Finite Logarithmic Order Meromorphic Solutions of Complex Linear Delay-
Differential Equations

Abdelkader DAHMANI and Benharrat BELAÏDI111Corresponding author

Department of Mathematics, Laboratory of Pure and Applied Mathematics,

University of Mostaganem (UMAB), B. P. 227 Mostaganem-Algeria

<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
Abstract. In this article, we study the growth of meromorphic solutions of
linear delay-differential equation of the form

$\sum_{i=0}^{n}\sum_{j=0}^{m}A_{ij}(z)f^{(j)}(z+c_{i})=F(z),$

where $A_{ij}(z)$ $(i=0,1,\ldots,n,j=0,1,\ldots,m,n,m\in\mathbb{N})$ and
$F(z)$ are meromorphic of finite logarithmic order, $c_{i}(i=0,\ldots,n)$ are
distinct non-zero complex constants. We extend those results obtained recently
by Chen and Zheng, Bellaama and Belaïdi to the logarithmic lower order.

2010 Mathematics Subject Classification: 30D35, 39B32, 39A10.

Keywords and phrases: Linear difference equation, linear delay-differential
equation, entire function, meromorphic function, logarithmic order,
logarithmic lower order.

##  1 Introduction and Main Results

Throughout this article, we assume the readers are familiar with the
fundamental results and standard notations of the Nevanlinna distribution
theory of meromorphic functions such as $m(r,f),$ $N(r,f),$ $M(r,f),$
$T(r,f),$ which can be found in $\left[13,14,25\right].$ The concepts of
logarithmic order and logarithmic type of entire or meromorphic functions were
introduced by Chern, $\left[9,10\right]$. Since then, many authors used them
in order to generalize previous results obtained on the growth of solutions of
linear difference equations and linear differential equations in which the
coefficients are entire or meromorphic functions in the complex plane
$\mathbb{C}$ of positive order different to zero, see for example
$\left[1,6,11,18,19,21,22\right]$, their new results were on the logarithmic
order, the logarithmic lower order and the logarithmic exponent of
convergence, where they considered the case when the coefficients are of zero
order see, for example, $\left[2,3,4,7,12,16,17,23\right]$. In this article,
we also use these concepts to investigate the lower logarithmic order of
solutions to more general homogeneous and non homogeneous linear delay-
differential equations, where we generalize those results obtained in
$\left[5,8\right]$. We start by stating some important definitions.

Definition 1.1 $\left(\left[3,10\right]\right)$ The logarithmic order and the
logarithmic lower order of a meromorphic function $f$ are defined by

$\rho_{\log}(f)=\limsup_{r\longrightarrow+\infty}\frac{\log T(r,f)}{\log\log
r},\quad\mu_{\log}(f)=\liminf_{r\longrightarrow+\infty}\frac{\log
T(r,f)}{\log\log r}.$

where $T(r,f)$ denotes the Nevanlinna character of the function $f$. If $f$ is
an entire function, then

$\rho_{\log}(f)=\limsup_{r\longrightarrow+\infty}\frac{\log\log
M(r,f)}{\log\log r}=\limsup_{r\longrightarrow+\infty}\frac{\log\log
T(r,f)}{\log\log r},$
$\mu_{\log}(f)=\liminf_{r\longrightarrow+\infty}\frac{\log\log
M(r,f)}{\log\log r}=\liminf_{r\longrightarrow+\infty}\frac{\log\log
T(r,f)}{\log\log r},$

where $M(r,f)$ denotes the maximum modulus of $f$ in the circle
$\left|z\right|=r$.

It is clear that, the logarithmic order of any non-constant rational function
$f$ is one, and thus, any transcendental meromorphic function in the plane has
logarithmic order no less than one. Moreover, any meromorphic function with
finite logarithmic order in the plane is of order zero.

Definition 1.2 $\left(\left[3,7\right]\right)$ The logarithmic type and the
logarithmic lower type of a meromorphic function $f$ are defined by

$\tau_{\log}(f)=\limsup_{r\longrightarrow+\infty}\frac{T(r,f)}{(\log{r})^{\rho_{\log}(f)}},\quad\underline{\tau}_{\log}(f)=\liminf_{r\longrightarrow+\infty}\frac{T(r,f)}{(\log{r})^{\mu_{\log}(f)}}.$

If $f$ is an entire function, then

$\tau_{\log}(f)=\limsup_{r\longrightarrow+\infty}\frac{\log
M(r,f)}{(\log{r})^{\rho_{\log}(f)}}=\limsup_{r\longrightarrow+\infty}\frac{\log
T(r,f)}{(\log{r})^{\rho_{\log}(f)}},$
$\underline{\tau}_{\log}(f)=\liminf_{r\longrightarrow+\infty}\frac{\log
M(r,f)}{(\log{r})^{\mu_{\log}(f)}}=\liminf_{r\longrightarrow+\infty}\frac{\log
M(r,f)}{(\log{r})^{\mu_{\log}(f)}}.$

It is clear that, the logarithmic type of any non-constant polynomial $P$
equals its degree $\deg P$, that any non-constant rational function is of
finite logarithmic type, and that any transcendental meromorphic function
whose logarithmic order equals one in the plane must be of infinite
logarithmic type.

Definition 1.3 $\left(\left[10\right]\right)$ Let $f$ be meromorphic function.
Then, the logarithmic exponent of convergence of poles of $f$ is defined by

$\lambda_{\log}\left(\frac{1}{f}\right)=\limsup_{r\longrightarrow+\infty}\frac{\log
n(r,f)}{\log\log r}=\limsup_{r\longrightarrow+\infty}\frac{\log
N(r,f)}{\log\log r}-1,$

where $n(r,f)$ denotes the number of poles and $N(r,f)$ is the counting
function of poles of $f$ in the disc $\left|z\right|\leq r$.

Definition 1.4 $\left(\left[25\right]\right)$ Let
$a\in\overline{\mathbb{C}}=\mathbb{C}\cup\\{\infty\\},$ the deficiency of $a$
with respect to a meromorphic function $f$ is given by

$\delta\left(a,f\right)=\underset{r\rightarrow+\infty}{\lim\inf}\frac{m\left(r,\frac{1}{f-a}\right)}{T\left(r,f\right)}=1-\underset{r\rightarrow+\infty}{\lim\sup}\frac{N\left(r,\frac{1}{f-a}\right)}{T\left(r,f\right)}.$

Recently, the research on the properties of meromorphic solutions of complex
delay-differential equations has become a subject of great interest from the
viewpoint of Nevanlinna theory and its difference analogues. In
$\left[20\right]$, Liu, Laine and Yang presented developments and new results
on complex delay-differential equations, an area with important and
interesting applications, which also gathers increasing attention (see,
$\left[4,5,8,24\right]$. In $\left[8\right]$, Chen and Zheng considered the
following homogeneous complex delay-differential equation

$\sum_{i=0}^{n}\sum_{j=0}^{m}A_{ij}(z)f^{(j)}(z+c_{i})=0,$ (1.1)

where $A_{ij}(z)$ $(i=0,1,\ldots,n,j=0,1,\ldots,m)$ and $F(z)$ are entire
functions of finite order, $c_{i}(i=0,\ldots,n)$ are distinct non-zero complex
constants, and they proved the following results.

Theorem A $\left(\left[8\right]\right)$ Let $A_{ij}(z)$
$(i=0,1,\ldots,n,j=0,1,\ldots,m)$ be entire functions, and
$a,l\in\left\\{0,1,...,n\right\\},$ $b\in\\{0,1,...,m\\}$ such that
$(a,b)\not=(l,0).$ If the following three assumptions hold simultaneously:

  1. 1.

$\max\\{\mu(A_{ab}),\rho(A_{ij})\colon(i,j)\neq(a,b),(l,0)\\}\leq\mu(A_{l0})<\infty,\mu(A_{l0})>0;$

  2. 2.

$\underline{\tau}_{M}(A_{l0})>\underline{\tau}_{M}(A_{ab}),$ when
$\mu(A_{l0})=\mu(A_{ab});$

  3. 3.

$\underline{\tau}_{M}(A_{l0})>\max\\{\tau_{M}((A_{ij}):\rho(A_{ij})=\mu(A_{l0})\colon(i,j)\neq(a,b),(l,0)\\},$
when $\mu(A_{l0})=\max\\{\rho(A_{ij})\colon(i,j)\neq(a,b),(l,0)\\},$

then any meromorphic solution $f(z)(\not\equiv 0)$ of $\left(1.1\right)$
satisfies $\rho(f)\geq\mu(A_{l0})+1.$

Theorem B $\left(\left[8\right]\right)$ Let $A_{ij}(z)$
$(i=0,1,\ldots,n,j=0,1,\ldots,m)$ be meromorphic functions, and
$a,l\in\left\\{0,1,...,n\right\\},$ $b\in\\{0,1,...,m\\}$ such that
$(a,b)\not=(l,0).$ If the following four assumptions hold simultaneously:

  1. 1.

$\delta(\infty,A_{l0})=\delta>0;$

  2. 2.

$\max\\{\mu(A_{ab}),\rho(A_{ij})\colon(i,j)\neq(a,b),(l,0)\\}\leq\mu(A_{l0})<\infty,\mu(A_{l0})>0;$

  3. 3.

$\delta\underline{\tau}(A_{l0})>\underline{\tau}(A_{ab}),$ when
$\mu(A_{l0})=\mu(A_{ab});$

  4. 4.

