1. Introduction
Consider the linear differential equation
\begin{equation}
f^{(k)}+a_{k-1}(z)f^{(k-1)}+\cdots +a_{1}(z)f^{\prime }+a_{0}(z)f=0,
\label{equ1}
\end{equation}
(1)
where \(k\geq 1\) is an integer, \(a_{0}(z),a_{1}(z),\dots ,a_{k-1}(z)\) are
analytic functions in the unit disc \(\mathbb{D}=\{z\in \mathbb{C}:|z|< 1\}\)
and \(a_{0}(z)\not\equiv 0\). The theory of complex differential equations in
the unit disc has been developed since 1980's, see [
1]. In the year
2000, Heittokangas in [
2] firstly investigated the growth and
oscillation theory of Equation (1) when the coefficients \(
a_{0}(z),a_{1}(z),\dots ,a_{k-1}(z)\) are analytic functions in the unit disc
\(\mathbb{D}\) by introducing the definition of the function spaces. His
results also gave some important tools for further investigations on the
theory of meromorphic solutions of Equation (1).
In this article, we investigate the growth of solutions of the Equation (1) when the coefficients \(a_{0}(z),a_{1}(z),\dots ,a_{k-1}(z)\) are
analytic in \(\mathbb{D}\), and we deal with the case that the coefficients
are fast growing in \(\mathbb{D}\). To define the order of fast-growth of
analytic functions, we define inductively for \(r\in \lbrack 0,+\infty ),\) \(
\exp _{0}r=r\), \(\exp _{1}r=e^{r}\) and \(\exp _{n+1}r=\exp \left( \exp
_{n}r\right) ,\) \(n\in \mathbb{N}\). For all \(r\) sufficiently large, we define
\(\log _{0}r=r,\) \(\log _{1}r=\log r\) and \(\log _{n+1}r=\log \left( \log
_{n}r\right) ,\) \(n\in \mathbb{N}\). Also, we need to be familiar with the
fundamental results and the standard notations of the Nevanlinna’s theory on
the complex plane \(\mathbb{C}\) and in the unit disc \(\mathbb{D}\), for more
details on Nevanlinna theory and its applications in complex differential
equations in complex plane and in unit disc, we refer to [2,3, 4, 5, 6, 7].
Before stating our main results, we recall definitions and preliminary
remarks concerning meromorphic and analytic functions in \(\mathbb{D}\). For
the definitions and more discussions, we refer the reader to [7, 8, 9, 10].
Let \(p\geq 1\) be an integer and \(f\) be a meromorphic function in \(\mathbb{D}\)
. Then, the iterated \(p\)-order of \(f\) is defined by
\begin{equation*}
\rho _{p}(f)=\limsup_{r\rightarrow 1^{-}}\frac{\log _{p}^{+}T(r,f)}{\log
\frac{1}{1-r}},
\end{equation*}
where \(\log _{1}^{+}r=\log ^{+}r=\max \{\log r;0\}\), \(\log _{p+1}^{+}r=\log
^{+}\left( \log _{p}^{+}r\right) \) and \(T(r,f)\) is the Nevanlinna
characteristic function. If \(f\) is analytic in \(\mathbb{D}\), then the
iterated \(p\)-order of \(f\) is defined by
\begin{equation*}
\rho _{M,p}(f)=\limsup_{r\rightarrow 1^{-}}\frac{\log _{p+1}^{+}M(r,f)}{\log
\frac{1}{1-r}},
\end{equation*}
where \(M(r,f)=\max \left\{ |f(z)|:|z|=r\right\} .\)
Remark 1.
For \(p=1\), \(\rho _{1}(f)\) is called order, see [2]. And for \(p=2,\)
\(\rho _{2}(f)\) is called hyper-order, see [11].
Remark 2.
It follows by [7, page 205] that if \(f\) is an analytic
function in \(\mathbb{D}\), then we have the inequalities
\begin{equation*}
\rho _{1}(f)\leq \rho _{M,1}(f)\leq \rho _{1}(f)+1
\end{equation*}
which are the best possible in the sense that there are analytic functions \(
g \) and \(h\) such that \(\rho _{1}(g)=\rho _{M,1}(g)\) and \(\rho _{M,1}(h)=\rho
_{1}(h)+1\), see [12]. However, it follows by [4,Proposition 2.2.2]
{laine} that \(\rho _{p}(f)=\rho _{M,p}(f)\) for \(p\geq 2\).
The iterated \(p\)-type of a meromorphic function \(f\) \ in \(\mathbb{D}\) with \(
0< \rho _{p}(f)< +\infty \) is defined by
\begin{equation*}
\tau _{p}(f)=\limsup_{r\rightarrow 1^{-}}\left( 1-r\right) ^{\rho
_{p}(f)}\log _{p-1}^{+}T(r,f),
\end{equation*}
and if \(f\) is analytic in \(\mathbb{D}\) with \(0< \rho _{M,p}(f)< +\infty \),
then the iterated \(p\)-type is defined by
\begin{equation*}
\tau _{M,p}(f)=\limsup_{r\rightarrow 1^{-}}\left( 1-r\right) ^{\rho
_{M,p}(f)}\log _{p}^{+}M(r,f).
\end{equation*}
Remark 3.
It follows by [4,Proposition 2.2.2] that \(\tau
_{p}(f)=\tau _{M,p}(f)\) for \(p\geq 2\).
2. Basic results
Heittokangas
et al. in [
10] proved the following results.
Theorem 1. ([10])
Let \(k\in \mathbb{N}\). If the coefficients \(a_{0}(z),a_{1}(z),
\dots ,a_{k-1}(z)\) are analytic in \(\mathbb{D}\) such that \(\rho
_{M,p}(a_{j})< \rho _{M,p}(a_{0})\) for all \(j=1,\dots ,k-1\), then all
solutions \(f\not\equiv 0\) of \((1)\) satisfy \(\rho _{M,p+1}(f)=\rho
_{M,p}(a_{0})\).
