuXu, ChyzhykovHeittokangasRattya}) \label{Defi1.1} Let $\varphi :%
\left[ 0,+\infty \right) \rightarrow \left( 0,+\infty \right) $ be a
non-decreasing unbounded function, the $\varphi $-order of a meromorphic
function $f$ is defined as
\begin{equation*}
\sigma \left( f,\varphi \right) =\underset{r\rightarrow \infty }{\lim \sup }%
\frac{\log T\left( r,f\right) }{\log \varphi \left( r\right) }.
\end{equation*}%
If $f$ is an entire function, then%
\begin{equation*}
\sigma \left( f,\varphi \right) =\underset{r\rightarrow \infty }{\lim \sup }%
\frac{\log \log M\left( r,f\right) }{\log \varphi \left( r\right) }.\text{\ }
\end{equation*}
\end{definition}

\begin{definition}
\label{Defi1.2}(\cite{ChyzhykovHeittokangasRattya}) If $f$ be a meromorphic
function satisfying $0<\sigma \left( f,\varphi \right) =\sigma <\infty .$
Then $\varphi $-type of $f$ is defined as%
\begin{equation*}
\tau \left( f,\varphi \right) =\underset{r\rightarrow \infty }{\lim \sup }%
\frac{T\left( r,f\right) }{\varphi \left( r\right) ^{\sigma }}.
\end{equation*}%
If $f$ is an entire function, then%
\begin{equation*}
\tau \left( f,\varphi \right) =\underset{r\rightarrow \infty }{\lim \sup }%
\frac{\log M\left( r,f\right) }{\varphi \left( r\right) ^{\sigma }}.
\end{equation*}
\end{definition}

\begin{remark}
\label{Remark1.2}\bigskip If $\varphi \left( r\right) =r$ in the Definitions %
\ref{Defi1.1} and \ref{Defi1.2}, then we obtain the standard definition of
the order and type of a function $f$ respectively.
\end{remark}

\begin{remark}
\label{Remark 1.2} Throughout this paper, we assume that $\varphi :\left[
0,\infty \right) \rightarrow \left( 0,\infty \right) $ is a non-decreasing
unbounded function and always satisfies the following two conditions without
special instruction:

$\left( i\right) $ $\underset{r\rightarrow +\infty }{\lim }\frac{\log \log r%
}{\log \varphi \left( r\right) }=0$.

$\left( ii\right) \underset{r\rightarrow +\infty }{\lim }\frac{\log \varphi
\left( \alpha r\right) }{\log \varphi \left( r\right) }=1$ for some $\alpha
>1.$
\end{remark}

Thus, a natural problem arises that: how to express the growth of solutions
of homogeneous and non-homogeneous linear difference Equations (\ref{Equa1})
and $\left( \ref{Equa1.1}\right) $ when the coefficients $A_{j}\left(
z\right) $ $(j=0,1,...,n)$ and $F\left( z\right) \left( \not\equiv 0\right) $
be entire functions of $\varphi $-order in a slow growth scale $\varphi $%
-order. The main purpose of this paper is to make use of the concept of $%
\varphi $-order due to Chyzhykov et al. \cite{ChyzhykovHeittokangasRattya}
to extend previous results for solutions to Equations (\ref{Equa1}) and $%
\left( \ref{Equa1.1}\right) $ in the complex plane $%
%TCIMACRO{\U{2102} }%
%BeginExpansion
\mathbb{C}
%EndExpansion
$.

\section{Main Results}

The main purpose of this paper is to used the concept of $\varphi $-order in
the complex plane $%
%TCIMACRO{\U{2102} }%
%BeginExpansion
\mathbb{C}
%EndExpansion
$ to investigate the growth of solutions of homogeneous and non-homogeneous
linear difference Equations (\ref{Equa1}) and $\left( \ref{Equa1.1}\right) $%
. In this direction we obtain the following results.

The Theorem \ref{Theorem2.1} investigate the order of meromorphic solutions
of homogeneous linear difference Equation ($\ref{Equa1}$) in the case when
there are more than one coefficients which have the maximal orders.

\begin{theorem}
\label{Theorem2.1}Let $H$ be a set of complex numbers satisfying $\overline{%
\log dens}\left\{ \left\vert z\right\vert :z\in H\right\} >0$ and let $%
A_{j}\left( z\right) $ $\left( j=0,1,...,n\right) $ be entire functions
satisfying
\begin{equation*}
\max \left\{ \sigma \left( A_{j},\varphi \right) ,j=0,1,...,n\right\} \leq
\sigma .
\end{equation*}%
If there exists an integer $l$ $\left( 0\leq l\leq n\right) $ such that for
some constants $\alpha $ and $\beta $ with $0\leq \beta <\alpha $ and $%
\varepsilon $ $\left( 0<\varepsilon <\sigma \right) $ sufficiently small, we
have%
\begin{equation}
T\left( r,A_{l}\right) \geq \exp \left\{ \alpha \left( \varphi \left(
r\right) \right) ^{\sigma -\varepsilon }\right\}  \tag{2.1}  \label{Equa 2.1}
\end{equation}%
and%
\begin{equation}
T\left( r,A_{j}\right) \leq \exp \left\{ \beta \left( \varphi \left(
r\right) \right) ^{\sigma -\varepsilon }\right\} ,\text{ }\left(
j\not=l\right) ,  \tag{2.2}  \label{Equa2.2}
\end{equation}%
as $z\rightarrow \infty $ for $z\in H,$ then every meromorphic solution $f$ $%
\left( \not\equiv 0\right) ~$of Equation $\left( \ref{Equa1}\right) $
satisfies $\sigma \left( f,\varphi \right) \geq \sigma \left( A_{l},\varphi
\right) +1$.