$\delta\underline{\tau}(A_{l0})>\max\\{\tau((A_{ij}):\rho(A_{ij})=\mu(A_{l0})\colon(i,j)\neq(a,b),(l,0)\\},$
when $\mu(A_{l0})=\max\\{\rho(A_{ij})\colon(i,j)\neq(a,b),(l,0)\\},$
then any meromorphic solution $f(z)(\not\equiv 0)$ of $\left(1.1\right)$
satisfies $\rho(f)\geq\mu(A_{l0})+1.$

Further, Bellaama and Belaïdi in $\left[5\right]$ extended the previous
results to the non homogeneous delay differential equation

$\sum_{i=0}^{n}\sum_{j=0}^{m}A_{ij}(z)f^{(j)}(z+c_{i})=F(z),$ (1.2)

where $A_{ij}(z)$ $(i=0,1,\ldots,n,j=0,1,\ldots,m,n,m\in\mathbb{N}),$
$\mathbb{N=}\left\\{0,1,2,\cdots\right\\}$ denote the set of natural numbers
and $F(z)$ are meromorphic or entire functions of finite logarithmic order,
$c_{i}(i=0,\ldots,n)$ are distinct non-zero complex constants, and obtained
the following theorems for the homogeneous and non-homogeneous cases.

Theorem C $\left(\left[5\right]\right)$ Consider the delay differential
equation with entire coefficients. Suppose that one of the coefficients, say
$A_{l0}$ with $\mu(A_{l0})>0$, is dominate in the sens that :

  1. 1.

$\max\\{\mu(A_{ab}),\rho(S)\\}\leq\mu(A_{l0})<\infty;$

  2. 2.

$\underline{\tau}_{M}(A_{l0})>\underline{\tau}_{M}(A_{ab}),$ whenever
$\mu(A_{l0})=\mu(A_{ab});$

  3. 3.

$\underline{\tau}_{M}(A_{l0})>\max\\{\tau_{M}(g):\rho(g)=\mu(A_{l0})\colon
g\in S\\},$ whenever $\mu(A_{l0})=\rho(S),$ where
$S:=\\{F,A_{ij}\colon(i,j)\neq(a,b),(l,0)\\}$ and
$\rho(S):=\max\\{\rho(g)\colon g\in S\\}.$
Then any meromorphic solution $f$ of $\left(1.2\right)$ satisfies
$\rho(f)\geq\mu(A_{l0})$ if $F(z)(\not\equiv 0).$ Further if $F(z)(\equiv 0),$
then any meromorphic solution $f(z)(\not\equiv 0)$ of $\left(1.1\right)$
satisfies $\rho(f)\geq\mu(A_{l0})+1.$

Theorem D $\left(\left[5\right]\right)$ Consider the delay differential
equation with meromorphic coefficients. Suppose that one of the coefficients,
say $A_{l0}$ with $\mu(A_{l0})>0$, is dominate in the sens that :

  1. 1.

$\max\\{\mu(A_{ab}),\rho(S)\\}\leq\mu(A_{l0})<\infty;$

  2. 2.

$\underline{\tau}(A_{l0})>\underline{\tau}(A_{ab}),$ whenever
$\mu(A_{l0})=\mu(A_{ab});$

  3. 3.

$\sum_{\rho(A_{ij})=\mu(A_{l0}),(i,j)\not=(l,0),(a,b)}\tau(A_{ij})+\tau(F)<\underline{\tau}(A_{l0})<\infty,$
whenever $\mu(A_{l0})=\rho(S);$

  4. 4.

$\sum_{\rho(A_{ij})=\mu(A_{l0}),(i,j)\not=(l,0),(a,b)}\tau(A_{ij})+\underline{\tau}(A_{ab})<\underline{\tau}(A_{l0})<\infty,$
whenever $\mu(A_{l0})=\mu(A_{ab})=\rho(S);$

  5. 5.

$\lambda\left(\frac{1}{A_{l0}}\right)<\mu(A_{l0})<\infty.$

Then any meromorphic solution $f$ of $\left(1.2\right)$ satisfies
$\rho(f)\geq\mu(A_{l0})$ if $F(z)(\not\equiv 0).$ Further if $F(z)(\equiv 0),$
then any meromorphic solution $f(z)(\not\equiv 0)$ of $\left(1.1\right)$
satisfies $\rho(f)\geq\mu(A_{l0})+1.$

Note that the case when the coefficients are of order zero is not included in
the above results and because the logarithmic order is an effective technique
to express the growth of solutions of the linear difference equations and the
linear differential equations even when the coefficients are zero order entire
or meromorphic functions, in this article, our main aim is to investigate the
logarithmic lower order of meromorphic solutions of equations
$\left(1.1\right)$ and $\left(1.2\right)$ to extend and improve the above
theorems. When the coefficients of $\left(1.1\right)$ and $\left(1.2\right)$
are meromorphic functions and there is one dominating coefficient by its
logarithmic lower order or by its logarithmic lower type, we get the following
two theorems.

Theorem 1.1 Let $A_{ij}(z)$ $(i=0,1,\ldots,n,j=0,1,\ldots,m,n,m\in\mathbb{N})$
be meromorphic functions, and $a,l\in\left\\{{0,1,...,n}\right\\},$
$b\in\\{0,1,...,m\\}$ such that $(a,b)\not=(l,0).$ Suppose that one of the
coefficients, say $A_{l0}$ with
$\lambda_{\log}\left(\frac{1}{A_{l0}}\right)+1<\mu_{\log}(A_{l0})<\infty$ is
dominate in the sens that :

$\left(\text{i}\right)$
$\max\\{\mu_{\log}(A_{ab}),\rho_{\log}(S)\\}\leq\mu_{\log}(A_{l0})<\infty;$

$\left(\text{ii}\right)$
$\underline{\tau}_{\log}(A_{l0})>\underline{\tau}_{\log}(A_{ab}),$ whenever
$\mu_{\log}(A_{l0})=\mu_{\log}(A_{ab});$

$\left(\text{iii}\right)$
$\sum_{\rho_{\log}(A_{ij})=\mu_{\log}(A_{l0}),(i,j)\not=(l,0),(a,b)}\tau_{\log}(A_{ij})+\tau_{\log}(F)<\underline{\tau}_{\log}(A_{l0})<\infty,$
whenever $\mu_{\log}(A_{l0})=\rho_{\log}(S);$

$\left(\text{iv}\right)$
$\sum_{\rho_{\log}(A_{ij})=\mu_{\log}(A_{l0}),(i,j)\not=(l,0),(a,b)}\tau_{\log}(A_{ij})+\tau_{\log}(F)+\underline{\tau}_{\log}(A_{ab})<\underline{\tau}_{\log}(A_{l0})<\infty,$
whenever $\mu_{\log}(A_{l0})=\mu_{\log}(A_{ab})=\rho_{\log}(S),$ where
$S:=\\{F,A_{ij}\colon(i,j)\neq(a,b),(l,0)\\}$ and
$\rho_{\log}(S):=\max\\{\rho_{\log}(g)\colon g\in S\\}.$

Then any meromorphic solution $f$ of $\left(1.2\right)$ satisfies
$\rho_{\log}(f)\geq\mu_{\log}(A_{l0})$ if $F(z)(\not\equiv 0).$ Further if
$F(z)(\equiv 0),$ then any meromorphic solution $f(z)(\not\equiv 0)$ of
$\left(1.1\right)$ satisfies $\rho_{\log}(f)\geq\mu_{\log}(A_{l0})+1.$

Theorem 1.2 Let $A_{ij}(z)$ $(i=0,1,\ldots,n,j=0,1,\ldots,m,n,m\in\mathbb{N})$
be meromorphic functions, and $a,l\in\left\\{{0,1,...,n}\right\\},$
$b\in\\{0,1,...,m\\}$ such that $(a,b)\not=(l,0).$ Suppose that one of the
coefficients, say $A_{l0}$ with $\mu(A_{l0})>0$ and $\delta(\infty,A_{l0})>0$
is dominate in the sens that :

$\left(\text{i}\right)$
$\max\\{\mu_{\log}(A_{ab}),\rho_{\log}(S)\\}\leq\mu_{\log}(A_{l0})<\infty;$

$\left(\text{ii}\right)$
$\delta\underline{\tau}_{\log}(A_{l0})>\underline{\tau}_{\log}(A_{ab}),$
whenever $\mu_{\log}(A_{l0})=\mu_{\log}(A_{ab});$

$\left(\text{iii}\right)$
$\sum_{\rho_{\log}(A_{ij})=\mu_{\log}(A_{l0}),(i,j)\not=(l,0),(a,b)}\tau_{\log}(A_{ij})+\tau_{\log}(F)<\delta\underline{\tau}_{\log}(A_{l0})<\infty,$
whenever $\mu_{\log}(A_{l0})=\rho_{\log}(S);$

$\left(\text{iv}\right)$
$\sum_{\rho_{\log}(A_{ij})=\mu_{\log}(A_{l0}),(i,j)\not=(l,0),(a,b)}\tau_{\log}(A_{ij})+\tau_{\log}(F)+\underline{\tau}_{\log}(A_{ab})<\delta\underline{\tau}_{\log}(A_{l0})<\infty,$
whenever $\mu_{\log}(A_{l0})=\mu_{\log}(A_{ab})=\rho_{\log}(S),$ where
$S:=\\{F,A_{ij}\colon(i,j)\neq(a,b),(l,0)\\}$ and
$\rho_{\log}(S):=\max\\{\rho_{\log}(g)\colon g\in S\\}.$

Then any meromorphic solution $f$ of $\left(1.2\right)$ satisfies
$\rho_{\log}(f)\geq\mu_{\log}(A_{l0})$ if $F(z)(\not\equiv 0).$ Further if
$F(z)(\equiv 0),$ then any meromorphic solution $f(z)(\not\equiv 0)$ of
$\left(1.1\right)$ satisfies $\rho_{\log}(f)\geq\mu_{\log}(A_{l0})+1.$

## Some lemmas

The following lemmas are important to our proofs.