Theorem 2.([10])
Let \(k\in \mathbb{N}\). If the coefficients \(a_{0}(z),a_{1}(z),
\dots ,a_{k-1}(z)\) are analytic in \(\mathbb{D}\) such that \(\rho
_{M,p}(a_{j})\leq \rho _{M,p}(a_{0})\) for all \(j=1,\dots ,k-1\) and
\begin{equation*}
\sum_{\rho _{M,p}(a_{j})=\rho _{M,p}(a_{0})}\tau _{M,p}(a_{j})< \tau
_{M,p}(a_{0}),
\end{equation*}
then all solutions \(f\not\equiv 0\) of \((1)\) satisfy \(\rho
_{M,p+1}(f)=\rho _{M,p}(a_{0})\).
Hamouda in [
13], gave an improvement of Theorem 2 as
follows.
Theorem 3.([13])
Let \(k\in \mathbb{N}\). If the coefficients \(a_{0}(z),a_{1}(z),
\dots ,a_{k-1}(z)\) are analytic in \(\mathbb{D}\) such that \(\rho
_{M,p}(a_{j})\leq \rho _{M,p}(a_{0})\) for all \(j=1,\dots ,k-1\) and
\begin{equation*}
\max \left\{ \tau _{M,p}(a_{j}):\rho _{M,p}(a_{j})=\rho
_{M,p}(a_{0})\right\} < \tau _{M,p}(a_{0}),
\end{equation*}
then all solutions \(f\not\equiv 0\) of \((1)\) satisfy \(\rho
_{M,p+1}(f)=\rho _{M,p}(a_{0})\).
Our proofs depend mainly upon the following lemmas. Before starting these
lemmas, we recall the concept of logarithmic measure. The logarithmic
measure of a set \(S\subset (0,1)\) is given by
\begin{equation*}
lm(S):=\int_{S}\frac{dt}{1-t}.
\end{equation*}
The set \(F\subset \lbrack 0,1)\) in all this paper is not necessarily the
same at each occurrence, but it is always of finite logarithmic measure,
that is \(lm(F)< +\infty \).
To avoid some problems of the exceptional sets, we need the following lemma.
Lemma 1. ([2, 14)
Let \(g:[0,1)\mapsto \mathbb{R}\) and \(h:[0,1)\mapsto \mathbb{R}\)
be monotone non-decreasing functions such that \(g(r)\leq h(r)\) holds outside
of an exceptional set \(F\subset \lbrack 0,1)\) of finite logarithmic measure.
Then there exists a \(d\in (0,1)\) such that if \(s(r)=1-d(1-r)\), then \(
g(r)\leq h(s(r))\) for all \(r\in \lbrack 0,1)\).
Lemma 2.[12, Theorem 3.1]
Let \(k\) and \(j\) be integers satisfying \(k>j\geq 0\), and let \(
\varepsilon >0\) and \(d\in (0,1)\). If \(f\) is a meromorphic in \(\mathbb{D}\)
such that \(f^{(j)}\not\equiv 0\), then
\begin{equation*}
\left\vert \frac{f^{(k)}(z)}{f^{(j)}(z)}\right\vert \leq \left[ \left( \frac{
1}{1-|z|}\right) ^{2+\varepsilon }\max \left\{ \log \frac{1}{1-|z|}
;T(s(|z|),f)\right\} \right] ^{k-j}
\end{equation*}
for \(|z|\not\in F,\) where \(F\subset \lbrack 0,1)\) is a set of finite
logarithmic measure, and where \(s(|z|)=1-d(1-r)\).
Lemma 3. ([10])
\label{lem2}Let \(f\) be a meromorphic function in the unit disc with \(\rho
_{p}(f):=\rho 0\)
be a given constant. Then, there exists a set \(F\subset (0,1)\) of finite
logarithmic measure such that for all \(z\) with \(|z|=r\not\in F\) and for all
integer \(j\geq 1\), we have:
- (i) If \(p=1\), then
\begin{equation}
\left\vert \frac{f^{(j)}(z)}{f(z)}\right\vert \leq \frac{1}{(1-r)^{j(\rho
+2+\varepsilon )}}. \label{conlem3}
\end{equation}
(2)
- (ii) If \(p\geq 2\), then
\begin{equation}
\left\vert \frac{f^{(j)}(z)}{f(z)}\right\vert \leq \exp _{p-1}\left\{ \frac{1
}{(1-r)^{\rho +\varepsilon }}\right\} . \label{conlem4}
\end{equation}
(3)
Lemma 4. ([10])
Let \(a_{0}(z),a_{1}(z),\dots ,a_{k-1}(z)\) be analytic functions
in the unit disc \(\mathbb{D}\). Then, every solution \(f\not\equiv 0\) of the
Equation \((1)\) satisfies
\begin{equation*}
\rho _{p+1}(f)=\rho _{M,p+1}(f)\leq \max \left\{ \rho
_{M,p}(a_{j}):j=0,\dots ,k-1\right\} .
\end{equation*}
Remark 4.
If \(p\geq 2\), then by Remark 2 and Lemma 4, we
obtain that very solution \(f\not\equiv 0\) of the Equation \((1)\)
satisfies
\begin{equation*}
\rho _{p+1}(f)\leq \max \left\{ \rho _{p}(a_{j}):j=0,\dots ,k-1\right\} .
\end{equation*}
Lemma 5. ([2, 3, 7])
Let \(f\) be a meromorphic function in the unit disc \(\mathbb{D}\)
and let \(k\in \mathbb{N}\). Then
\begin{equation*}
m\left( r,\frac{f^{(k)}}{f}\right) =S(r,f),
\end{equation*}
where \(S(r,f)=O\left( \log ^{+}T(r,f)+\log \left( \frac{1}{1-r}\right)
\right) \), possibly outside a set \(F\subset \lbrack 0,1)\) with finite
logarithmic measure.