\begin{remark}
\label{Remark2.1}By the assumptions of Theorem \ref{Theorem2.1}, we obtain
that $\sigma \left( A_{l},\varphi \right) =\sigma .$ Indeed, we have $\sigma
\left( A_{l},\varphi \right) \leq \sigma ,$ suppose that $\sigma \left(
A_{l},\varphi \right) =\eta <\sigma .$ Then by Definition \ref{Defi1.1} of $%
\varphi $-order and $\left( \ref{Equa 2.1}\right) ,$ we have for any given $%
\varepsilon \left( 0<\varepsilon <\frac{\sigma -\eta }{2}\right) $%
\begin{equation}
\exp \left\{ \alpha \left( \varphi \left( r\right) \right) ^{\sigma
-\varepsilon }\right\} \leq T\left( r,A_{l}\right) \leq \exp \left\{ \left(
\varphi \left( r\right) \right) ^{\eta +\varepsilon }\right\} ,  \tag{2.3}
\label{Equa2.3}
\end{equation}%
as $\left\vert z\right\vert =r\rightarrow \infty $ for $z\in H.$ So by $%
\varepsilon \left( 0<\varepsilon <\frac{\sigma -\eta }{2}\right) $ we get a
contradiction from $\left( \ref{Equa2.3}\right) $ as $r\rightarrow \infty .$
Hence $\sigma \left( A_{l},\varphi \right) =\sigma .$
\end{remark}
\end{theorem}

The following example illustrate the sharpness of Theorem \ref{Theorem2.1}.

\begin{example}
\label{Example2.1}The function $f\left( z\right) =e^{z^{2}-3z}$ satisfies
the equation%
\begin{equation*}
e^{-z}f\left( z+2\right) +e^{z}f\left( z+1\right) -2e^{3z-2}f\left( z\right)
=0.
\end{equation*}%
Here $A_{2}\left( z\right) =e^{-z},~A_{1}\left( z\right) =e^{z},~A_{0}\left(
z\right) =-2e^{3z-2},$ we take $\varphi \left( z\right) =z,$ then we obtain
that $\sigma \left( A_{2},\varphi \right) =\sigma \left( A_{1},\varphi
\right) =\sigma \left( A_{0},\varphi \right) =1.$ Now set $H=\left\{ z:\arg
z=\pi \right\} $ and $l=2,$ then it is clear that $\overline{dens}\left\{
\left\vert z\right\vert =r:z\in H\right\} =1>0.$ Moreover, $A_{2}\left(
z\right) ,~A_{1}\left( z\right) $ and$~A_{0}\left( z\right) $ satisfy the
assumptions $\left( \ref{Equa 2.1}\right) $ and $\left( \ref{Equa2.2}\right)
$ of Theorem \ref{Theorem2.1}. Therefore, we get $\sigma \left( f,\varphi
\right) =2=\sigma \left( A_{2},\varphi \right) +1.$
\end{example}

Secondly, we consider the growth of entire solutions of non-homogeneous
linear difference Equation $\left( \ref{Equa1.1}\right) $. Note that the
above result may not be applicable to the Equation $\left( \ref{Equa1.1}%
\right) $ to which Equation ($\ref{Equa1}$) is the corresponding homogeneous
equation (see the following Example \ref{Example2.2}). But we can obtain
similar results with some additional conditions.

\begin{theorem}
\label{Theorem2.2} Let $A_{j}\left( z\right) $ $\left( j=0,1,...,n\right) $
and $F\left( z\right) \left( \not\equiv 0\right) $ be entire functions such
that there exists an integer $l$ $\left( 0\leq l\leq n\right) $ satisfying%
\begin{equation}
b=\max \left\{ \sigma \left( A_{j},\varphi \right) ,\sigma \left( F,\varphi
\right) ,j\not=l,\right\} <\sigma \left( A_{l},\varphi \right) <\frac{1}{2},
\tag{2.4}  \label{Equa2.4}
\end{equation}%
then every nontrivial entire solution $f$ $\left( \not\equiv 0\right) ~$of
Equation $\left( \ref{Equa1.1}\right) $ satisfies $\sigma \left( f,\varphi
\right) \geq \sigma \left( A_{l},\varphi \right) +1$.
\end{theorem}

\begin{example}
\label{Example2.2}Take $\varphi \left( z\right) =z$ and the function $%
f\left( z\right) =e^{z}$ satisfies the equation%
\begin{equation*}
f\left( z+2\right) -ef\left( z+1\right) +f\left( z\right) =e^{z}
\end{equation*}%
and%
\begin{equation*}
f\left( z+2\right) -ef\left( z+1\right) +e^{-z}f\left( z\right) =1.
\end{equation*}%
Though there is only one dominant coefficient such that the assumptions in
Theorems \ref{Theorem2.1} hold, we cannot get similar results in the
non-homogeneous equation case.
\end{example}

\begin{theorem}
\label{Theorem2.3} Let $A_{j}\left( z\right) $ $\left( i=0,1,...,n\right) $
and $F\left( z\right) \left( \not\equiv 0\right) $ entire functions such
that there exists an integer $\left( 0\leq l\leq n\right) $ satisfying%
\begin{equation*}
b=\max \left\{ \sigma \left( A_{j},\varphi \right) ,\sigma \left( F,\varphi
\right) ,j\not=l,\right\} <\sigma \left( A_{l},\varphi \right) <\infty .
\end{equation*}%
Also suppose that $A_{l}\left( z\right) =\sum_{n=1}^{\infty }C_{\lambda
_{n}}z^{\lambda _{n}}$ satisfies that the sequence of exponents $\left\{
\lambda _{n}\right\} $ satisfies the Fabry gap condition $\frac{\lambda _{n}%
}{n}\rightarrow \infty $ as $n\rightarrow \infty ,$ then every nontrivial
entire solution $f$ $\left( \not\equiv 0\right) ~$of Equation $\left( \ref%
{Equa1.1}\right) $ satisfies $\sigma \left( f,\varphi \right) \geq \sigma
\left( A_{l},\varphi \right) +1$.
\end{theorem}