Lemma 2.1 ($\left[15\right]$). Let $k$ and $j$ be integers such that $k>j\geq
0.$ Let $f$ be a meromorphic function in the plane $\mathbb{C}$ such that
$f^{(j)}$ does not vanish identically. Then, there exists an $r_{0}>1$ such
that

$m(r,\frac{f^{(k)}}{f^{(j)}})\leq(k-j)\log^{+}\frac{\rho(T(\rho,f))}{r(\rho-r)}+\log\frac{k!}{j!}+5.3078(k-j),$

for all $r_{0}<r<\rho<+\infty.$ If $f$ is of finite order $s$, then

$\limsup\limits_{r\rightarrow+\infty}\frac{m(r,\frac{f^{(k)}}{f^{(j)}})}{\log
r}\leq\max\\{0,(k-j)(s-1)\\}.$

Remark 2.1. It is shown in $\left[13,\text{ p}.\text{ 66}\right]$, that for an
arbitrary complex number $c\neq 0$, the following inequalities

$\left(1+o\left(1\right)\right)T\left(r-\left|c\right|,f\left(z\right)\right)\leq
T\left(r,f\left(z+c\right)\right)\leq\left(1+o\left(1\right)\right)T\left(r+\left|c\right|,f\left(z\right)\right)$

hold as $r\rightarrow+\infty$ for a general meromorphic function
$f\left(z\right)$. Therefore, it is easy to obtain that

$\rho_{\log}(f+c)=\rho_{\log}(f),\text{ }\mu_{\log}(f+c)=\mu_{\log}(f).$

Lemma 2.2 $\left(\left[3\right]\right)$ Let $f$ be a meromorphic function with
$1\leq\mu_{\log}\left(f\right)<+\infty.$ Then there exists a set
$E_{1}\subset\left(1,+\infty\right)$ with infinite logarithmic measure such
that for any given $\varepsilon>0$ and $r\in
E_{1}\subset\left(1,+\infty\right),$ we have

$T\left(r,f\right)<\left(\log
r\right)^{\mu_{\log}\left(f\right)+\varepsilon}.$

Lemma 2.3 Let $f$ be a meromorphic function with
$1\leq\mu_{\log}\left(f\right)<+\infty.$ Then there exists a set
$E_{2}\subset\left(1,+\infty\right)$ with infinite logarithmic measure such
that

$\underline{\tau}_{\log}(f)=\underset{\underset{r\in
E_{2}}{r\rightarrow+\infty}}{\lim}\frac{T(r,f)}{(\log r)^{\mu_{\log}(f)}}.$

Consequently, for any given $\varepsilon>0$ and all sufficiently large $r\in
E_{2},$ we have

$T\left(r,f\right)<\left(\underline{\tau}_{\log}(f)+\varepsilon\right)\left(\log
r\right)^{\mu_{\log}\left(f\right)},$

Proof. By the definition of the logarithmic lower type, there exists a
sequence $\left\\{r_{n}\right\\}_{n=1}^{\infty}$ tending to $\infty$
satisfying $\left(1+\frac{1}{n}\right)r_{n}<r_{n+1},$ and

$\underline{\tau}_{\log}(f)=\underset{r_{n}\rightarrow+\infty}{\lim}\frac{T(r_{n},f)}{(\log
r_{n})^{\mu_{\log}(f)}}.$

Then for any given $\varepsilon>0,$ there exists an $n_{1}$ such that for
$n\geq n_{1}$ and any $r\in\left[\frac{n}{n+1}r_{n},r_{n}\right],$ we have

$\frac{T(\frac{n}{n+1}r_{n},f)}{(\log
r_{n})^{\mu_{\log}(f)}}\leq\frac{T(r,f)}{(\log
r)^{\mu_{\log}(f)}}\leq\frac{T(r_{n},f)}{(\log\frac{n}{n+1}r_{n})^{\mu_{\log}(f)}}.$

It follows that

$\left(\frac{\log\frac{n}{n+1}r_{n}{}}{\log
r_{n}}\right)^{\mu_{\log}(f)}\frac{T(\frac{n}{n+1}r_{n},f)}{(\log\frac{n}{n+1}r_{n})^{\mu_{\log}(f)}}\leq\frac{T(r,f)}{(\log
r)^{\mu_{\log}(f)}}$ $\leq\frac{T(r_{n},f)}{(\log
r_{n})^{\mu_{\log}(f)}}\left(\frac{\log
r_{n}{}}{\log\frac{n}{n+1}r_{n}}\right)^{\mu_{\log}(f)}.$ (2.1)

Set

$E_{2}=\bigcup\limits_{n=n_{1}}^{+\infty}\left[\frac{n}{n+1}r_{n},r_{n}\right].$

Then from $\left(2.1\right)$, we obtain

$\underset{\underset{r\in
E_{2}}{r\rightarrow+\infty}}{\lim}\frac{T(r,f)}{(\log
r)^{\mu_{\log}(f)}}=\underset{r_{n}\rightarrow+\infty}{\lim}\frac{T(r_{n},f)}{(\log
r_{n})^{\mu_{\log}(f)}}=\underline{\tau}_{\log}(f),$

so for any given $\varepsilon>0$ and all sufficiently large $r\in E_{2},$ we
get

$T\left(r,f\right)<\left(\underline{\tau}_{\log}(f)+\varepsilon\right)\left(\log
r\right)^{\mu_{\log}\left(f\right)},$

where
$lm\left(E_{2}\right)=\int\limits_{E_{2}}\frac{dr}{r}=\sum\limits_{n=n_{1}}^{+\infty}\int\limits_{\frac{n}{n+1}r_{n}}^{r_{n}}\frac{dt}{t}=\sum\limits_{n=n_{1}}^{+\infty}\log\left(1+\frac{1}{n}\right)=+\infty.$

Lemma 2.4 $\left(\left[3\right]\right)$ Let $\eta_{1},\eta_{2}$ be two
arbitrary complex numbers such that $\eta_{1}\neq\eta_{2}$ and let $f$ be a
finite logarithmic order meromorphic function. Let $\rho$ be the logarithmic
order of $f$. Then for each $\varepsilon>0,$ we have

$m\left(r,\frac{f\left(z+\eta_{1}\right)}{f\left(z+\eta_{2}\right)}\right)=O\left(\left(\log
r\right)^{\rho-1+\varepsilon}\right).$

##  2 Proof of Theorem 1.1

Let $f$ be a meromorphic solution of $\left(1.2\right)$. If $f$ has infinite
logarithmic order, then the result holds. Now, we suppose that
$\rho_{\log}(f)<\infty$. We divide $(1.2)$ by $f(z+c_{l})$ to get

$-A_{l0}(z)=\sum_{i=0,i\neq
l,a}^{n}\sum_{j=0}^{m}A_{ij}\frac{f^{(j)}(z+c_{i})}{f(z+c_{i})}\frac{f(z+c_{i})}{f(z+c_{l})}$
$+\sum_{j=0,j\neq
b}^{m}A_{aj}\frac{f^{(j)}(z+c_{a})}{f(z+c_{a})}\frac{f(z+c_{a})}{f(z+c_{l})}+\sum_{j=1}^{m}A_{lj}\frac{f^{(j)}(z+c_{l})}{f(z+c_{l})}$
$+A_{ab}\frac{f^{(b)}(z+c_{a})}{f(z+c_{a})}\frac{f(z+c_{a})}{f(z+c_{l})}-\frac{F(z)}{f(z+c_{l})}.$
(3.1)

By $\left(3.1\right)$ and Remark 2.1, for sufficiently large $r$, we have

$T(r,A_{l0})=m(r,A_{l0})+N(r,A_{l0})\leq\sum_{i=0,i\neq
l,a}^{n}\sum_{j=0}^{m}m(r,A_{ij})+m(r,A_{ab})$
$+\sum_{j=1}^{m}m(r,A_{lj})+\sum_{j=0,j\neq b}^{m}m(r,A_{aj})+\sum_{i=0,i\neq
l,a}^{n}\sum_{j=0}^{m}m\left(r,\frac{f^{(j)}(z+c_{i})}{f(z+c_{i})}\right)$
$+\sum_{i=0,i\neq
l,a}^{n}m\left(r,\frac{f(z+c_{i})}{f(z+c_{l})}\right)+\sum_{j=1}^{m}m\left(r,\frac{f^{(j)}(z+c_{a})}{f(z+c_{a})}\right)+2m\left(r,\frac{f(z+c_{a})}{f(z+c_{l})}\right)$
$+\sum_{j=1}^{m}m\left(r,\frac{f^{(j)}(z+c_{l})}{f(z+c_{l})}\right)+m\left(r,F\right)+m\left(r,\frac{1}{f(z+c_{l})}\right)+N(r,A_{l0})+O(1)$
$\leq\sum_{i=0,i\neq
l,a}^{n}\sum_{j=0}^{m}T(r,A_{ij})+T(r,A_{ab})+\sum_{j=1}^{m}T(r,A_{lj})+\sum_{j=0,j\neq
b}^{m}T(r,A_{aj})$ $+\sum_{i=0,i\neq
l,a}^{n}\sum_{j=1}^{m}m\left(r,\frac{f^{(j)}(z+c_{i})}{f(z+c_{i})}\right)+\sum_{i=0,i\neq
l,a}^{n}m\left(r,\frac{f(z+c_{i})}{f(z+c_{l})}\right)$
$+\sum_{j=1}^{m}m\left(r,\frac{f^{(j)}(z+c_{a})}{f(z+c_{a})}\right)+2m\left(r,\frac{f(z+c_{a})}{f(z+c_{l})}\right)$
$+\sum_{j=1}^{m}m\left(r,\frac{f^{(j)}(z+c_{l})}{f(z+c_{l})}\right)+T\left(r,F\right)+2T\left(2r,f\right)+N(r,A_{l0})+O(1).$
(3.2)