Lemma 6. ([15])
Let \(f\) be a meromorphic function in the unit
disc \(\mathbb{D}\) for which \(i\left( f\right) =p\geq 1\)
and \(\rho _{p}\left( f\right) =\rho 0,\)
\begin{equation*}
m\left( r,\frac{f^{\left( k\right) }}{f}\right) =O\left( \exp _{p-2}\left\{
\frac{1}{1-r}\right\} ^{\rho +\varepsilon }\right)
\end{equation*}
holds for all \(r\) outside a set \(F\) \(\subset \lbrack 0,1)\)
with \(\int_{F}\frac{dr}{1-r}< +\infty .\)
Lemma 7.
For an integer \(p\geq 2\), let \(f\) be a meromorphic function in \(
\mathbb{D}\) such that \(0< \rho _{p}(f)=\rho < +\infty \) (see Definition 7), \(0< \tau _{p}(f)=\tau
< +\infty \) and \(0< \tau _{p}^{\ast }(f)=\tau ^{\ast }< +\infty \). Then for any
given \(\eta \eta \exp \left\{ \frac{\tau }{\left( 1-r\right) ^{\rho }}
\right\} .
\end{equation*}
Proof.
By the definition of \(\tau _{p}^{\ast }(f)\), there exists an increasing
sequence \(\{r_{m}\}_{m=1}^{+\infty }\subset \lbrack 0,1)\) satisfying \(\frac{1
}{m}+\left( 1-\frac{1}{m}\right) r_{m}< r_{m+1},\;(r_{m}\longrightarrow
1^{-}, \) \(m\longrightarrow +\infty )\) and
\begin{equation*}
\lim_{m\rightarrow +\infty }\frac{\log _{p-2}T(r_{m},f)}{\exp \left\{ \frac{
\tau }{\left( 1-r_{m}\right) ^{\rho }}\right\} }=\tau ^{\ast }.
\end{equation*}
Then, for any given \(0< \varepsilon < \tau ^{\ast }\), there exists a positive
integer \(m_{0}\) such that for all \(m\geq m_{0}\), we have
\begin{equation}
\log _{p-2}T(r_{m},f)>(\tau ^{\ast }-\varepsilon )\exp \left\{ \frac{\tau }{
\left( 1-r_{m}\right) ^{\rho }}\right\} . \label{for1}
\end{equation}
(4)
For \(r\in \left[ r_{m},\frac{1}{m}+\left( 1-\frac{1}{m}\right) r_{m}\right] \), we get
\begin{equation*}
\lim_{m\rightarrow +\infty }\frac{\exp \left\{ \tau \left[ \left( 1-\frac{1}{
m}\right) \left( \frac{1}{1-r}\right) \right] ^{\rho }\right\} }{\exp
\left\{ \frac{\tau }{\left( 1-r\right) ^{\rho }}\right\} }=1.
\end{equation*}
Then for any given \(0< \eta < \tau ^{\ast }-\varepsilon \), there exists a
positive integer \(m_{1}\) such that for all \(m\geq m_{1}\), and for all \(r\in
\left[ r_{m},\frac{1}{m}+\left( 1-\frac{1}{m}\right) r_{m}\right] \), we have
\begin{equation}
\frac{\exp \left\{ \tau \left[ \left( 1-\frac{1}{m}\right) \left( \frac{1}{
1-r}\right) \right] ^{\rho }\right\} }{\exp \left\{ \frac{\tau }{\left(
1-r\right) ^{\rho }}\right\} }>\frac{\eta }{\tau ^{\ast }-\varepsilon }.
\label{for2}
\end{equation}
(5)
By (4) and (5), for all \(m\geq m_{2}=\max \left\{ {
m_{0};m_{1}}\right\} \) and for all \(r\in \left[ r_{m},\frac{1}{m}+\left( 1-
\frac{1}{m}\right) r_{m}\right] \), we have
\begin{eqnarray*}
\log _{p-2}T(r,f)&\geq& \log _{p-2}^{+}T(r_{m},f)>(\tau ^{\ast }-\varepsilon
)\exp \left\{ \frac{\tau }{\left( 1-r_{m}\right) ^{\rho }}\right\}\\
&\geq& (\tau ^{\ast }-\varepsilon )\exp \left\{ \tau \left[ \left( 1-\frac{1}{m
}\right) \left( \frac{1}{1-r}\right) \right] ^{\rho }\right\} >\eta \exp
\left\{ \frac{\tau }{\left( 1-r\right) ^{\rho }}\right\} .
\end{eqnarray*}
Set
\begin{equation*}
E=\bigcup_{m=m_{2}}^{+\infty }\left[ r_{m},\frac{1}{m}+\left( 1-\frac{1}{m}
\right) r_{m}\right] .
\end{equation*}
Then
\begin{equation*}
\int_{E}\frac{dt}{1-t}=\sum_{m=m_{2}}^{+\infty }\int_{r_{m}}^{\frac{1}{m}
+\left( 1-\frac{1}{m}\right) r_{m}}\frac{dt}{1-t}=\sum_{m=m_{2}}^{+\infty
}\log \frac{m}{m-1}=+\infty .
\end{equation*}
By using similar reasoning as in the proof of Lemma 7, we easily
obtain the following lemma.
Lemma 8.
For an integer \(p\geq 2\), let \(f\) be a meromorphic function in
\(\mathbb{D}\) such that \(0< \rho _{M,p}(f)=\rho < +\infty \), \(0< \tau
_{M,p}(f)=\tau < +\infty \) and \(0< \tau _{M,p}^{\ast }(f)=\tau ^{\ast
}< +\infty \). Then for any given \(\eta \eta \exp \left\{ \frac{\tau }{\left( 1-r\right) ^{\rho }}
\right\} .