\section{Preliminary Lemmas}

To prove the above theorems, we need some lemmas as follows.

\begin{lemma}
\label{Lemma3.1}\cite{ChaingFeng} Let $f$ be a meromorphic function, $\eta $
be a non-zero complex number and let $\gamma >1$ and $\varepsilon >0$ be
given real constants. Then there exist a subset $E_{1}\subset \left(
1,+\infty \right) $ of finite logarithmic measure and a constant $A$
depending only on $\gamma $ and $\eta ,$ such that for all $\left\vert
z\right\vert =r\notin E_{1}\cup \left[ 0,1\right] ,$ we have%
\begin{equation*}
\left\vert \log \left\vert \frac{f\left( z+\eta \right) }{f\left( z\right) }%
\right\vert \right\vert \leq A\left( \frac{T\left( \gamma r,f\right) }{r}+%
\frac{n\left( \gamma r\right) }{r}\log ^{\gamma }r\log ^{+}n\left( \gamma
r\right) \right) ,
\end{equation*}%
where $n\left( t\right) =n\left( t,\infty ,f\right) +n\left( t,\infty ,\frac{%
1}{f}\right) .$
\end{lemma}

\begin{lemma}
\label{Lemma3.2}\cite{Gundersen} Let $f$ be a transcendental meromorphic
function and let $j$ be a non-negative integer, let $a$ be a value in the
extended complex plane and let $\alpha >1$ be a real constant. Then there
exists a constant $R>0$ such that for all $r>R,$ we have%
\begin{equation*}
n\left( r,a,f^{\left( j\right) }\right) \leq \frac{2j+6}{\log \alpha }%
T\left( \alpha r,f\right) .
\end{equation*}
\end{lemma}

\begin{lemma}
\label{Lemma3.6}Let $f$ be a meromorphic function and $\eta $ be a non-zero
complex number and let $\varepsilon >0$ be given real constants. Then there
exists a subset $E_{2}\subset \left( 1,+\infty \right) $ of finite
logarithmic measure, such that if $f$ has finite $\varphi $-order $\sigma $,
then for all $\left\vert z\right\vert =r\notin E_{2}\cup \left[ 0,1\right] ,$
we have%
\begin{equation*}
\exp \left\{ -\frac{\left( \varphi \left( r\right) \right) ^{\sigma
+\varepsilon }}{r}\right\} \leq \left\vert \frac{f\left( z+\eta \right) }{%
f\left( z\right) }\right\vert \leq \exp \left\{ \frac{\left( \varphi \left(
r\right) \right) ^{\sigma +\varepsilon }}{r}\right\} .
\end{equation*}

\begin{proof}
By Lemma \ref{Lemma3.1}, there exist a subset there exist a subset $%
E_{2}\subset \left( 1,+\infty \right) $ of finite logarithmic measure and a
constant $A$ depending only on $\gamma $ and $\eta ,$ such that for all $%
\left\vert z\right\vert =r\notin E_{2}\cup \left[ 0,1\right] ,$ we have%
\begin{equation}
\left\vert \log \left\vert \frac{f\left( z+\eta \right) }{f\left( z\right) }%
\right\vert \right\vert \leq A\left( \frac{T\left( \gamma r,f\right) }{r}+%
\frac{n\left( \gamma r\right) }{r}\log ^{\gamma }r\log ^{+}n\left( \gamma
r\right) \right) ,  \tag{3.1}  \label{Equa3.4}
\end{equation}%
where $n\left( t\right) =n\left( t,\infty ,f\right) +n\left( t,\infty ,\frac{%
1}{f}\right) .$

Using $\left( \ref{Equa3.4}\right) $ and Lemma \ref{Lemma3.2}, we obtain
that
\begin{equation*}
\left\vert \log \left\vert \frac{f\left( z+\eta \right) }{f\left( z\right) }%
\right\vert \right\vert \leq A\left( \frac{T\left( \gamma r,f\right) }{r}+%
\frac{12}{\log \alpha }\frac{T\left( \alpha \gamma r,f\right) }{r}\log
^{\gamma }r\log ^{+}\left( \frac{12}{\log \alpha }T\left( \alpha \gamma
r,f\right) \right) \right)
\end{equation*}

\begin{equation}
\leq B\left( \frac{T\left( \beta r,f\right) }{r}+\frac{\log ^{\beta }r}{r}%
T\left( \beta r,f\right) \log T\left( \beta r,f\right) \right) ,  \tag{3.2}
\label{Equa3.5}
\end{equation}%
for all $\left\vert z\right\vert =r\not\in \left[ 0,1\right] \cup E_{2}$
with $m_{l}\left( E_{2}\right) <+\infty ,$ where $B>0$ is some constant and $%
\beta =\alpha \gamma >1.$