From Lemma 2.1, for sufficiently large $r$, we obtain

$m\left(r,\frac{f^{(j)}(z+c_{i})}{f(z+c_{i})}\right)\leq
2j\log^{+}T\left(2r,f\right),\ (i=0,1,...,n,j=1,...,m).$ (3.3)

By Lemma 2.4, for any given $\varepsilon>0$ and all sufficiently large $r$, we
have

$m\left(r,\frac{f(z+c_{i})}{f(z+c_{l})}\right)=O\left((\log
r)^{\rho_{\log}(f)-1+\varepsilon}\right),\ (i=0,1,...,n,i\neq l).$ (3.4)

From the definition of $\lambda_{\log}\left(\frac{1}{A_{l0}}\right)$, for any
given $\varepsilon>0$ with sufficiently large $r$, we have

$N(r,A_{l0})\leq\left(\log
r\right)^{\lambda_{\log}\left(\frac{1}{A_{l0}}\right)+1+\varepsilon}.$ (3.5)

By using the assumptions $\left(3.3\right)-\left(3.5\right)$, we may rewrite
$\left(3.2\right)$ as

$T(r,A_{l0})\leq\sum_{i=0,i\neq l,a}^{n}\sum_{j=0}^{m}T(r,A_{ij})+T(r,A_{ab})$
$+\sum_{j=1}^{m}T(r,A_{lj})+\sum_{j=0,j\neq
b}^{m}T(r,A_{aj})+O\left(\log^{+}T\left(2r,f\right)\right)$ $+O\left((\log
r)^{\rho_{\log}(f)-1+\varepsilon}\right)+T\left(r,F\right)+2T\left(2r,f\right)+(\log
r)^{\lambda_{\log}\left(\frac{1}{A_{l0}}\right)+1+\varepsilon}.$ (3.6)

This proof is also divided into four cases:

Case $\left(\mathbf{i}\right)$: If
$\max\\{\mu_{\log}(A_{ab}),\rho_{\log}(S)\\}<\mu_{\log}(A_{l0}),$ then by the
definitions of $\mu_{\log}(A_{l0})$ and $\rho_{\log}(S)$ for any given
$\varepsilon>0$ and all sufficiently large $r$, we have

$T(r,A_{l0})\geq(\log r)^{\mu_{\log}(A_{l0})-\varepsilon},$ (3.7)
$T(r,g)\leq(\log r)^{\rho_{\log}(S)+\varepsilon},\quad g\in S.$ (3.8)

By the definition of $\mu_{\log}(A_{ab})$ and Lemma 2.2, there exists a subset
$E_{1}\subset(1,+\infty)$ of infinite logarithmic measure such that for any
given $\varepsilon>0$ and for all sufficiently large $r\in E_{1}$, we have

$T(r,A_{ab})\leq(\log r)^{\mu_{\log}(A_{ab})+\varepsilon}.$ (3.9)

We set $\rho=\max\\{\mu_{\log}(A_{ab}),\rho_{\log}(S)\\},$ then from
$\left(3.8\right)$ and $\left(3.9\right)$, it follows

$\max\left\\{T(r,A_{ab}),T(r,g)\right\\}\leq(\log r)^{\rho+\varepsilon}.$
(3.10)

Also, from the definition of $\rho_{\log}(f)$ for any given $\varepsilon>0$
and all sufficiently large $r$, we have

$T(r,f)\leq\left(\log r\right)^{\rho_{\log}(f)+\varepsilon}.$ (3.11)

By substituting $\left(3.7\right),$ $\left(3.10\right)$ and
$\left(3.11\right)$ into $\left(3.6\right)$, for any given $\varepsilon>0$ and
all sufficiently large $r\in E_{1}$, we get

$(\log r)^{\mu_{\log}(A_{l0})-\varepsilon}\leq O\left((\log
r)^{\rho+\varepsilon}\right)+O\left(\log\left(\log
r\right)\right)+O\left((\log r)^{\rho_{\log}(f)-1+\varepsilon}\right)$
$+O\left(\left(\log r\right)^{\rho_{\log}(f)+\varepsilon}\right)+(\log
r)^{\lambda_{\log}\left(\frac{1}{A_{l0}}\right)+1+\varepsilon}.$ (3.12)

Now, we choose sufficiently small $\varepsilon$ satisfying

$0<3\varepsilon<\min\left\\{\mu_{\log}(A_{l0})-\rho,\mu_{\log}(A_{l0})-\lambda_{\log}\left(\frac{1}{A_{l0}}\right)-1\right\\},$

for all sufficiently large $r\in E_{1}$, it follows from $\left(3.12\right)$
that

$(\log r)^{\mu_{\log}(A_{l0})-2\varepsilon}\leq\left(\log
r\right)^{\rho_{\log}(f)+\varepsilon},$

that means, $\mu_{\log}(A_{l0})-3\varepsilon\leq\rho_{\log}(f)$ and since
$\varepsilon>0$ is arbitrary, then $\rho_{\log}(f)\geq\mu_{\log}(A_{l0}).$

Similarly, for the homogeneous case, by $\left(1.1\right)$ and
$\left(3.3\right)-\left(3.5\right)$, we obtain

$T(r,A_{l0})\leq\sum_{i=0,i\neq
l,a}^{n}\sum_{j=0}^{m}T(r,A_{ij})+T(r,A_{ab})+\sum_{j=1}^{m}T(r,A_{lj})+\sum_{j=0,j\neq
b}^{m}T(r,A_{aj})$ $+O\left(\log\left(\log r\right)\right)+O\left((\log
r)^{\rho_{\log}(f)-1+\varepsilon}\right)+(\log
r)^{\lambda_{\log}\left(\frac{1}{A_{l0}}\right)+1+\varepsilon}.$ (3.13)

Then, by substituting $\left(3.7\right)$ and $\left(3.10\right)$ into
$\left(3.13\right)$, for all sufficiently large $r\in E_{1}$, we have

$(\log r)^{\mu_{\log}(A_{l0})-\varepsilon}\leq O\left((\log
r)^{\rho+\varepsilon}\right)+O\left(\log\left(\log r\right)\right)$
$+O\left((\log r)^{\rho_{\log}(f)-1+\varepsilon}\right)+(\log
r)^{\lambda_{\log}\left(\frac{1}{A_{l0}}\right)+1+\varepsilon}.$ (3.14)

For sufficiently small $\varepsilon$ satisfying

$0<3\varepsilon<\min\left\\{\mu_{\log}(A_{l0})-\rho,\mu_{\log}(A_{l0})-\lambda_{\log}\left(\frac{1}{A_{l0}}\right)-1\right\\},$

and all sufficiently large $r\in E_{1}$, we deduce from $\left(3.14\right)$
that

$(\log r)^{\mu_{\log}(A_{l0})-2\varepsilon}\leq\log
r)^{\rho_{\log}(f)-1+\varepsilon},$

that is, $\mu_{\log}(A_{l0})-3\varepsilon\leq\rho_{\log}(f)-1$ and since
$\varepsilon>0$ is arbitrary, then $\rho_{\log}(f)\geq\mu_{\log}(A_{l0})+1.$

Case $\left(\mathbf{ii}\right)$: If
$\beta=\rho_{\log}(S)<\mu_{\log}(A_{l0})=\mu_{\log}(A_{ab})$ and
$\underline{\tau}_{\log}(A_{l0})>\underline{\tau}_{\log}(A_{ab}),$ then by the
definition of $\underline{\tau}_{\log}(A_{l0}),$ for any given $\varepsilon>0$
and all sufficiently large $r$, we have

$T(r,A_{l0})\geq(\underline{\tau}_{\log}(A_{l0})-\varepsilon)(\log
r)^{\mu_{\log}(A_{l0})}.$ (3.15)

Also from the definition of $\underline{\tau}_{\log}(A_{ab})$ and Lemma 2.3,
there exists a subset $E_{2}\subset(1,+\infty)$ of infinite logarithmic
measure such that for any given $\varepsilon>0$ and for all sufficiently large
$r\in E_{2}$, we obtain

$T(r,A_{ab})\leq(\underline{\tau}_{\log}(A_{ab})+\varepsilon)(\log
r)^{\mu_{\log}(A_{ab})}=(\underline{\tau}_{\log}(A_{ab})+\varepsilon)(\log
r)^{\mu_{\log}(A_{l0})}.$ (3.16)

By substituting $\left(3.8\right),\left(3.11\right),\left(3.15\right)$ and
$\left(3.16\right)$ into $\left(3.6\right)$, for all sufficiently large $r\in
E_{2}$, we get

$(\underline{\tau}_{\log}(A_{l0})-\varepsilon)(\log
r)^{\mu_{\log}(A_{l0})}\leq O\left((\log r)^{\beta+\varepsilon}\right)$
$+(\underline{\tau}_{\log}(A_{ab})+\varepsilon)(\log
r)^{\mu_{\log}(A_{l0})}+O\left(\log\left(\log r\right)\right)+O\left((\log
r)^{\rho_{\log}(f)-1+\varepsilon}\right)$ $+O\left(\left(\log
r\right)^{\rho_{\log}(f)+\varepsilon}\right)+(\log
r)^{\lambda_{\log}\left(\frac{1}{A_{l0}}\right)+1+\varepsilon}.$ (3.17)