\end{equation*}
Lemma 9. ([16])
Let \(f\) be a solution of Equation \(\left(
1\right) ,\)where the coefficients \(a_{j}\left( z\right) \) \(\left( j=0,…,k-1\right) \) are analytic functions in the disc
\(\Delta _{R}=\left\{ z\in\mathbb{C}:\left\vert z\right\vert < R\right\} ,\) \(00.\) If \(z_{\theta }=\nu e^{i\theta }\in \Delta _{R}\)
is such that \(a_{j}\left( z_{\theta }\right) \neq 0\) for
some \(j=0,…,k-1,\) then for all \(\nu < r0\) is a constant satisfying
\begin{equation*}
C\leq \left( 1+\varepsilon \right) \underset{j=0,…,k-1}{\max }\left( \frac{
\left\vert f^{\left( j\right) }\left( z_{\theta }\right) \right\vert }{
\left( n_{c}\right) ^{j}\underset{n=0,…,k-1}{\max }\left\vert a_{n}\left(
z_{\theta }\right) \right\vert ^{\frac{j}{k-n}}}\right) .
\end{equation*}
Lemma 10.
Let \(\{a_{j}(z)\}_{0\leq j\leq k-1}\) be analytic functions
in the disc \(\mathbb{D}\) such that \(0< p< \infty \) and \(0< \max \{\rho
_{M,p}(a_{j}):j=1,\dots ,k-1\}\leq \rho _{M,p}\left( a_{0}\right) =\rho
< \infty \) and \(\max \{\tau _{M,p}(a_{j}):j=1,\dots ,k-1\}\leq \tau
_{M,p}\left( a_{0}\right) =\tau < \infty \). Then, every solution \(f\not\equiv
0\) \textit{\ }of the Equation \(\left( 1\right) \) with \(\rho
_{p+1}(f)=\rho \) satisfies \(\tau _{p+1}(f)\leq \tau \).
Proof.
Let \(f\not\equiv 0\) be a solution of \(\left( 1\right) \) with \(\rho
_{p+1}(f)=\rho .\) Let \(\theta _{0}\in \left[ 0,2\pi \right) \) be such that \(
\left\vert f\left( re^{i\theta _{0}}\right) \right\vert =M\left( r,f\right)
. \) By Lemma 9, we have
\begin{eqnarray}
M\left( r,f\right) &\leq& C\exp \left( n_{c}\overset{r}{\underset{\nu }{\int }}
\underset{j=0,…,k-1}{\max }\left\vert a_{j}\left( te^{i\theta }\right)
\right\vert ^{\frac{1}{k-j}}dt\right)\nonumber\\
&\leq& C\exp \left( n_{c}\overset{r}{\underset{\nu }{\int }}\underset{
j=0,…,k-1}{\max }\left( M\left( r,a_{j}\right) \right) ^{\frac{1}{k-j}
}dt\right)\nonumber\\
&\leq& C\exp \left( n_{c}\left( r-\nu \right) \underset{j=0,…,k-1}{\max }
\left\{ M\left( r,a_{j}\right) \right\} \right) . \label{eq3.10Hu}
\end{eqnarray}
(6)
We have \(\max \left\{ \rho _{M,p}(a_{j}):j=0,1,…,k-1\right\} =\rho
_{M,p}\left( a_{0}\right) =\rho \). By the definition of \(\tau _{M,p}(a_{j})\)
, for any given \(\varepsilon >0\) and \(r\rightarrow 1^{-},\) we obtain
\begin{equation}
M(r,a_{j})\leq \exp _{p}\left\{ \frac{\tau _{M,p}(a_{j})+\frac{\varepsilon }{
2}}{\left( 1-r\right) ^{\rho _{M,p}(a_{j})}}\right\}
\leq \exp _{p}\left\{ \frac{\tau +\frac{\varepsilon }{2}}{\left( 1-r\right)
^{\rho }}\right\} \text{ }\left( j=0,1,…,k-1\right) . \label{eq3.12Hu}
\end{equation}
(7)
By (6) and (7), we have for \(r\rightarrow 1^{-}\)
\begin{equation}
M\left( r,f\right) \leq \exp _{p+1}\left\{ \frac{\tau +\varepsilon }{\left(
1-r\right) ^{\rho }}\right\} . \label{eq3.14Hu}
\end{equation}
(8)
Then, it follows from (8), arbitrariness of \(\varepsilon >0\)
that \(\tau _{p+1}(f)=\tau _{M,p+1}(f)\leq \tau .\)
3. Main results
In this article, we aim to answer the following questions:
- What can be said about the growth of solutions of the Equation \((1)\) in the case when \(\rho _{M,p}(a_{j})\leq \rho _{M,p}(a_{0})\) for
all \(j=1,\dots ,k-1\) and
\begin{equation*}
\max \left\{ \tau _{M,p}(a_{j}):\rho _{M,p}(a_{j})=\rho
_{M,p}(a_{0})\right\} \leq \tau _{M,p}(a_{0})?
\end{equation*}
- What happened when we replace \(\rho _{M,p}\) and \(\tau _{M,p}\) by \(\rho
_{p}\) and \(\tau _{p}\)?
As the first result, we give an improvement to Theorems 1 and 2 of [
13].
Theorem 4.