Again, since $f$ has finite $\varphi $-order $\sigma \left( f,\varphi
\right) =\sigma <+\infty ,$ so given $\varepsilon \left( 0<\varepsilon
<2\right) ,$ for sufficiently large $r,$ we have%
\begin{equation}
T\left( r,f\right) <\left( \varphi \left( r\right) \right) ^{\sigma +\frac{%
\varepsilon }{2}}.  \tag{3.3}  \label{Equa3.6}
\end{equation}%
Then by substituting $\left( \ref{Equa3.6}\right) $ into $\left( \ref{Equa3.5}%
\right) ,\,\ $we get that%
\begin{equation*}
\left\vert \log \left\vert \frac{f\left( z+\eta \right) }{f\left( z\right) }%
\right\vert \right\vert \leq B\left( \frac{\left( \varphi \left( \beta
r\right) \right) ^{\sigma +\frac{\varepsilon }{2}}}{r}+\frac{\log ^{\beta }r%
}{r}\left( \varphi \left( \beta r\right) \right) ^{\sigma +\frac{\varepsilon
}{2}}\log \left( \varphi \left( \beta r\right) \right) ^{\sigma +\frac{%
\varepsilon }{2}}\right)
\end{equation*}
\begin{equation}
\leq \frac{\left( \varphi \left( r\right) \right)
^{\sigma +\varepsilon }}{r}.  \tag{3.4}  \label{Equa3.7}
\end{equation}%
From $\left( \ref{Equa3.7}\right) $, we obtain that
\begin{equation*}
\exp \left\{ -\frac{\left( \varphi \left( r\right) \right) ^{\sigma
+\varepsilon }}{r}\right\} \leq \left\vert \frac{f\left( z+\eta \right) }{%
f\left( z\right) }\right\vert \leq \exp \left\{ \frac{\left( \varphi \left(
r\right) \right) ^{\sigma +\varepsilon }}{r}\right\} .
\end{equation*}%
This proves the lemma.
\end{proof}
\end{lemma}

\begin{lemma}
\label{Lemma 3.1}Let $\eta _{1},\eta _{2}$ be two arbitrary complex numbers
such that $\eta _{1}\neq \eta _{2}$ and let $f$ be a meromorphic function of
finite $\varphi $-order $\sigma $ and let $\varepsilon >0$ be given. Then
there exists a subset $E_{3}\subset \left( 1,+\infty \right) $ of finite
logarithmic measure such that for all $\left\vert z\right\vert =r\notin %
\left[ 0,1\right] \cup E_{3},$ we have%
\begin{equation*}
\exp \left\{ -\frac{\left( \varphi \left( r\right) \right) ^{\sigma
+\varepsilon }}{r}\right\} \leq \left\vert \frac{f\left( z+\eta _{1}\right)
}{f\left( z+\eta _{2}\right) }\right\vert \leq \exp \left\{ \frac{\left(
\varphi \left( r\right) \right) ^{\sigma +\varepsilon }}{r}\right\} .
\end{equation*}

\begin{proof}
We can write
\begin{equation*}
\left\vert \frac{f\left( z+\eta _{1}\right) }{f\left( z+\eta _{2}\right) }%
\right\vert =\left\vert \frac{f\left( z+\eta _{2}+\eta _{1}-\eta _{2}\right)
}{f\left( z+\eta _{2}\right) }\right\vert ,\text{\ }\left( \eta _{1}\neq
\eta _{2}\right) .
\end{equation*}%
Then by using Lemma \ref{Lemma3.6}, there exists a subset $E_{3}\subset
\left( 1,+\infty \right) $ such that for any $\varepsilon >0$ and all $%
\left\vert z+\eta _{2}\right\vert =R\notin E_{3}\cup \left[ 0,1\right] ,$
with $m_{l}\left( E_{3}\right) <\infty ,$ we get%
\begin{eqnarray*}
\exp \left\{ -\frac{\left( \varphi \left( r\right) \right) ^{\sigma
+\varepsilon }}{r}\right\} &\leq &\exp \left\{ -\frac{\left( \varphi \left(
\left\vert z\right\vert +\left\vert \eta _{2}\right\vert \right) \right)
^{\sigma +\frac{\varepsilon }{2}}}{\left\vert z+\eta _{2}\right\vert }%
\right\} \\
&=&\exp \left\{ -\frac{\left( \varphi \left( R\right) \right) ^{\sigma +%
\frac{\varepsilon }{2}}}{R}\right\} \leq \left\vert \frac{f\left( z+\eta
_{1}\right) }{f\left( z+\eta _{2}\right) }\right\vert \\
&=&\left\vert \frac{f\left( z+\eta _{2}+\eta _{1}-\eta _{2}\right) }{f\left(
z+\eta _{2}\right) }\right\vert \leq \exp \left\{ \frac{\left( \varphi
\left( R\right) \right) ^{\sigma +\frac{\varepsilon }{2}}}{R}\right\} \\
&\leq &\exp \left\{ \frac{\left( \varphi \left( \left\vert z\right\vert
+\left\vert \eta _{2}\right\vert \right) \right) ^{\sigma +\varepsilon }}{%
\left\vert z+\eta _{2}\right\vert }\right\} \leq \exp \left\{ \frac{\left(
\varphi \left( r\right) \right) ^{\sigma +\varepsilon }}{r}\right\} ,
\end{eqnarray*}%
where $\left\vert z\right\vert =r\notin \left[ 0,1\right] \cup E_{3}$.