Now, we choose sufficiently small $\varepsilon$ satisfying
$0<2\varepsilon<\min\\{\mu_{\log}(A_{l0})-\beta,\mu_{\log}(A_{l0})-\lambda_{\log}\left(\frac{1}{A_{l0}}\right)-1,\underline{\tau}_{\log}(A_{l0})-\underline{\tau}_{\log}(A_{ab})\\},$
for all sufficiently large $r\in E_{2}$, it follows from $\left(3.17\right)$
that

$(\underline{\tau}_{\log}(A_{l0})-\underline{\tau}_{\log}(A_{ab})-2\varepsilon)(\log
r)^{\mu_{\log}(A_{l0})-\varepsilon}\leq\left(\log
r\right)^{\rho_{\log}(f)+\varepsilon},$

this means, $\mu_{\log}(A_{l0})-2\varepsilon\leq\rho_{\log}(f)$ and since
$\varepsilon>0$ is arbitrary, then $\rho_{\log}(f)\geq\mu_{\log}(A_{l0}).$

Next, for the homogeneous case, by substituting
$\left(3.8\right),\left(3.15\right)$ and $\left(3.16\right)$ into
$\left(3.13\right)$, for all sufficiently large $r\in E_{2}$, we have

$(\underline{\tau}_{\log}(A_{l0})-\varepsilon)(\log
r)^{\mu_{\log}(A_{l0})}\leq O\left((\log
r)^{\beta+\varepsilon}\right)+(\underline{\tau}_{\log}(A_{ab})+\varepsilon)(\log
r)^{\mu_{\log}(A_{l0})}$ $+O\left(\log\left(\log r\right)\right)+O\left((\log
r)^{\rho_{\log}(f)-1+\varepsilon}\right)+(\log
r)^{\lambda_{\log}\left(\frac{1}{A_{l0}}\right)+1+\varepsilon}.$ (3.18)

Now, we choose sufficiently small $\varepsilon$ satisfying
$0<2\varepsilon<\min\\{\mu_{\log}(A_{l0})-\beta,\mu_{\log}(A_{l0})-\lambda_{\log}\left(\frac{1}{A_{l0}}\right)-1,\underline{\tau}_{\log}(A_{l0})-\underline{\tau}_{\log}(A_{ab})\\},$
for all sufficiently large $r\in E_{2}$, we deduce from $\left(3.18\right)$
that

$(\underline{\tau}_{\log}(A_{l0})-\underline{\tau}_{\log}(A_{ab})-2\varepsilon)(\log
r)^{\mu_{\log}(A_{l0})-\varepsilon}\leq(\log
r)^{\rho_{\log}(f)-1+\varepsilon},$

that is, $\mu_{\log}(A_{l0})-2\varepsilon\leq\rho_{\log}(f)-1$ and since
$\varepsilon>0$ is arbitrary, then $\rho_{\log}(f)\geq\mu_{\log}(A_{l0})+1.$

Case $\left(\mathbf{iii}\right)$: When
$\mu_{\log}(A_{ab})<\mu_{\log}(A_{l0})=\rho_{\log}(S)$ and

$\tau_{1}=\sum_{\rho_{\log}(A_{ij})=\mu_{\log}(A_{l0}),(i,j)\not=(l,0),(a,b)}\tau_{\log}(A_{ij})+\tau_{\log}(F)$
$=\tau+\tau_{\log}(F)<\underline{\tau}_{\log}(A_{l0}),\text{
}\tau=\sum_{\rho_{\log}(A_{ij})=\mu_{\log}(A_{l0}),(i,j)\not=(l,0),(a,b)}\tau_{\log}(A_{ij}).$

Then, there exists a subset
$J\subseteq\\{0,1,\dots,n\\}\times\\{0,1,\dots,m\\}\setminus\left\\{(l,0),(a,b)\right\\}$
such that for all $(i,j)\in J,$ when
$\rho_{\log}(A_{ij})=\mu_{\log}\left(A_{l0}\right),$ we have
$\underset{(i,j)\in
J}{\sum}\tau_{\log}\left(A_{ij}\right)<\underline{\tau}_{\log}\left(A_{l0}\right)-\tau_{\log}(F),$
and for
$(i,j)\in\Pi=\\{0,1,\dots,n\\}\times\\{0,1,\dots,m\\}\setminus\left(J\cup\left\\{\,(l,0),(a,b)\right\\}\right)$
we have $\rho_{\log}\left(A_{ij}\right)<\mu_{\log}\left(A_{l0}\right).$ Hence,
for any given $\varepsilon>0$ and all sufficiently large $r,$ we get

$T\left(r,A_{ij}\right)\leq\left\\{\begin{array}[]{c}\left(\tau_{\log}(A_{ij})+\varepsilon\right)\left(\log
r\right)^{\mu_{\log}(A_{l0})},\text{ if }(i,j)\in J,\\\ \left(\log
r\right)^{\rho_{\log}(A_{ij})+\varepsilon}\leq\left(\log
r\right)^{\mu_{\log}(A_{l0})-\varepsilon},\text{ if
}(i,j)\in\Pi\end{array}\right.$ (3.19)

and

$T\left(r,F\right)\leq\left\\{\begin{array}[]{c}\left(\tau_{\log}(F)+\varepsilon\right)\left(\log
r\right)^{\mu_{\log}(A_{l0})},\text{ if }\rho_{\log}(F)=\mu_{\log}(A_{l0}),\\\
\left(\log r\right)^{\rho_{\log}(F)+\varepsilon}\leq\left(\log
r\right)^{\mu_{\log}(A_{l0})-\varepsilon},\text{ if
}\rho_{\log}(F)<\mu_{\log}(A_{l0}).\end{array}\right.$ (3.20)

By substituting $\left(3.9\right),$ $\left(3.11\right),$ $\left(3.15\right),$
$\left(3.19\right)$ and $\left(3.20\right)$ into $\left(3.6\right)$, for all
sufficiently large $r\in E_{1},$ we get

$\left(\underline{\tau}_{\log}(A_{l0})-\varepsilon\right)\left(\log
r\right)^{\mu_{\log}(A_{l0})}\leq\underset{(i,j)\in
J}{\sum}\left(\tau_{\log}\left(A_{ij}\right)+\varepsilon\right)\left(\log
r\right)^{\mu_{\log}\left(A_{l0}\right)}$
$+\underset{(i,j)\in\Pi}{\sum}\left(\log
r\right)^{\mu_{\log}(A_{l0})-\varepsilon}+(\log
r)^{\mu_{\log}(A_{ab})+\varepsilon}+O\left(\log\left(\log
r\right)\right)+O\left((\log r)^{\rho_{\log}(f)-1+\varepsilon}\right)$
$+\left(\tau_{\log}(F)+\varepsilon\right)\left(\log
r\right)^{\mu_{\log}(A_{l0})}+O\left(\left(\log
r\right)^{\rho_{\log}(f)+\varepsilon}\right)+(\log
r)^{\lambda_{\log}\left(\frac{1}{A_{l0}}\right)+1+\varepsilon}$
$\leq\left(\tau_{1}+\left(mn+m+n\right)\varepsilon\right)\left(\log
r\right)^{\mu_{\log}\left(A_{l0}\right)}+O\left(\log
r\right)^{\mu_{\log}(A_{l0})-\varepsilon}$ $+(\log
r)^{\mu_{\log}(A_{ab})+\varepsilon}+O\left(\log\left(\log
r\right)\right)+O\left((\log r)^{\rho_{\log}(f)-1+\varepsilon}\right)$
$+O\left(\left(\log r\right)^{\rho_{\log}(f)+\varepsilon}\right)+(\log
r)^{\lambda_{\log}\left(\frac{1}{A_{l0}}\right)+1+\varepsilon}.$ (3.21)

We may choose sufficiently small $\varepsilon$ satisfying
$0<2\varepsilon<\min\\{\mu_{\log}(A_{l0})-\mu_{\log}(A_{ab}),\mu_{\log}(A_{l0})-\lambda_{\log}\left(\frac{1}{A_{l0}}\right)-1,\frac{\underline{\tau}_{\log}(A_{l0})-\tau_{1}}{mn+m+n+1}\\},$
for all sufficiently large $r\in E_{1},$ by $\left(3.21\right)$ we have

$(\underline{\tau}_{\log}(A_{l0})-\tau_{1}-\left(mn+m+n+1\right)\varepsilon)(\log
r)^{\mu_{\log}(A_{l0})-\varepsilon}\leq\left(\log
r\right)^{\rho_{\log}(f)+\varepsilon},$

this means, $\mu_{\log}(A_{l0})-2\varepsilon\leq\rho_{\log}(f)$ and since
$\varepsilon>0$ is arbitrary, then $\rho_{\log}(f)\geq\mu_{\log}(A_{l0}).$

Further, for the homogeneous case, by substituting $\left(3.9\right)$,
$\left(3.15\right),$ $\left(3.19\right)$ and $\left(3.20\right)$ into
$\left(3.13\right)$, for all sufficiently large $r\in E_{1}$, we get

$(\underline{\tau}_{\log}(A_{l0})-\tau-\left(nm+m+n\right)\varepsilon)(\log
r)^{\mu_{\log}(A_{l0})}\leq O\left((\log
r)^{\mu_{\log}(A_{l0})-\varepsilon}\right)$ $+(\log
r)^{\mu_{\log}(A_{ab})+\varepsilon}+O\left(\log\left(\log
r\right)\right)+O\left(\left(\log
r\right)^{\rho_{\log}(f)-1+\varepsilon}\right)+(\log
r)^{\lambda_{\log}\left(\frac{1}{A_{l0}}\right)+1+\varepsilon}.$ (3.22)