Let \(a_{0}(z),a_{1}(z),\dots ,a_{k-1}(z)\) be meromorphic
functions in the unit disc \(\mathbb{D}\), and \(a_{0}(z)\not\equiv 0\). Suppose
that there exist a point \(\omega \in \partial \mathbb{D}\), a curve \(\gamma \)
tending to \(\omega \) and a set \(F_{1}\subset (0,1)\) of finite logarithmic
measure such that for \(z\in \gamma \) and \(|z|=r\not\in F_{1},\) we have for
the largest integer \(p\geq 1\)
\begin{equation}
\lim_{z\rightarrow \omega }\frac{\displaystyle{\sum_{j=1}^{k-1}\left\vert
a_{j}(z)\right\vert }+1}{\left\vert a_{0}(z)\right\vert }\exp _{p}\left\{
\frac{\lambda }{(1-r)^{\alpha }}\right\} =0 \label{conthm1}
\end{equation}
(9)
for all \(\lambda >0\) and \(\alpha >0\). Then, every nontrivial meromorphic
solution \(f\) of the Equation \((1)\) satisfies \(\rho
_{p+1}(f)=+\infty \).
Proof.
Suppose that \(f\not\equiv 0\) is a solution of the Equation (1) with
\(\rho _{p+1}(f)=\rho < +\infty \). By (1), \(f\) satisfies
\begin{equation}
1\leq \frac{1}{|a_{0}(z)|}\left\vert \frac{f^{(k)}}{f}\right\vert
+\sum_{j=1}^{k-1}\frac{|a_{j}(z)|}{|a_{0}(z)|}\left\vert \frac{f^{(j)}}{f}
\right\vert . \label{11}
\end{equation}
(10)
For \(p\geq 1\), by (3), for all \(z\) satisfying \(|z|=r\not\in F\) (\(
F\) has finite logarithmic measure), we obtain
\begin{equation}
\left\vert \frac{f^{(j)}(z)}{f(z)}\right\vert \leq \exp _{p}\left\{ \frac{1}{
(1-|z|)^{\alpha }}\right\} , \label{12}
\end{equation}
(11)
where \(\alpha >0\) is a constant which depends on \(\rho \), \(\varepsilon \) and
\(j=1,\dots ,k\). By substituting (10) into (11), it yields
\begin{equation}
1\leq \frac{\displaystyle{\sum_{j=1}^{k-1}\left\vert a_{j}(z)\right\vert }+1
}{\left\vert a_{0}(z)\right\vert }\exp _{p}\left\{ \frac{1}{(1-|z|)^{\alpha }
}\right\} . \label{13}
\end{equation}
(12)
By (9), for \(z\in \gamma \) such that \(|z|=r\not\in F_{1}\) (\(
F_{1} \) has finite logarithmic measure), we know that as \(z\rightarrow
\omega \)
\begin{equation}
\frac{\displaystyle{\sum_{j=1}^{k-1}\left\vert a_{j}(z)\right\vert }+1}{
\left\vert a_{0}(z)\right\vert }\exp _{p}\left\{ \frac{1}{(1-|z|)^{\alpha }}
\right\} \longrightarrow 0. \label{14}
\end{equation}
(13)
Thus, for all \(z\in \gamma \) with \(|z|=r\not\in F_{1}\cup F\), by (12)
and (13), we get a contradiction. Hence, every meromorphic solution \(
f\not\equiv 0\) of (1) has an infinite \((p+1)\)-order.
Remark 5.
In [13], under the same hypotheses of Theorem 4, Hamouda
obtained that \(\rho _{p+1}(f)\geq \alpha .\)
In all the next, we consider \(p\in \mathbb{N}\backslash \{1\}\). In trying to
give an answer on the above questions, we prove the following results.
Theorem 5.
Let \(a_{0}(z),a_{1}(z),\dots ,a_{k-1}(z)\) be analytic functions
in the unit disc \(\mathbb{D}\) satisfying \(\rho _{M,p}(a_{j})\leq \rho
_{M,p}(a_{0})=\rho \) \((0< \rho < +\infty )\) and \(\tau _{M,p}(a_{j})\leq \tau
_{M,p}(a_{0})=\tau \) \((0< \tau < +\infty )\) for all \(j=1,\dots ,k-1\). Suppose
that there exist two positive real numbers \(\alpha \) and \(\beta \) with \(
0\leq \beta < \alpha \), such that
\begin{equation}
\left\vert a_{0}(z)\right\vert \geq \exp _{p-1}\left\{ \alpha \exp \frac{
\tau }{\left( 1-r\right) ^{\rho }}\right\} \label{con1}
\end{equation}
(14)
and
\begin{equation}
\left\vert a_{j}(z)\right\vert \leq \exp _{p-1}\left\{ \beta \exp \frac{\tau
}{\left( 1-r\right) ^{\rho }}\right\} ,\text{ }j=1,\dots ,k-1 \label{con2}
\end{equation}
(15)
as \(|z|=r\rightarrow 1^{-}\) for \(r\in E_{1}\) (\(E_{1}\) is of infinite
logarithmic measure). Then, every solution \(f\not\equiv 0\) of the Equation \(
\left( 1\right) \) satisfies \(\rho _{p}\left( f\right) =+\infty ,\) \(
\rho _{p+1}\left( f\right) =\rho \) and \(d^{\rho }\tau \leq \tau
_{p+1}(f)\leq \tau ,\) \(d\in \left( 0,1\right) \).
Proof.
Let \(f\not\equiv 0\) be a solution of the Equation (1). By (1), \(f\) satisfies
\begin{equation}
\left\vert a_{0}(z)\right\vert \leq \left\vert \frac{f^{(k)}}{f}\right\vert
+\sum_{j=1}^{k-1}|a_{j}(z)|\left\vert \frac{f^{(j)}}{f}\right\vert .
\label{pr1}
\end{equation}
(16)
By hypotheses of Theorem 5 and Lemma 4, we know that \(
\rho _{p+1}(f)\leq \rho \). Suppose that \(\rho _{p+1}(f)=\rho _{1}< \rho \).