This proves the lemma.
\end{proof}
\end{lemma}

\begin{lemma}
\label{Lemma3.2/}\cite{DattaBiswas}Let $\eta _{1},\eta _{2}$ be two
arbitrary complex numbers such that $\eta _{1}\neq \eta _{2},$ and let $f$
be a meromorphic function of finite $\varphi $-order. Let $\sigma $ be the $%
\varphi $-order of $f(z)$. Then for each $\varepsilon >0,$ we have
\begin{equation*}
m\left( r,\frac{f\left( z+\eta _{1}\right) }{f\left( z+\eta _{2}\right) }%
\right) =O\left( \left( \varphi \left( r\right) \right) ^{\sigma
-1+\varepsilon }\right) .
\end{equation*}
\end{lemma}

\begin{lemma}
\label{Lemma 3.2}\cite{Kovaria} Let $f\left( z\right) =\sum_{n=1}^{\infty
}C_{\lambda _{n}}z^{\lambda _{n}}$be an entire function and the sequence of
exponents $\left\{ \lambda _{n}\right\} $ satisfies the Fabry gap condition $%
\frac{\lambda _{n}}{n}\rightarrow \infty $ as $n\rightarrow \infty $. Then
for any given $\varepsilon >0$%
\begin{equation*}
\log L\left( r,f\right) >\left( 1-\varepsilon \right) \log M\left(
r,f\right) ,
\end{equation*}%
holds outside a set $E_{4}$ of finite logarithmic measure, where $M\left(
r,f\right) =\underset{\left\vert z\right\vert =r}{\sup }\left\vert f\left(
z\right) \right\vert $ and $L\left( r,f\right) =\underset{\left\vert
z\right\vert =r}{\inf }\left\vert f\left( z\right) \right\vert .$
\end{lemma}

\begin{lemma}
\label{Lemma 3.3} Let $f\left( z\right) $ be an entire function of finite $%
\varphi $-order satisfying $0<\sigma \left( f,\varphi \right) <\infty ,$
where $\varphi \left( r\right) $ only satisfies $\underset{r\rightarrow
+\infty }{\lim }\frac{\log \varphi \left( \alpha r\right) }{\log \varphi
\left( r\right) }=1$ for some $\alpha >1.$\ Then for any given $\beta
<\sigma \left( f,\varphi \right) ,$ there exists a set $E_{5}\subset \left(
1,\infty \right) $ having infinite logarithmic measure such that for all $%
\left\vert z\right\vert =r\in E_{5}$ we have%
\begin{equation*}
M\left( r,f\right) >\exp \left\{ \left( \varphi \left( r\right) \right)
^{\beta }\right\} .
\end{equation*}

\begin{proof}
By the Definition \ref{Defi1.1}\ of the $\varphi $-order, there exists an
increasing sequence $\left\{ r_{n}\right\} $ $\left( r_{n}\rightarrow \infty
\right) $ satisfying $\left( 1+\frac{1}{n}\right) r_{n}<r_{n+1}$ and
\begin{equation*}
\sigma \left( f,\varphi \right) =\underset{r_{n}\rightarrow \infty }{\lim }%
\frac{\log \log M\left( r_{n},f\right) }{\log \varphi \left( r_{n}\right) }.
\end{equation*}%
Then, there exists a positive integer $n_{0}$ such that for all $n\geq n_{0}$
and for any given $\varepsilon >0$, we have
\begin{equation}
M\left( r_{n},f\right) >\exp \left\{ \left( \varphi \left( r_{n}\right)
\right) ^{\sigma \left( f,\varphi \right) -\varepsilon }\right\} .  \tag{3.5}
\label{Equa3.1}
\end{equation}%
Now we have \qquad
\begin{equation*}
\underset{n\rightarrow \infty }{\lim }\frac{\log \varphi \left( \left( 1+%
\frac{1}{n}\right) r\right) }{\log \varphi \left( r\right) }=1.
\end{equation*}%
Since $\beta <\sigma \left( f,\varphi \right) ,$ then we can choose
sufficiently small $\varepsilon >0$ to satisfy $0<\varepsilon <\sigma \left(
f,\varphi \right) -\beta ,$\ so there exists a positive integer $n_{1}$ such
that for all $n>n_{1}$, we have%
\begin{equation*}
\frac{\log \varphi \left( \left( 1+\frac{1}{n}\right) r\right) }{\log
\varphi \left( r\right) }>\frac{\beta }{\sigma \left( f,\varphi \right)
-\varepsilon },
\end{equation*}%
which implies that%
\begin{equation*}
\left( \sigma \left( f,\varphi \right) -\varepsilon \right) \log \varphi
\left( \left( 1+\frac{1}{n}\right) r\right) >\beta \log \varphi \left(
r\right)
\end{equation*}%
\begin{equation}
\Rightarrow \left( \varphi \left( \left( 1+\frac{1}{n}\right) r\right)
\right) ^{\left( \sigma \left( f,\varphi \right) -\varepsilon \right)
}>\varphi \left( r\right) ^{\beta }.  \tag{3.6}  \label{Equa3.2}
\end{equation}%
Taking $n\geq n_{2}=\max \left\{ n_{0},n_{1}\right\} $ and $E_{5}=\overset{%
\infty }{\underset{n=n_{2}}{\bigcup }}I_{n}$, where $I_{n}=\left[
r_{n},\left( 1+\frac{1}{n}\right) r_{n}\right] .$ Then by $\left( \ref%
{Equa3.1}\right) $ and $\left( \ref{Equa3.2}\right) ,$ we get for $r\in %
\left[ r_{n},\left( 1+\frac{1}{n}\right) r_{n}\right] $ that%
\begin{equation*}
M\left( r,f\right) \geq M\left( r_{n},f\right) >\exp \left\{ \left( \varphi
\left( r_{n}\right) \right) ^{\sigma \left( f,\varphi \right) -\varepsilon
}\right\}
\end{equation*}
\begin{equation*}
\geq \exp \left\{ \left( \varphi \left( \left( 1+\frac{1}{n}%
\right) r\right) \right) ^{\sigma \left( f,\varphi \right) -\varepsilon
}\right\} >\exp \left\{ \varphi \left( r\right) ^{\beta }\right\} .
\end{equation*}%
Now we obtain that