We may choose sufficiently small $\varepsilon$ satisfying
$0<2\varepsilon<\min\\{\mu_{\log}(A_{l0})-\mu_{\log}(A_{ab}),\mu_{\log}(A_{l0})-\lambda_{\log}\left(\frac{1}{A_{l0}}\right)-1,\frac{\underline{\tau}_{\log}(A_{l0})-\tau}{nm+m+n}\\},$
for all sufficiently large $r\in E_{1}$, by $\left(3.22\right)$ we have

$(\underline{\tau}_{\log}(A_{l0})-\tau-\left(nm+m+n\right)\varepsilon)(\log
r)^{\mu_{\log}(A_{l0})-\varepsilon}\leq\left(\log
r\right)^{\rho_{\log}(f)-1+\varepsilon},$

that is, $\mu_{\log}(A_{l0})-2\varepsilon\leq\rho_{\log}(f)-1$ and since
$\varepsilon>0$ is arbitrary, then $\rho_{\log}(f)\geq\mu_{\log}(A_{l0})+1.$

Case $\left(\mathbf{iv}\right)$: When
$\mu_{\log}(A_{l0})=\mu_{\log}(A_{ab})=\rho_{\log}(S)$ and

$\tau_{3}=\sum_{\rho_{\log}(A_{ij})=\mu_{\log}(A_{l0}),(i,j)\not=(l,0),(a,b)}\tau_{\log}(A_{ij})+\tau_{\log}(F)+\underline{\tau}_{\log}(A_{ab})$
$=\tau_{2}+\tau_{\log}(F)<\underline{\tau}_{\log}(A_{l0}),$
$\tau_{2}=\sum_{\rho_{\log}(A_{ij})=\mu_{\log}(A_{l0}),(i,j)\not=(l,0),(a,b)}\tau_{\log}(A_{ij})+\underline{\tau}_{\log}(A_{ab}).$

Then, by substituting $\left(3.11\right)$, $\left(3.15\right)$,
$\left(3.16\right),$ $\left(3.19\right)$ and $\left(3.20\right)$ into
$\left(3.6\right)$, for all sufficiently large $r\in E_{1}$, we have

$(\underline{\tau}_{\log}(A_{l0})-\tau_{3}-\left(mn+m+n+2\right)\varepsilon)(\log
r)^{\mu_{\log}(A_{l0})}\leq O\left((\log
r)^{\mu_{\log}(A_{l0})-\varepsilon}\right)$ $+O\left(\log\left(\log
r\right)\right)+O\left(\left(\log
r\right)^{\rho_{\log}(f)-1+\varepsilon}\right)$ $+O\left(\left(\log
r\right)^{\rho_{\log}(f)+\varepsilon}\right)+(\log
r)^{\lambda_{\log}\left(\frac{1}{A_{l0}}\right)+1+\varepsilon}.$ (3.23)

Now, we may choose sufficiently small $\varepsilon$ satisfying
$0<2\varepsilon<\min\\{\mu_{\log}(A_{l0})-\lambda_{\log}\left(\frac{1}{A_{l0}}\right)-1,\frac{\underline{\tau}_{\log}(A_{l0})-\tau_{3}}{mn+m+n+2}\\},$
for all sufficiently large $r\in E_{1},$ we deduce from $\left(3.23\right)$
that

$(\underline{\tau}_{\log}(A_{l0})-\tau_{3}-\left(mn+m+n+2\right)\varepsilon)(\log
r)^{\mu_{\log}(A_{l0})-\varepsilon}\leq\left(\log
r\right)^{\rho_{\log}(f)+\varepsilon},$

this means, $\mu_{\log}(A_{l0})-2\varepsilon\leq\rho_{\log}(f)$ and since
$\varepsilon>0$ is arbitrary, then $\rho_{\log}(f)\geq\mu_{\log}(A_{l0}).$

Further, for the homogeneous case, by substituting $\left(3.15\right)$,
$\left(3.16\right),$ $\left(3.19\right)$ and $\left(3.20\right)$ into
$\left(3.13\right)$, for all sufficiently large $r\in E_{1}$, we get

$(\underline{\tau}_{\log}(A_{l0})-\tau_{2}-\left(mn+m+n+1\right)\varepsilon)(\log
r)^{\mu_{\log}(A_{l0})}\leq O\left((\log
r)^{\mu_{\log}(A_{l0})-\varepsilon}\right)$ $+O\left(\log\left(\log
r\right)\right)+O\left(\left(\log
r\right)^{\rho_{\log}(f)-1+\varepsilon}\right)+(\log
r)^{\lambda_{\log}\left(\frac{1}{A_{l0}}\right)+1+\varepsilon}.$ (3.24)

Therefore, for $\varepsilon$ satisfying
$0<2\varepsilon<\min\\{\mu_{\log}(A_{l0})-\lambda_{\log}\left(\frac{1}{A_{l0}}\right)-1,\frac{\underline{\tau}_{\log}(A_{l0})-\tau_{2}}{mn+m+n+1}\\}$
and for all sufficiently large $r\in E_{1}$, by $\left(3.24\right)$ we have

$(\underline{\tau}_{\log}(A_{l0})-\tau_{2}-\left(mn+m+n+1\right)\varepsilon)(\log
r)^{\mu_{\log}(A_{l0})-\varepsilon}\leq\left(\log
r\right)^{\rho_{\log}(f)-1+\varepsilon},$

that is, $\mu_{\log}(A_{l0})-2\varepsilon\leq\rho_{\log}(f)-1$ and since
$\varepsilon>0$ is arbitrary, then $\rho_{\log}(f)\geq\mu_{\log}(A_{l0})+1.$
The proof of Theorem 1.1 is complete.

##  3 Proof of Theorem 1.2

Let $f$ be a meromorphic solution of $\left(1.2\right)$. If $f$ has infinite
logarithmic order, then the result holds. Now, we suppose that
$\rho_{\log}(f)<\infty$. By $\left(3.1\right)$ and Remark 2.1, for
sufficiently large $r$, we have

$m(r,A_{l0})\leq\sum_{i=0,i\neq l,a}^{n}\sum_{j=0}^{m}m(r,A_{ij})+m(r,A_{ab})$
$+\sum_{j=1}^{m}m(r,A_{lj})+\sum_{j=0,j\neq b}^{m}m(r,A_{aj})+\sum_{i=0,i\neq
l,a}^{n}\sum_{j=0}^{m}m\left(r,\frac{f^{(j)}(z+c_{i})}{f(z+c_{i})}\right)$
$+\sum_{i=0,i\neq
l,a}^{n}m\left(r,\frac{f(z+c_{i})}{f(z+c_{l})}\right)+\sum_{j=1}^{m}m\left(r,\frac{f^{(j)}(z+c_{a})}{f(z+c_{a})}\right)+2m\left(r,\frac{f(z+c_{a})}{f(z+c_{l})}\right)$
$+\sum_{j=1}^{m}m\left(r,\frac{f^{(j)}(z+c_{a})}{f(z+c_{a})}\right)+m\left(r,\frac{F(z)}{f(z+c_{l})}\right)+O(1)$
$\leq\sum_{i=0,i\neq
l,a}^{n}\sum_{j=0}^{m}T(r,A_{ij})+T(r,A_{ab})+\sum_{j=1}^{m}T(r,A_{lj})$
$+\sum_{j=0,j\neq b}^{m}T(r,A_{aj})+\sum_{i=0,i\neq
l,a}^{n}\sum_{j=0}^{m}m\left(r,\frac{f^{(j)}(z+c_{i})}{f(z+c_{i})}\right)$
$+\sum_{i=0,i\neq
l,a}^{n}m\left(r,\frac{f(z+c_{i})}{f(z+c_{l})}\right)+\sum_{j=1}^{m}m\left(r,\frac{f^{(j)}(z+c_{a})}{f(z+c_{a})}\right)+2m\left(r,\frac{f(z+c_{a})}{f(z+c_{l})}\right)$
$+\sum_{j=1}^{m}m\left(r,\frac{f^{(j)}(z+c_{a})}{f(z+c_{a})}\right)+T(r,F)+2T(2r,f)+O(1).$
(4.1)

By substituting $\left(3.3\right)$ and $\left(3.4\right)$ into
$\left(4.1\right)$, for any given $\varepsilon>0$ and all sufficiently large
$r$, we obtain

$m(r,A_{l0})\leq\sum_{i=0,i\neq
l,a}^{n}\sum_{j=0}^{m}T(r,A_{ij})+T(r,A_{ab})+\sum_{j=1}^{m}T(r,A_{lj})+\sum_{j=0,j\neq
b}^{m}T(r,A_{aj})$ $+O\left(\log^{+}T\left(2r,f\right)\right)+O\left((\log
r)^{\rho_{\log}(f)-1+\varepsilon}\right)+T(r,F)+2T(2r,f).$ (4.2)

Let us set

$\delta=\delta(\infty,A_{l0})>0.$ (4.3)

Now, we divide this proof into four cases:

Case $\left(\mathbf{i}\right)$: If
$\max\\{\mu_{\log}(A_{ab}),\rho_{\log}(S)\\}<\mu_{\log}(A_{l0}),$ then by the
definition of $\mu_{\log}(A_{l0})$ and $\left(4.3\right)$, for any given
$\varepsilon>0$ and all sufficiently large $r$, we have