Then, by (3) for all \(0< \varepsilon < \rho -\rho _{1}\), we have
\begin{equation}
\left\vert \frac{f^{(j)}(z)}{f(z)}\right\vert \leq \exp _{p}\left\{ \frac{1}{
(1-r)^{\rho _{1}+\varepsilon }}\right\} ,\;j=1,\dots ,k, \label{pr2}
\end{equation}
(17)
where \(|z|=r\not\in F\). By substituting (14), (15) and (17) into (16) we obtain
\begin{equation}
\exp _{p-1}\left\{ \alpha \exp \frac{\tau }{\left( 1-r\right) ^{\rho }}
\right\} \leq k\exp _{p-1}\left\{ \beta \exp \frac{\tau }{\left( 1-r\right)
^{\rho }}\right\} \exp _{p}\left\{ \frac{1}{(1-r)^{\rho _{1}+\varepsilon }}
\right\} , \label{pr3}
\end{equation}
(18)
for all \(r\in E_{1}\backslash F\). Hence, we get
\begin{equation}
(\alpha -\beta )\exp \left\{ \frac{\tau }{\left( 1-r\right) ^{\rho }}
\right\} \leq \exp \left\{ \frac{1}{(1-r)^{\rho _{1}+\varepsilon }}\right\}
+C_{1} \label{pr4}
\end{equation}
(19)
for some constant \(C_{1}>0,\) which is a contradiction as \(|z|=r\rightarrow
1^{-}\), \(r\in E_{1}\backslash F\), since \(\alpha >\beta \geq 0\) and \(\rho
>\rho _{1}+\varepsilon \). Thus, \(\rho _{p+1}(f)=\rho \).
Now, by Lemma 2, we have for \(j=1,\dots ,k\)
\begin{equation}
\left\vert \frac{f^{(j)}(z)}{f(z)}\right\vert \leq \left[ \left( \frac{1}{
1-|z|}\right) ^{2+\varepsilon }\max \left\{ \log \frac{1}{1-|z|}
;T(s(|z|),f)\right\} \right] ^{k} \label{pr5}
\end{equation}
(20)
for all \(r\not\in F_{2}\cup \lbrack 0,1]\). From, (14), (15),
(16) and (20), we obtain
\begin{equation}
\exp _{p-1}\left\{ \alpha \exp \frac{\tau }{\left( 1-r\right) ^{\rho }}
\right\} \leq k\exp _{p-1}\left\{ \beta \exp \frac{\tau }{\left( 1-r\right)
^{\rho }}\right\} \left( \frac{1}{1-r}\right) ^{\left( 2+\varepsilon \right)
k}T^{k}(s(r),f) \label{pr6}
\end{equation}
(21)
for all \(r\in E_{1}\backslash (F_{2}\cup \lbrack 0,1])\). Hence
\begin{equation}
\log (\alpha -\beta )+\frac{\tau }{\left( 1-r\right) ^{\rho }}\leq \log _{p}
\frac{1}{1-r}+\log _{p}T(s(r),f)+C_{2} \label{pr7}
\end{equation}
(22)
for some constant \(C_{2}>0\) and for all \(r\in E_{1}\backslash (F_{2}\cup
\lbrack 0,1])\). Setting \(R=s(r)=1-d(1-r)\), \(d\in (0,1)\). We have \(1-r=\frac{
1-R}{d},\) \(d\in (0,1)\). Then by Lemma 1, we obtain we obtain for \(
R\longrightarrow 1^{-}\)
\begin{equation}
\log (\alpha -\beta )+\frac{d^{\rho }\tau }{\left( 1-R\right) ^{\rho }}\leq
\log _{p}\frac{d}{1-R}+\log _{p}T(R,f)+C_{2}. \label{pr8}
\end{equation}
(23)
Since \(0< \rho _{p+1}(f)=\rho < \infty ,\) from (23) we deduce that
\begin{equation*}
\tau _{p+1}(f)=\underset{R\rightarrow 1^{-}}{\lim \sup }\frac{\log
_{p}^{+}T(R,f)}{\frac{1}{\left( 1-R\right) ^{\rho }}}\geq d^{\rho }\tau .
\end{equation*}
By Lemma 10, we conclude that \(d^{\rho }\tau \leq \tau _{p+1}(f)\leq
\tau .\)
Theorem 6.
Let \(a_{0}(z),a_{1}(z),\dots ,a_{k-1}(z)\) be analytic functions
in the unit disc \(\mathbb{D}\) satisfying \(\rho _{p}(a_{j})\leq \rho
_{p}(a_{0})=\rho \) \((0< \rho < +\infty )\) and \(\tau _{p}(a_{j})\leq \tau
_{p}(a_{0})=\tau \) \((0< \tau < +\infty )\) for all \(j=1,\dots ,k-1\). Suppose
that there exist two positive real numbers \(\alpha \) and \(\beta \) with \(
0\leq \beta < \alpha \), such that
\begin{equation}
m(r,a_{0})\geq \exp _{p-2}\left\{ \alpha \exp \frac{\tau }{\left( 1-r\right)
^{\rho }}\right\} \label{con3}
\end{equation}
(24)
and
\begin{equation}
m(r,a_{j})\leq \exp _{p-2}\left\{ \beta \exp \frac{\tau }{\left( 1-r\right)
^{\rho }}\right\} ,\text{ }j=1,\dots ,k-1 \label{con4}
\end{equation}
(25)
as \(|z|=r\rightarrow 1^{-}\) for \(r\in E_{2}\) (\(E_{2}\) is of infinite
logarithmic measure). Then, every solution \(f\not\equiv 0\) of the Equation \(
\left( 1\right) \) satisfies \(\ \rho _{p}\left( f\right) =+\infty ,\)
\(\rho _{p+1}\left( f\right) =\rho \) and \(d^{\rho }\tau \leq \tau
_{p+1}(f)\leq \tau ,\) \(d\in \left( 0,1\right) \).
Proof.