$m_{l}\left( E_{5}\right) =\overset{\infty }{\underset{n=n_{2}}{\sum }}%
\int_{I_{n}}\frac{dr}{r}=\overset{\infty }{\underset{n=n_{2}}{\sum }}\left(
\log \frac{1}{1-r}\right) _{r_{n}}^{\left( 1+\frac{1}{n}\right) r_{n}}=%
\overset{\infty }{\underset{n=n_{2}}{\sum }}\log \frac{n+1}{n}=\infty .$

This proves the lemma.
\end{proof}
\end{lemma}

\begin{lemma}
\label{Lemma 3.4}Let $f\left( z\right) =\sum_{n=1}^{\infty }C_{\lambda
_{n}}z^{\lambda _{n}}$be an entire function with $0<\sigma \left( f,\varphi
\right) <\infty $ where $\varphi \left( r\right) $ only satisfies $\underset{%
r\rightarrow +\infty }{\lim }\frac{\log \varphi \left( \alpha r\right) }{%
\log \varphi \left( r\right) }=1$ for some $\alpha >1.$ If the sequence of
exponents $\left\{ \lambda _{n}\right\} $ satisfies the Fabry gap condition $%
\frac{\lambda _{n}}{n}\rightarrow \infty $ as $n\rightarrow \infty $. Then
for any given $\beta <\sigma \left( f,\varphi \right) ,$ there exists a set $%
E_{6}\subset \left( 1,\infty \right) $ having infinite logarithmic measure
such that for all $\left\vert z\right\vert =r\in E_{6}$ we have%
\begin{equation*}
\left\vert f\left( z\right) \right\vert >\exp \left\{ \varphi \left(
r\right) ^{\beta }\right\} .
\end{equation*}

\begin{proof}
By Lemma \ref{Lemma 3.2}, for any $\varepsilon >0,$ there exists a set $%
E_{4} $ of finite logarithmic measure such that for all $\left\vert
z\right\vert =r\not\in E_{4},$ we have%
\begin{equation*}
\log L\left( r,f\right) >\left( 1-\varepsilon \right) \log M\left(
r,f\right) ,
\end{equation*}%
which implies that%
\begin{equation*}
L\left( r,f\right) >\left[ M\left( r,f\right) \right] ^{\left( 1-\varepsilon
\right) }.
\end{equation*}%
For any given $\beta <\sigma \left( f,\varphi \right) ,$ we can choose $%
\delta >0$ such that $\beta <\delta <\sigma \left( f,\varphi \right) $ and
sufficiently small $\varepsilon $ satisfying $0<\varepsilon <\frac{\delta
-\beta }{2}.$ Then by Lemma \ref{Lemma 3.3}, there exists a set $E_{5}$ of
infinite logarithmic measure such that for all $\left\vert z\right\vert
=r\in E_{5},$ we have%
\begin{equation*}
\left\vert f\left( z\right) \right\vert >L\left( r,f\right) >\left[ M\left(
r,f\right) \right] ^{\left( 1-\varepsilon \right) }>\left( \exp \left\{
\varphi \left( r\right) ^{\beta }\right\} \right) ^{\left( 1-\varepsilon
\right) }>\exp \left\{ \varphi \left( r\right) ^{\beta }\right\},
\end{equation*}%
where $E_{6}=E_{5}\backslash E_{4}$ is a set with infinite logarithmic
measure.

Thus the lemma is established.
\end{proof}
\end{lemma}

\section{Proof of Main Results}

\begin{proof}[Proof of Theorem 2.1]
By Remark \ref{Remark2.1}, we know that $\sigma \left( A_{l},\varphi \right)
=\sigma .$ Let $f\not\equiv 0$ be a meromorphic solution of Equation $\left( %
\ref{Equa1}\right) .$ Now let us suppose that $\sigma \left( f,\varphi
\right) <\sigma \left( A_{l},\varphi \right) +1=\sigma +1<\infty .$ From the
conditions of Theorem \ref{Theorem2.1}, there \ is a set $H$ of complex
numbers satisfying $\overline{\log dens}\left\{ \left\vert z\right\vert
:z\in H\right\} >0$ such that for $z\in H,$ we have $\left( \ref{Equa 2.1}%
\right) $ and $\left( \ref{Equa2.2}\right) $ as $\left\vert z\right\vert
=r\rightarrow \infty .$ Set $H_{1}=\left\{ \left\vert z\right\vert =r:z\in
H\right\} ,$ since $\overline{\log dens}\left\{ \left\vert z\right\vert
:z\in H\right\} >0$, then by Proposition \ref{Proposition1.1}, $H_{1}$ is a
set with $\int_{H_{1}}\frac{dr}{r}=\infty $.