$m(r,A_{l0})\geq\frac{\delta}{2}T(r,A_{l0})\geq\frac{\delta}{2}(\log
r)^{\mu_{\log}(A_{l0})-\frac{\varepsilon}{2}}\geq(\log
r)^{\mu_{\log}(A_{l0})-\varepsilon}.$ (4.4)

By substituting $\left(3.10\right),$ $\left(3.11\right)$ and
$\left(4.4\right)$ into $\left(4.2\right)$, for all sufficiently large $r$, we
get

$(\log r)^{\mu_{\log}(A_{l0})-\varepsilon}\leq O\left((\log
r)^{\rho+\varepsilon}\right)+O(\log\left(\log r\right))$ $+O\left((\log
r)^{\rho_{\log}(f)-1+\varepsilon}\right)+O\left((\log
r)^{\rho_{\log}(f)+\varepsilon}\right).$ (4.5)

Now, we choose sufficiently small $\varepsilon$ satisfying
$0<3\varepsilon<\mu_{\log}(A_{l0})-\rho,$ for all sufficiently large $r$, it
follows from $\left(4.5\right)$ that

$(\log r)^{\mu_{\log}(A_{l0})-2\varepsilon}\leq(\log
r)^{\rho_{\log}(f)+\varepsilon},$

this means, $\mu_{\log}(A_{l0})-3\varepsilon\leq\rho_{\log}(f)$ and since
$\varepsilon>0$ is arbitrary, then $\rho_{\log}(f)\geq\mu_{\log}(A_{l0}).$

Similarly, for the homogeneous case, by $\left(1.1\right)$ and
$\left(3.3\right)$ and $\left(3.4\right)$, we obtain

$m(r,A_{l0})\leq\sum_{i=0,i\neq
l,a}^{n}\sum_{j=0}^{m}T(r,A_{ij})+T(r,A_{ab})+\sum_{j=1}^{m}T(r,A_{lj})+\sum_{j=0,j\neq
b}^{m}T(r,A_{aj})$ $+O(\log\left(\log r\right))+O\left((\log
r)^{\rho_{\log}(f)-1+\varepsilon}\right).$ (4.6)

Then, by substituting $\left(3.10\right)$ and $\left(4.4\right)$ into
$\left(4.6\right)$, for all sufficiently large $r$, we have

$(\log r)^{\mu_{\log}(A_{l0})-\varepsilon}\leq O\left((\log
r)^{\rho+\varepsilon}\right)+O(\log\left(\log r\right))+O\left((\log
r)^{\rho_{\log}(f)-1+\varepsilon}\right).$ (4.7)

For the above $\varepsilon$ and all sufficiently large $r$, we deduce from
$\left(4.7\right)$ that

$(\log r)^{\mu_{\log}(A_{l0})-2\varepsilon}\leq(\log
r)^{\rho_{\log}(f)-1+\varepsilon},$

that is, $\mu_{\log}(A_{l0})-3\varepsilon\leq\rho_{\log}(f)-1$ and since
$\varepsilon>0$ is arbitrary, then $\rho_{\log}(f)\geq\mu_{\log}(A_{l0})+1.$

Case $\left(\mathbf{ii}\right)$: If
$\beta=\rho_{\log}(S)<\mu_{\log}(A_{l0})=\mu_{\log}(A_{ab})$ and
$\delta\underline{\tau}_{\log}(A_{l0})>\underline{\tau}_{\log}(A_{ab}),$ then
by the definition of $\underline{\tau}_{\log}(A_{l0})$ and $\left(4.3\right)$,
for any given $\varepsilon>0$ and all sufficiently large $r$, we have

$m(r,A_{l0})\geq(\delta-\varepsilon)T(r,A_{l0})\geq(\delta-\varepsilon)(\underline{\tau}_{\log}(A_{l0})-\varepsilon)(\log
r)^{\mu_{\log}(A_{l0})}$
$\geq\left(\delta\underline{\tau}_{\log}(A_{l0})-(\underline{\tau}_{\log}(A_{l0})+\delta)\varepsilon+\varepsilon^{2}\right)(\log
r)^{\mu_{\log}(A_{l0})}$
$\geq\left(\delta\underline{\tau}_{\log}(A_{l0})-(\underline{\tau}_{\log}(A_{l0})+\delta)\varepsilon\right)(\log
r)^{\mu_{\log}(A_{l0})}.$ (4.8)

By substituting $\left(3.8\right),$ $\left(3.11\right),$ $\left(3.16\right)$
and $\left(4.8\right)$ into $\left(4.2\right)$, for all sufficiently large
$r\in E_{2}$, we get

$\left(\delta\underline{\tau}_{\log}(A_{l0})-\underline{\tau}_{\log}\left(A_{ab}\right)-(\underline{\tau}_{\log}(A_{l0})+\delta+1)\varepsilon\right)(\log
r)^{\mu_{\log}(A_{l0})}\leq O\left((\log r)^{\beta+\varepsilon}\right)$
$+O(\log\left(\log r\right))+O\left((\log
r)^{\rho_{\log}(f)-1+\varepsilon}\right)+O\left((\log
r)^{\rho_{\log}(f)+\varepsilon}\right).$ (4.9)

Now, we choose sufficiently small $\varepsilon$ satisfying
$0<2\varepsilon<\min\\{\mu_{\log}(A_{l0})-\beta,\frac{\delta\underline{\tau}_{\log}(A_{l0})-\underline{\tau}_{\log}(A_{ab})}{\underline{\tau}_{\log}(A_{l0})+\delta+1}\\},$
for all sufficiently large $r\in E_{2}$, by $\left(4.9\right)$, we obtain

$\left(\delta\underline{\tau}_{\log}(A_{l0})-\underline{\tau}_{\log}\left(A_{ab}\right)-(\underline{\tau}_{\log}(A_{l0})+\delta+1)\varepsilon\right)(\log
r)^{\mu_{\log}(A_{l0})-\varepsilon}$ $\leq(\log
r)^{\rho_{\log}(f)+\varepsilon},$

this means, $\mu_{\log}(A_{l0})-2\varepsilon\leq\rho_{\log}(f)$ and since
$\varepsilon>0$ is arbitrary, then $\rho_{\log}(f)\geq\mu_{\log}(A_{l0}).$

Next, for the homogeneous case, by substituting $\left(3.8\right),$
$\left(3.11\right),$ $\left(3.16\right)$ and $\left(4.8\right)$ into
$\left(4.6\right)$, for all sufficiently large $r\in E_{2}$, we have

$\left(\delta\underline{\tau}_{\log}(A_{l0})-\underline{\tau}_{\log}\left(A_{ab}\right)-(\underline{\tau}_{\log}(A_{l0})+\delta+1)\varepsilon\right)(\log
r)^{\mu_{\log}(A_{l0})}\leq O\left((\log r)^{\beta+\varepsilon}\right)$
$+O(\log\left(\log r\right))+O\left((\log
r)^{\rho_{\log}(f)-1+\varepsilon}\right).$ (4.10)

For the above $\varepsilon$ and all sufficiently large $r\in E_{2}$, from
$\left(4.10\right),$ we obtain

$\left(\delta\underline{\tau}_{\log}(A_{l0})-\underline{\tau}_{\log}\left(A_{ab}\right)-(\underline{\tau}_{\log}(A_{l0})+\delta+1)\varepsilon\right)(\log
r)^{\mu_{\log}(A_{l0})-\varepsilon}$ $\leq(\log
r)^{\rho_{\log}(f)-1+\varepsilon},$

that is, $\mu_{\log}(A_{l0})-2\varepsilon\leq\rho_{\log}(f)-1$ and since
$\varepsilon>0$ is arbitrary, then $\rho_{\log}(f)\geq\mu_{\log}(A_{l0})+1.$

Case $\left(\mathbf{iii}\right)$: When
$\mu_{\log}(A_{ab})<\mu_{\log}(A_{l0})=\rho_{\log}(S)$ and

$\tau_{1}=\sum_{\rho_{\log}(A_{ij})=\mu_{\log}(A_{l0}),(i,j)\not=(l,0),(a,b)}\tau_{\log}(A_{ij})+\tau_{\log}(F)<\delta\underline{\tau}_{\log}(A_{l0}).$

Then, by substituting $\left(3.9\right),$ $\left(3.11\right),$
$\left(3.19\right),$ $\left(3.20\right)$ and $\left(4.8\right)$ into
$\left(4.2\right)$, for all sufficiently large $r\in E_{1}$, we get

$(\delta\underline{\tau}_{\log}(A_{l0})-\tau_{1}-\left(\underline{\tau}_{\log}(A_{l0})+\delta+mn+m+n+1\right)\varepsilon)(\log
r)^{\mu_{\log}(A_{l0})}$ $\leq O\left((\log
r)^{\mu_{\log}(A_{l0})-\varepsilon}\right)+(\log
r)^{\mu_{\log}(A_{ab})+\varepsilon}+O(\log\left(\log r\right))$ $+O\left((\log
r)^{\rho_{\log}(f)-1+\varepsilon}\right)+O\left((\log
r)^{\rho_{\log}(f)+\varepsilon}\right).$ (4.11)

We may choose sufficiently small $\varepsilon$ satisfying
$0<2\varepsilon<\min\\{\mu_{\log}(A_{l0})-\mu_{\log}(A_{ab}),\frac{\underline{\tau}_{\log}(A_{l0})-\tau_{1}}{\underline{\tau}_{\log}(A_{l0})+\delta+mn+m+n+1}\\},$
for all sufficiently large $r\in E_{1}$, by $\left(4.11\right)$, we obtain