Let \(f\not\equiv 0\) be a solution of the Equation (1). By (1) we can write
\begin{equation}
a_{0}(z)=-\left( \frac{f^{(k)}}{f}+\sum_{j=1}^{k-1}a_{j}(z)\frac{f^{(j)}}{f}
\right) . \label{21}
\end{equation}
(26)
By hypotheses of Theorem 6 and Lemma 4, we know that \(
\rho _{p+1}(f)\leq \rho \). Suppose that \(\rho _{p+1}(f)=\rho _{1}< \rho \).
Then by Lemma 6, for all \(0< \varepsilon < \rho -\rho _{1}\) and for
all \(|z|=r\notin F\), we have for \(j=1,\dots ,k\)
\begin{equation}
m\left( r,\frac{f^{(j)}}{f}\right) =O\left( \exp _{p-1}\left\{ \frac{1}{
(1-r)^{\rho _{1}+\varepsilon }}\right\} \right) . \label{22}
\end{equation}
(27)
Now, let \(p\geq 2\), it follows by (24), (25), (26) and
(27) that
\begin{eqnarray}
\exp _{p-2}\left\{ \alpha \exp \frac{\tau }{\left( 1-r\right) ^{\rho }}
\right\} &\leq& m(r,a_{0})\notag\\&\leq& \sum_{j=1}^{k-1}m(r,a_{j})+\sum_{j=1}^{k-1}m\left( r,\frac{f^{(j)}}{f}
\right) +O\left( 1\right)\notag\\
&\leq& (k-1)\exp _{p-2}\left\{ \beta \exp \frac{\tau }{\left( 1-r\right)
^{\rho }}\right\} +M\exp _{p-1}\left\{ \frac{1}{(1-r)^{\rho _{1}+\varepsilon
}}\right\} \label{24}
\end{eqnarray}
(28)
holds for all \(z\) satisfying \(|z|=r\in E_{2}\backslash F\) as \(r\rightarrow
+\infty ,\) and \(M>0\) is some constant. Hence, from (\ref{24}) we obtain
\begin{equation*}
(\alpha -\beta )\exp \left\{ \frac{\tau }{\left( 1-r\right) ^{\rho }}
\right\} \leq \exp \left\{ \frac{1}{(1-r)^{\rho _{1}+\varepsilon }}\right\}
+C_{3}
\end{equation*}
for some constant \(C_{3}>0,\) which is a contradiction as \(|z|=r\rightarrow
1^{-}\), \(r\in E_{2}\backslash F\), since \(\alpha >\beta \geq 0\) and \(\rho
>\rho _{1}+\varepsilon \). Thus, \(\rho _{p+1}(f)=\rho \).
Now, it follows by (24), (25), (26) and Lemma 5 that
\begin{eqnarray}
\exp _{p-2}\left\{ \alpha \exp \frac{\tau }{\left( 1-r\right) ^{\rho }}
\right\} &\leq& m(r,a_{0})\notag\\
&\leq& \sum_{j=1}^{k-1}m(r,a_{j})+\sum_{j=1}^{k-1}m\left( r,\frac{f^{(j)}}{f}
\right) +O\left( 1\right)\notag\\
&\leq& (k-1)\exp _{p-2}\left\{ \beta \exp \frac{\tau }{\left( 1-r\right)
^{\rho }}\right\} +O\left( \log ^{+}T(r,f)+\log \left( \frac{1}{1-r}\right)
\right) \label{25}
\end{eqnarray}
(29)
for all sufficiently large \(|z|=r\in E_{2}\backslash F\). Then, for all
sufficiently large \(|z|=r\in E_{2}\backslash F\)
\begin{equation}
\log (\alpha -\beta )+\frac{\tau }{\left( 1-r\right) ^{\rho }}\leq \log
_{p}^{+}T(r,f)+\log _{p}\left( \frac{1}{1-r}\right) +C_{4} \label{26}
\end{equation}
(30)
for some constant \(C_{4}>0.\) Then by Lemma 1, we obtain for \(
|z|=r\in E_{2},\) \(s\left( r\right) \rightarrow 1^{-}\)
\begin{equation}
\log (\alpha -\beta )+\frac{\tau }{\left( 1-r\right) ^{\rho }}\leq \log
_{p}^{+}T(s\left( r\right) ,f)+\log _{p}\left( \frac{1}{1-s\left( r\right) }
\right) +C_{4}, \label{27}
\end{equation}
(31)
where \(s(r)=1-d(1-r)\), \(d\in (0,1)\). Hence, by (31) we obtain
\begin{equation*}
\tau _{p+1}(f)=\underset{s\left( r\right) \rightarrow 1^{-}}{\lim \sup }
\frac{\log _{p}^{+}T(s\left( r\right) ,f)}{\frac{1}{\left( 1-s\left(
r\right) \right) ^{\rho }}}\geq d^{\rho }\tau .
\end{equation*}
By Lemma 10, we conclude that \(d^{\rho }\tau \leq \tau _{p+1}(f)\leq
\tau .\)
Hamouda in [
17], to study the growth of meromorphic solutions of
differential equations with finite iterated \(p\)-order in complex plane,
introduced new type of growth (see[
17, p. 46]). According to the
definition of this new type of growth, we introduce a new definition of type
of growth that we note \(\tau _{p}^{\ast }(f)\) related to iterated \(p\)-order
of meromorphic function \(f\) in the unit disc, as follows.
Definition 7.
For \(p\geq 2,\) let \(f\) be a meromorphic function of finite iterated \(p\)
-order in \(\mathbb{D}\) such that \(0< \rho _{p}\left( f\right) =\rho < +\infty \)
and \(0< \tau _{p}\left( f\right) =\tau < +\infty \), we define \(\tau _{p}^{\ast
}(f)\) by
\begin{equation*}
\tau _{p}^{\ast }(f)=\limsup_{r\rightarrow 1^{-}}\frac{\log _{p-2}^{+}T(r,f)
}{\exp \left\{ \frac{\tau }{\left( 1-r\right) ^{\rho }}\right\} }.