We divide Equation $\left( \ref{Equa1}\right) $ by $f\left( z+l\right) $ to
get%
\begin{equation}
-A_{l}\left( z\right) =\sum_{\substack{ j=0  \\ i\not=l}}^{n}A_{j}\left(
z\right) \frac{f\left( z+j\right) }{f\left( z+l\right) }.  \tag{4.1}
\label{Equa4.1/}
\end{equation}%
Since $A_{j}\left( z\right) $ $\left( j=0,1,...,n\right) $ are entire
functions, then by Equation $\left( \ref{Equa4.1/}\right) ,$ we get that%
\begin{equation*}
m\left( r,A_{l}\right) =T\left( r,A_{l}\right) \leq \sum_{\substack{ j=0  \\ %
i\not=l}}^{n}m\left( r,A_{j}\right) +\sum_{\substack{ j=0  \\ i\not=l}}%
^{n}m\left( r,\frac{f\left( z+j\right) }{f\left( z+l\right) }\right)
+O\left( 1\right)
\end{equation*}%
\begin{equation}
=\sum_{\substack{ j=0  \\ i\not=l}}^{n}T\left( r,A_{j}\right) +\sum
_{\substack{ j=0  \\ i\not=l}}^{n}m\left( r,\frac{f\left( z+j\right) }{%
f\left( z+l\right) }\right) +O\left( 1\right).  \tag{4.2}  \label{Equ4.2}
\end{equation}%
Now by Lemma \ref{Lemma3.2/}, for any $\varepsilon $ $\left( 0<\varepsilon <%
\frac{\sigma +1-\sigma \left( f,\varphi \right) }{2}\right) ,$ we have
\begin{equation}
m\left( r,\frac{f\left( z+j\right) }{f\left( z+l\right) }\right) =O\left(
\left( \varphi \left( r\right) \right) ^{\sigma \left( f,\varphi \right)
-1+\varepsilon }\right) .  \tag{4.3}  \label{Equ4.3}
\end{equation}%
Substituting $\left( \ref{Equa 2.1}\right) ,$ $\left( \ref{Equa2.2}\right) $
and $\left( \ref{Equ4.3}\right) $ into $\left( \ref{Equ4.2}\right) ,$ we get
for $\left\vert z\right\vert =r\rightarrow \infty ,~z\in H$ that%
\begin{equation*}
\exp \left\{ \alpha \left( \varphi \left( r\right) \right) ^{\sigma
-\varepsilon }\right\} \leq n\exp \left\{ \beta \left( \varphi \left(
r\right) \right) ^{\sigma -\varepsilon }\right\} +O\left( \left( \varphi
\left( r\right) \right) ^{\sigma \left( f,\varphi \right) -1+\varepsilon
}\right)
\end{equation*}%
\begin{equation*}
\Rightarrow \text{ }\exp \left\{ \left( \varphi \left( r\right) \right)
^{\sigma -\varepsilon }\right\} \left\{ \exp \left( \alpha \right) -\exp
\left( \beta \right) \right\} \leq O\left( 1\right) \left( \varphi \left(
r\right) \right) ^{\sigma \left( f,\varphi \right) -1+\varepsilon }.
\end{equation*}%
Since, $\left( \exp \left( \alpha \right) -\exp \left( \beta \right) \right)
>0,$ so it follows that%
\begin{equation}
1\leq O\left( 1\right) \left( \varphi \left( r\right) \right) ^{\sigma
\left( f,\varphi \right) -1+2\varepsilon -\sigma }\rightarrow 0\text{ as }%
r\rightarrow \infty ,  \tag{4.4}  \label{Equ4.4}
\end{equation}%
which is a contradiction since $0<\varepsilon <\frac{\sigma +1-\sigma \left(
f,\varphi \right) }{2}$. Hence, we get $\sigma \left( f,\varphi \right) \geq
\sigma \left( A_{l},\varphi \right) +1$.

This completes the proof of the theorem.\qquad \qquad \qquad \qquad \qquad
\end{proof}