$(\delta\underline{\tau}_{\log}(A_{l0})-\tau_{1}-\left(\underline{\tau}_{\log}(A_{l0})+\delta+mn+m+n+1\right)\varepsilon)(\log
r)^{\mu_{\log}(A_{l0})-\varepsilon}$ $\leq(\log
r)^{\rho_{\log}(f)+\varepsilon},$

this means, $\mu_{\log}(A_{l0})-2\varepsilon\leq\rho_{\log}(f)$ and since
$\varepsilon>0$ is arbitrary, then $\rho_{\log}(f)\geq\mu_{\log}(A_{l0}).$

Further, for the homogeneous case, by substituting $\left(3.9\right),$
$\left(3.19\right),$ $\left(3.20\right)$ and $\left(4.8\right)$ into
$\left(4.6\right)$, for all sufficiently large $r\in E_{1}$, we get

$(\delta\underline{\tau}_{\log}(A_{l0})-\tau-\left(\underline{\tau}_{\log}(A_{l0})+\delta+nm+m+n\right)\varepsilon)(\log
r)^{\mu_{\log}(A_{l0})}$ $\leq O\left((\log
r)^{\mu_{\log}(A_{l0})-\varepsilon}\right)+(\log
r)^{\mu_{\log}(A_{ab})+\varepsilon}+O(\log\left(\log r\right))+O\left((\log
r)^{\rho_{\log}(f)-1+\varepsilon}\right).$ (4.12)

For $\varepsilon$ sufficiently small satisfying

$0<2\varepsilon<\min\left\\{\mu_{\log}(A_{l0})-\mu_{\log}(A_{ab}),\frac{\underline{\tau}_{\log}(A_{l0})-\tau}{\underline{\tau}_{\log}(A_{l0})+\delta+nm+m+n}\right\\},$

and for all sufficiently large $r\in E_{1}$, from $\left(4.12\right)$ we
conclude

$(\delta\underline{\tau}_{\log}(A_{l0})-\tau-\left(\underline{\tau}_{\log}(A_{l0})+\delta+nm+m+n\right)\varepsilon)(\log
r)^{\mu_{\log}(A_{l0})}\leq(\log r)^{\rho_{\log}(f)-1+\varepsilon},$

that is, $\mu_{\log}(A_{l0})-2\varepsilon\leq\rho_{\log}(f)-1$ and since
$\varepsilon>0$ is arbitrary, then $\rho_{\log}(f)\geq\mu_{\log}(A_{l0})+1.$

Case $\left(\mathbf{iv}\right)$: When
$\mu_{\log}(A_{l0})=\mu_{\log}(A_{ab})=\rho_{\log}(S)$ and

$\tau_{3}=\sum_{\rho_{\log}(A_{ij})=\mu_{\log}(A_{l0}),(i,j)\not=(l,0),(a,b)}\tau_{\log}(A_{ij})+\tau_{\log}(F)+\underline{\tau}_{\log}(A_{ab})<\underline{\tau}_{\log}(A_{l0}).$

Then, by substituting $\left(3.9\right),$ $\left(3.11\right),$
$\left(3.19\right),$ $\left(3.20\right)$ and $\left(4.8\right)$ into
$\left(4.2\right)$, for all sufficiently large $r\in E_{1}$, we get

$(\delta\underline{\tau}_{\log}(A_{l0})-\tau_{3}-\left(\underline{\tau}_{\log}(A_{l0})+\delta+mn+m+n+2\right)\varepsilon)(\log
r)^{\mu_{\log}(A_{l0})}$ $\leq O\left((\log
r)^{\mu_{\log}(A_{l0})-\varepsilon}\right)+O(\log\left(\log
r\right))+O\left((\log r)^{\rho_{\log}(f)-1+\varepsilon}\right)+O\left((\log
r)^{\rho_{\log}(f)+\varepsilon}\right).$ (4.13)

Now, we may choose sufficiently small $\varepsilon$ satisfying
$0<2\varepsilon<\frac{\delta\underline{\tau}_{\log}(A_{l0})-\tau_{3}}{\underline{\tau}_{\log}(A_{l0})+\delta+mn+m+n+2},$
for all sufficiently large $r\in E_{1}$, we deduce from $\left(4.13\right)$
that

$(\delta\underline{\tau}_{\log}(A_{l0})-\tau_{3}-\left(\underline{\tau}_{\log}(A_{l0})+\delta+mn+m+n+2\right)\varepsilon)(\log
r)^{\mu_{\log}(A_{l0})-\varepsilon}$ $\leq(\log
r)^{\rho_{\log}(f)+\varepsilon},$

this means, $\mu_{\log}(A_{l0})-2\varepsilon\leq\rho_{\log}(f)$ and since
$\varepsilon>0$ is arbitrary, then $\rho_{\log}(f)\geq\mu_{\log}(A_{l0}).$

Also for the homogeneous case, by substituting $\left(3.9\right),$
$\left(3.19\right),$ $\left(3.20\right)$ and $\left(4.8\right)$ into
$\left(4.6\right)$, for all sufficiently large $r\in E_{1}$, we have

$(\delta\underline{\tau}_{\log}(A_{l0})-\tau_{2}-\left(\underline{\tau}_{\log}(A_{l0})+\delta+mn+m+n+1)\right)\varepsilon)(\log
r)^{\mu_{\log}(A_{l0})}$ $\leq O\left((\log
r)^{\mu_{\log}(A_{l0})-\varepsilon}\right)+O(\log\left(\log
r\right))+O\left((\log r)^{\rho_{\log}(f)-1+\varepsilon}\right).$ (4.14)

Thus, for sufficiently small $\varepsilon$ satisfying
$0<2\varepsilon<\frac{\delta\underline{\tau}_{\log}(A_{l0})-\tau_{2}}{\underline{\tau}_{\log}(A_{l0})+\delta+mn+m+n+1},$
for all sufficiently large $r\in E_{1}$, from $\left(4.14\right)$ we obtain

$(\delta\underline{\tau}_{\log}(A_{l0})-\tau_{2}-\left(\underline{\tau}_{\log}(A_{l0})+\delta+mn+m+n+1\right)\varepsilon)(\log
r)^{\mu_{\log}(A_{l0})-\varepsilon}$ $\leq(\log
r)^{\rho_{\log}(f)-1+\varepsilon},$

that is, $\mu_{\log}(A_{l0})-2\varepsilon\leq\rho_{\log}(f)-1$ and since
$\varepsilon>0$ is arbitrary, then $\rho_{\log}(f)\geq\mu_{\log}(A_{l0})+1$
which completes the proof of Theorem 1.2.

## Example

The following example is for illustrating the sharpness of some assertions in
Theorem 1.2.

Example 5.1 For Theorem 1.2, we consider the meromorphic function

$f(z)=\frac{1}{z^{5}}$ (5.1)

which is a solution to the delay-differential equation

$A_{20}(z)f(z-2i)+A_{11}(z)f^{\prime}(z+i)+A_{10}(z)f(z+i)$
$+A_{01}(z)f^{\prime}(z)+A_{00}(z)f(z)=F(z),$ (5.2)

where $A_{20}(z)=\frac{2}{3}(z-2i)^{4},$ $A_{11}(z)=2e,$
$A_{10}(z)=\frac{10e}{z+i},$ $A_{01}(z)=\frac{i}{2},$
$A_{00}(z)=\frac{5i}{2z}$ and $F(z)=\frac{2}{3(z-2i)}.$ Obviously, $A_{ij}(z)$
$\left(i=0,1,2,j=0,1\right)$ and $F(z)$ satisfy the conditions in Case
$\left(\text{iii}\right)$ of Theorem 2.1 such that

$\delta(\infty,A_{20})=1>0,$
$\mu_{\log}(A_{11})=0<\max\\{\rho_{\log}(F),\rho_{\log}(A_{ij}),(i,j)\not=(1,1),(2,0)\\}=\mu_{\log}(A_{20})=1$

and

$\sum_{\rho_{\log}(A_{ij})=\mu_{\log}(A_{20}),(i,j)\not=(1,1),(2,0)}\tau_{\log}(A_{ij})+\tau_{\log}(F)=3<\delta\underline{\tau}_{\log}(A_{20})=4.$

We see that $f$ satisfies

$\mu_{\log}(f)=1=\rho_{\log}(A_{20}).$

The meromorphic function $f(z)=\frac{1}{z^{5}}$ is a solution of equation
$\left(5.2\right)$ for the coefficients $A_{20}(z)=3(z-2i)^{7},$
$A_{11}(z)=\frac{1}{z-i},$ $A_{10}(z)=\frac{5}{z^{2}+1},$
$A_{01}(z)=\frac{i}{2},$ $A_{00}(z)=\frac{5i}{2z}$ and $F(z)=3(z-2i)^{2}.$
Clearly, $A_{ij}(z)$ $\left(i=0,1,2,j=0,1\right)$ and $F(z)$ satisfy the
conditions in Case $\left(\text{iv}\right)$ of Theorem 1.2 such that

$\delta(\infty,A_{20})=1>0,$
$\mu_{\log}(A_{11})=\max\\{\rho_{\log}(F),\rho_{\log}(A_{ij}),(i,j)\not=(1,1),(2,0)\\}=\mu_{\log}(A_{20})=1$

and

$\sum_{\rho_{\log}(A_{ij})=\mu_{\log}(A_{20}),(i,j)\not=(1,1),(2,0)}\tau_{\log}(A_{ij})+\tau_{\log}(F)+\underline{\tau}_{\log}(A_{11})=6<\delta\underline{\tau}_{\log}(A_{20})=7.$

We see that $f$ satisfies $\rho_{\log}(f)=1=\mu_{\log}(A_{20}).$

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