\end{equation*}
If \(f\)\ is an analytic function in \(\mathbb{D}\) with \(0< \tau _{M,p}\left(
f\right) =\tau _{M}< +\infty \), we also define
\begin{equation*}
\tau _{M,p}^{\ast }(f)=\limsup_{r\rightarrow 1^{-}}\frac{\log
_{p-1}^{+}M(r,f)}{\exp \left\{ \frac{\tau _{M}}{\left( 1-r\right) ^{\rho }}
\right\} }.
\end{equation*}
The following theorems improves and extends Theorems 2 and 3.
Theorem 8.
Let \(a_{0}(z),a_{1}(z),\dots ,a_{k-1}(z)\) be analytic functions
in the unit disc \(\mathbb{D}\) satisfying \(\rho _{M,p}(a_{j})\leq \rho
_{M,p}(a_{0})=\rho \) \((0< \rho < +\infty )\) and \(\tau _{M,p}(a_{j})\leq \tau
_{M,p}(a_{0})=\tau \) \((0< \tau < +\infty )\) for all \(j=1,\dots ,k-1\) and \
\begin{equation*}
\max \left\{ \tau _{M,p}^{\ast }(a_{j}):j=1,\dots ,k-1\right\} < \tau
_{M,p}^{\ast }(a_{0}).
\end{equation*}
Then all solutions \(f\not\equiv 0\) of \((1)\) satisfy \(\ \rho
_{p}\left( f\right) =+\infty ,\) \(\rho _{p+1}\left( f\right) =\rho \) and \(
d^{\rho }\tau \leq \tau _{p+1}(f)\leq \tau ,\) \(d\in \left( 0,1\right) \).
Proof.
Suppose that all coefficients of the Equation (1) satisfy the
hypotheses of Theorem 8. Now, let \(\alpha \) and \(\beta \) be two
real numbers such that
\begin{equation*}
\max \left\{ \tau _{M,p}^{\ast }(a_{j}):j=1,\dots ,k-1\right\} < \beta
< \alpha < \tau _{M,p}^{\ast }(a_{0}).
\end{equation*}
Because all coefficients are analytic, then for \(r\longrightarrow 1^{-}\)
\begin{equation}
|a_{j}(z)|\leq \exp _{p-1}\left\{ \beta \exp \frac{\tau }{\left( 1-r\right)
^{\rho }}\right\} ,\quad j=1,\dots ,k-1 \label{113}
\end{equation}
(32)
and by Lemma 8, we have
\begin{equation}
M\left( r,a_{0}\right) =|a_{0}(z)|>\exp _{p-1}\left\{ \alpha \exp \frac{\tau
}{\left( 1-r\right) ^{\rho }}\right\} \label{114}
\end{equation}
(33)
for all \(r\in E\) (\(E\) is a set of infinite logarithmic measure). \ From (32) and (33), and by Theorem 5, we obtain the result.
Theorem 9.
Let \(a_{0}(z),a_{1}(z),\dots ,a_{k-1}(z)\) be analytic functions
in the unit disc \(\mathbb{D}\) satisfying \(\rho _{p}(a_{j})\leq \rho
_{p}(a_{0})=\rho \) \((0< \rho < +\infty )\), \(\tau _{p}(a_{j})\leq \tau
_{p}(a_{0})=\tau \) \((0< \tau < +\infty )\) for all \(j=1,\dots ,k-1\) and \
\begin{equation*}
\max \left\{ \tau _{p}^{\ast }(a_{j}):j=1,\dots ,k-1\right\} < \tau
_{p}^{\ast }(a_{0}).
\end{equation*}
Then all solutions \(f\not\equiv 0\) of \((1)\) satisfy \(\ \rho
_{p}\left( f\right) =+\infty ,\) \(\rho _{p+1}\left( f\right) =\rho \) and \(
d^{\rho }\tau \leq \tau _{p+1}(f)\leq \tau ,\) \(d\in \left( 0,1\right) \).
Proof.
Suppose that all coefficients of the Equation (1) satisfy the
hypotheses of Theorem 9. Now, let \(\alpha \) and \(\beta \) be two
real numbers such that
\begin{equation*}
\max \left\{ \tau _{p}^{\ast }(a_{j}):j=1,\dots ,k-1\right\} < \beta < \alpha
< \tau _{p}^{\ast }(a_{0}).
\end{equation*}
Since all coefficients are analytic, then for \(r\longrightarrow 1^{-}\)
\begin{equation}
m\left( r,a_{j}\right) \leq \exp _{p-2}\left\{ \beta \exp \frac{\tau }{
\left( 1-r\right) ^{\rho }}\right\} ,\quad j=1,\dots ,k-1 \label{115}
\end{equation}
(34)
and by Lemma 7, we have
\begin{equation}
T\left( r,a_{0}\right) =m\left( r,a_{0}\right) >\exp _{p-2}\left\{ \alpha
\exp \frac{\tau }{\left( 1-r\right) ^{\rho }}\right\} \label{116}
\end{equation}
(35)
for all \(r\in E\) (\(E\) is a set of infinite logarithmic measure). From (34) and (35), and by Theorem 6, we obtain the result.
For some related results in the whole complex plane, see [
18].
Acknowledgments
The authors would like to thank the referee for his/her valuables remarks,
which led to an improvement of the presentation of this paper. This paper is
supported by University of Mostaganem (UMAB) (PRFU Project Code
C00L03UN270120180005).
Author Contributions
All authors contributed equally to the writing of this paper. All authors read and approved the
final manuscript.
Competing Interests
The authors do not have any competing interests in the manuscript.