\begin{proof}[Proof of Theorem 2.2]
If $\sigma \left( f,\varphi \right) =\infty ,$ then the result is trivial.
Now let us suppose that $\sigma \left( f,\varphi \right) <\sigma \left(
A_{l},\varphi \right) +1<\infty .$ We divide Equation $\left( \ref{Equa1.1}%
\right) $ by $f\left( z+l\right) $ to get%
\begin{equation*}
-A_{l}\left( z\right) =\sum_{\substack{ j=0  \\ i\not=l}}^{n}A_{j}\left(
z\right) \frac{f\left( z+j\right) }{f\left( z+l\right) }-\frac{F\left(
z\right) }{f\left( z\right) }\cdot \frac{f\left( z\right) }{f\left(
z+l\right) },
\end{equation*}%
which implies that%
\begin{equation}
\left\vert A_{l}\left( z\right) \right\vert \leq \sum_{\substack{ j=0  \\ %
i\not=l}}^{n}\left\vert A_{j}\left( z\right) \right\vert \left\vert \frac{%
f\left( z+j\right) }{f\left( z+l\right) }\right\vert +\left\vert \frac{%
F\left( z\right) }{f\left( z\right) }\right\vert \cdot \left\vert \frac{%
f\left( z\right) }{f\left( z+l\right) }\right\vert .  \tag{4.5}
\label{Equa4.1}
\end{equation}%
By Lemma \ref{Lemma 3.1}, for any given $\varepsilon \left( 0<\varepsilon <%
\frac{\sigma \left( A_{l},\varphi \right) +1-\sigma \left( f,\varphi \right)
}{2}\right) ,$ there exists a subset $E_{3}\subset \left( 1,\infty \right) $
of finite logarithmic measure such that for all $r\not\in \left[ 0,1\right]
\cup E_{3}$, we have%
\begin{equation}
\left\vert \frac{f\left( z+j\right) }{f\left( z+l\right) }\right\vert \leq
\exp \left\{ \frac{\left( \varphi \left( r\right) \right) ^{\sigma \left(
f,\varphi \right) +\varepsilon }}{r}\right\} ,\text{ \ }\left(
j=0,1,...,n,j\not=l\right) .  \tag{4.6}  \label{Equa4.2}
\end{equation}%
Now by the assumption (\ref{Equa2.4}), we have that for sufficiently large $%
r,$%
\begin{equation}
\left\vert A_{j}\left( z\right) \right\vert \leq \exp \left\{ \left( \varphi
\left( r\right) \right) ^{\sigma \left( A_{l},\varphi \right) +\varepsilon
}\right\} ,\text{ \ }\left( j=0,1,...,n,j\not=l\right) ,  \tag{4.7}
\label{Equa4.3}
\end{equation}%
and%
\begin{equation*}
\left\vert F\left( z\right) \right\vert \leq \exp \left\{ \left( \varphi
\left( r\right) \right) ^{\sigma \left( A_{l},\varphi \right) +\varepsilon
}\right\} .
\end{equation*}%
Since $M\left( r,f\right) >1$ for sufficiently large $r,$ we have that%
\begin{equation}
\frac{\left\vert F\left( z\right) \right\vert }{M\left( r,f\right) }\leq
\left\vert F\left( z\right) \right\vert \leq \exp \left\{ \left( \varphi
\left( r\right) \right) ^{\sigma \left( A_{l},\varphi \right) +\varepsilon
}\right\} .  \tag{4.8}  \label{Equa4.4}
\end{equation}%
Now by the Definition $\ref{Defi1.1}$ of $\varphi $-order and for above $%
\varepsilon >0,$ we get that%
\begin{equation}
\left\vert A_{l}\left( z\right) \right\vert \geq \exp \left\{ \left( \varphi
\left( r\right) \right) ^{\sigma \left( A_{l},\varphi \right) -\varepsilon
}\right\} .  \tag{4.9}  \label{Equa4.5}
\end{equation}%
Substituting (\ref{Equa4.2})-(\ref{Equa4.5}) into (\ref{Equa4.1}) for all $%
r\not\in \left[ 0,1\right] \cup E_{3}$ and $\left\vert f\left( z\right)
\right\vert =M\left( r,f\right) ,$ we have%
\begin{equation*}
\exp \left\{ \left( \varphi \left( r\right) \right) ^{\sigma \left(
A_{l},\varphi \right) -\varepsilon }\right\} \leq \left\vert A_{l}\left(
z\right) \right\vert 
\end{equation*}
\begin{equation}
\leq \left( n+1\right) \exp \left\{ \left( \varphi
\left( r\right) \right) ^{\sigma \left( A_{l},\varphi \right) +\varepsilon
}\right\} \cdot \exp \left\{ \frac{\left( \varphi \left( r\right) \right)
^{\sigma \left( f,\varphi \right) +\varepsilon }}{r}\right\} .  \tag{4.10}
\label{Equa4.6}
\end{equation}%
Since $\varepsilon $ $\left( 0<\varepsilon <\frac{\sigma \left(
A_{l},\varphi \right) +1-\sigma \left( f,\varphi \right) }{2}\right) ,$ so
we obtain a contradiction from $\left( \ref{Equa4.6}\right) $ by applying
the same procedure we applied in $\left( \ref{Equ4.4}\right) .$ Hence we get
that $\sigma \left( f,\varphi \right) \geq \sigma \left( A_{l},\varphi
\right) +1.$

This proves the theorem.
\end{proof}

\begin{proof}[Proof of Theorem 2.3]
If $\sigma \left( f,\varphi \right) =\infty ,$ then the result is trivial.
Now let us suppose that $\sigma \left( f,\varphi \right) <\sigma \left(
A_{l},\varphi \right) +1<\infty .$ Now by Lemma \ref{Lemma 3.4}, there
exists a set $E_{6}\subset \left( 1,\infty \right) $ having infinite
logarithmic measure such that for all $\left\vert z\right\vert =r\in E_{6}$
we have%
\begin{equation}
\left\vert A_{l}\left( z\right) \right\vert >\exp \left\{ \left( \varphi
\left( r\right) \right) ^{\beta }\right\} .  \tag{4.11}  \label{Equa4.7}
\end{equation}%
Substituting (\ref{Equa4.2})-(\ref{Equa4.4}) and (\ref{Equa4.7}) into (\ref%
{Equa4.1}) for all $r\in E_{6}\backslash \left[ 0,1\right] \cup E_{3}$ and $%
\left\vert f\left( z\right) \right\vert =M\left( r,f\right) ,$ we have\qquad
\begin{equation}
\exp \left\{ \left( \varphi \left( r\right) \right) ^{\beta }\right\} \leq
\left\vert A_{l}\left( z\right) \right\vert \leq \left( n+1\right) \exp
\left\{ \left( \varphi \left( r\right) \right) ^{\sigma \left( A_{l},\varphi
\right) +\varepsilon }\right\} \cdot \exp \left\{ \frac{\left( \varphi
\left( r\right) \right) ^{\sigma \left( f,\varphi \right) +\varepsilon }}{r}%
\right\} .  \tag{4.12}  \label{Equa4.8}
\end{equation}%
We we get a contradiction from $\left( \ref{Equa4.8}\right) $ by applying
the same procedure we applied in $\left( \ref{Equ4.4}\right) .$ Hence we get
that $\sigma \left( f,\varphi \right) \geq \sigma \left( A_{l},\varphi
\right) +1.$\qquad

This proves the theorem.
\end{proof}

\subsection*{Acknowledgement}The authors are grateful to the referee for his valuable suggestions which has considerably improved the presentation of the paper.


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