\documentclass[reqno]{amsart} \usepackage{hyperref} \usepackage{amssymb} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2013 (2013), No. 16, pp. 1--14.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2013 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2013/16\hfil Differential polynomials] {Growth and oscillation of differential polynomials generated by complex differential equations} % Some results on certain differential polynomials \author[Z. Latreuch, B. Bela\"idi \hfil EJDE-2013/??\hfilneg] {Zinela\^{a}bidine Latreuch, Benharrat Bela\"idi} \address{Zinela\^{a}bidine Latreuch\newline Department of Mathematics, Laboratory of Pure and Applied Mathematics, University of Mostaganem (UMAB), B. P. 227 Mostaganem, Algeria} \email{z.latreuch@gmail.com} \address{Benharrat Bela\"idi \newline Department of Mathematics, Laboratory of Pure and Applied Mathematics, University of Mostaganem (UMAB), B. P. 227 Mostaganem, Algeria} \email{belaidi@univ-mosta.dz} \thanks{Submitted July 27, 2012. Published January 21, 2013.} \subjclass[2000]{34M10, 30D35} \keywords{Linear differential equations; finite order; hyper-order; \hfill\break\indent sequence of zeros; exponent of convergence; hyper-exponent of convergence} \begin{abstract} The main purpose of this article is to study the controllability of solutions to the linear differential equation $$ f^{(k)}+A(z) f=0\quad (k\geqslant 2) . $$ We study the growth and oscillation of higher-order differential polynomials with meromorphic coefficients generated by solutions of the above differential equation. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{definition}[theorem]{Definition} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction and main results} In this article, we assume that the reader is familiar with the fundamental results and the standard notations of the Nevanlinna's value distribution theory \cite{h1,y1}. In addition, we will use $\lambda (f) $ and $\overline{\lambda }(f) $ to denote respectively the exponents of convergence of the zero-sequence and distinct zeros of a meromorphic function $f$, $\rho (f) $ to denote the order of growth of $f$. A meromorphic function $\varphi (z) $ is called a small function with respect to $f(z) $ if $T(r,\varphi ) =o(T(r,f) ) $ as $r\to +\infty $ except possibly a set of $r$ of finite linear measure, where $T(r,f) $ is the Nevanlinna characteristic function of $f$. \begin{definition}[\cite{k1,y1}] \label{def1.1}\rm Let $f$ be a meromorphic function. Then the hyper-order $\rho _2(f)$ of $f(z)$ is defined as \[ \rho _2(f) =\limsup_{r\to +\infty } \frac{\log \log T(r,f) }{\log r}. \] \end{definition} \begin{definition}[\cite{h1,l3}] \label{def1.2}\rm The type of a meromorphic function $f$ of order $\rho $ $(0<\rho <\infty ) $ is defined as \[ \tau (f) =\limsup_{r\to +\infty }\frac{T(r,f) }{r^{\rho }}. \] \end{definition} \begin{definition}[\cite{c3,y1}] \label{def1.3}\rm Let $f$ be a meromorphic function. Then the hyper-exponent of convergence of zeros sequence of $f(z) $ is defined as \[ \lambda _2(f) =\limsup_{r\to +\infty }\frac{\log \log N(r,\frac{1}{f}) }{\log r}, \] where $N(r,\frac{1}{f}) $ is the counting function of zeros of $f(z) $ in $\{ z:|z| 0,\quad \rho (A) =\rho <+\infty . $$ Then every meromorphic solution $f\not\equiv 0$ of \eqref{e1.1} satisfies that $f, f',f'',\dots ,f^{(k) }$ have infinitely many fixed points and \begin{gather*} \overline{\lambda }(f^{(j) }-z) =\rho (f) =+\infty ,\quad (j=0,1,\dots ,k),\\ \overline{\lambda }_2(f^{(j) }-z) =\rho _2(f) =\rho \quad (j=0,1,\dots ,k) . \end{gather*} \end{theorem} Let $\mathcal{L}(\mathbf{G}) $ denote a differential subfield of the field $\mathcal{M}(\mathbf{G}) $ of meromorphic functions in a domain $\mathbf{G}\subset\mathbb{C}$. If $\mathbf{G}=\mathbb{C}$, we simply denote $\mathcal{L}$ instead of $\mathcal{L}(\mathbb{C})$. Special case of such differential subfield \[ \mathcal{L}_{p+1,\rho }\text{ }\mathbf{=}\{ g\text{ meromorphic: }\rho _{p+1}(g) <\rho \} , \] where $\rho $ is a positive constant. Laine and Rieppo \cite{l2} investigated the fixed points and iterated order of the second order differential equation \eqref{e1.3} and obtained the following result. \begin{theorem}[\cite{l2}] \label{thmD} Let $A(z)$ be a transcendental meromorphic function of finite iterated order $\rho _{p}(A) =\rho >0$ such that $\delta (\infty ,A) =\delta >0$, and let $f$ be a transcendental meromorphic solution of equation \eqref{e1.3}. Suppose, moreover, that either: \begin{itemize} \item[(i)] all poles of $f$ are of uniformly bounded multiplicity or that \item[(ii)] $\delta (\infty ,f) >0$. \end{itemize} Then $\rho _{p+1}(f) =\rho _{p}(A) =\rho $. Moreover, let \begin{equation} P[ f] =P(f,f',\dots ,f^{(m) }) =\sum_{j=0}^{m}p_{j}f^{(j) } \label{e1.4} \end{equation} be a linear differential polynomial with coefficients $p_{j}\in \mathcal{L}_{p+1,\rho }$, assuming that at least one of the coefficients $p_{j}$ does vanish identically. Then for the fixed points of $P[ f] $, we have $\overline{\lambda }_{p+1}(P[ f] -z) =\rho$, provided that neither $P[ f] $ nor $P[ f] -z$ vanishes identically. \end{theorem} \begin{remark}[{\cite[p. 904]{l2}}] \label{rmk1.1}\rm In Theorem \ref{thmD}, in order to study $P[ f] $, the authors consider $m\leqslant 1$. Indeed, if $m\geqslant 2$, we obtain, by repeated differentiation of \eqref{e1.3}, that $f^{(k)}=q_{k,0}f+q_{k,1}f'$, $q_{k,0}$, $q_{k,1} \in \mathcal{L}_{p+1,\rho }$ for $k=2,\dots ,m$. Substitution into \eqref{e1.4} yields the required reduction. \end{remark} The main purpose of this paper is to study the growth and oscillation of the differential polynomial \eqref{e1.2} generated by meromorphic solutions of equation \eqref{e1.1}. The method used in the proofs of our theorems is simple, and different, from the method in Laine and Rieppo \cite{l2}. Before we state our results, we define the sequence of functions $\alpha _{i,j}$ $(j=0,\dots ,k-1)$ by \begin{equation} \alpha _{i,j}=\begin{cases} \alpha _{i,j-1}'+\alpha _{i-1,j-1},&\text{for }i=1,\dots ,k-1,\\ \alpha _{0,j-1}'-A\alpha _{k-1,j-1}, &\text{for }i=0 \end{cases} \label{e1.5} \end{equation} and \begin{equation} \alpha _{i,0}=\begin{cases} d_{i}, &\text{for }i=1,\dots ,k-1, \\ d_0-d_{k}A,&\text{for }i=0. \end{cases} \label{e1.6} \end{equation} We define also \begin{equation} h=\begin{vmatrix} \alpha _{0,0} & \alpha _{1,0} & \dots & \alpha _{k-1,0} \\ \alpha _{0,1} & \alpha _{1,1} & \dots & \alpha _{k-1,1} \\ \vdots & & & \vdots \\ \alpha _{0,k-1} & \alpha _{1,k-1} & \dots & \alpha _{k-1,k-1} \end{vmatrix} \label{e1.7} \end{equation} and \begin{equation} \psi (z) =C_0\varphi +C_1\varphi '+\dots +C_{k-1}\varphi ^{(k-1) }, \label{e1.8} \end{equation} where $C_{j}$ $(j=0,\dots ,k-1) $ are finite order meromorphic functions depending on $\alpha _{i,j}$ and $\varphi \not\equiv 0$ is a meromorphic function with $\rho (\varphi ) <\infty$. \begin{theorem} \label{thm1.1} Let $A(z) $ be a meromorphic function of finite order. Let $d_{j}(z) $ $(j=0,1,\dots ,k) $ be finite order meromorphic functions that are not all vanishing identically such that $h\not\equiv 0$. If $f(z) $ is an infinite order meromorphic solution of \eqref{e1.1} with $\rho _2(f)=\rho$, then the differential polynomial \eqref{e1.2} satisfies \[ \rho (g_{f}) =\rho (f) =\infty \] and \[ \rho _2(g_{f}) =\rho _2(f) =\rho . \] Furthermore, if $f$ is a finite order meromorphic solution of \eqref{e1.1} such that \begin{equation} \rho (f) >\max \{ \rho (A) ,\rho (d_{j}) \text{ }(j=0,1,\dots ,k) \} , \label{e1.9} \end{equation} then \[ \rho (g_{f}) =\rho (f) . \] \end{theorem} \begin{remark} \label{rmk1.2}\rm In Theorem \ref{thm1.1}, if we do not have the condition $h\not\equiv 0$, then the conclusions of Theorem \ref{thm1.1} cannot hold. For example, if we take $d_{k}=1,d_0=A$ and $d_{j}\equiv 0$ $(j=1,\dots ,k-1) $, then $h\equiv 0$. It follows that $g_{f}\equiv 0$ and $\rho (g_{f}) =0$. So, if $f(z) $ is an infinite order meromorphic solution of \eqref{e1.1}, then $\rho (g_{f})=0<\rho (f) =\infty $, and if $f$ is a finite order meromorphic solution of \eqref{e1.1} such that \eqref{e1.9} holds, then $\rho (g_{f}) =0<\rho (f) $. \end{remark} \begin{theorem} \label{thm1.2} Under the hypotheses of Theorem \ref{thm1.1}, let $\varphi (z) \not\equiv 0$ be a meromorphic function with finite order such that $\psi (z) $ is not a solution of \eqref{e1.1}. If $f(z) $ is an infinite order meromorphic solution of \eqref{e1.1} with $\rho _2(f) =\rho$, then the differential polynomial \eqref{e1.2} satisfies \begin{gather*} \overline{\lambda }(g_{f}-\varphi ) =\lambda ( g_{f}-\varphi ) =\rho (f) =\infty, \\ \overline{\lambda }_2(g_{f}-\varphi ) =\lambda _2( g_{f}-\varphi ) =\rho _2(f) =\rho . \end{gather*} Furthermore, if $f$ is a finite order meromorphic solution of \eqref{e1.1} such that \begin{equation} \rho (f) >\max \{ \rho (A) ,\rho (\varphi) ,\rho (d_{j})\quad (j=0,1,\dots ,k)\} , \label{e1.10} \end{equation} then \[ \overline{\lambda }(g_{f}-\varphi ) =\lambda (g_{f}-\varphi ) =\rho (f) . \] \end{theorem} \begin{corollary} \label{coro1.1} Let $A(z) $ be a transcendental entire function of finite order and let $d_{j}(z) $ $(j=0,1,\dots ,k) $ be finite order entire functions that are not all vanishing identically such that $h\not\equiv 0$. If $f\not\equiv 0$ is a solution of \eqref{e1.1}, then the differential polynomial \eqref{e1.2} satisfies \begin{gather*} \rho (g_{f}) =\rho (f) =\infty,\\ \rho _2(g_{f}) =\rho _2(f) =\rho (A) =\rho . \end{gather*} \end{corollary} \begin{corollary} \label{coro1.2} Under the hypotheses of Corollary \ref{coro1.1}, let $\varphi (z) \not\equiv 0$ be an entire function with finite order such that $\psi (z) \not\equiv 0$. Then the differential polynomial \eqref{e1.2} satisfies \begin{gather*} \overline{\lambda }(g_{f}-\varphi ) =\lambda (g_{f}-\varphi ) =\rho (f) =\infty,\\ \overline{\lambda }_2(g_{f}-\varphi ) =\lambda _2(g_{f}-\varphi ) =\rho _2(f) =\rho (A) . \end{gather*} \end{corollary} \begin{corollary} \label{coro1.3} Let $A(z) $ be a nonconstant polynomial and for $j=0,1,\dots ,k$ let $d_{j}(z) $ be nonconstant polynomials that are not all vanishing identically such that $h\not\equiv 0$. If $f\not\equiv 0$ is a solution of \eqref{e1.1}, then the differential polynomial \eqref{e1.2} satisfies \[ \rho (g_{f}) =\rho (f) =\frac{\deg (A) +k}{k}. \] \end{corollary} \begin{corollary} \label{coro1.4} Let $A(z) $ be a transcendental meromorphic function of finite order $\rho (A)>0 $ such that $\delta (\infty ,A) =\delta >0$, and let $f\not\equiv 0$ be a meromorphic solution of \eqref{e1.1}. Suppose, moreover, that either: \begin{itemize} \item[(i)] all poles of $f$ are uniformly bounded multiplicity, or \item[(ii)] $\delta (\infty ,f) >0$. \end{itemize} Let $d_{j}(z) $ $(j=0,1,\dots ,k) $ be finite order meromorphic functions that are not all vanishing identically such that $h\not\equiv 0$. Then the differential polynomial \eqref{e1.2} satisfies $\rho (g_{f}) =\rho (f) =\infty $ and $\rho_2(g_{f}) =\rho _2(f) =\rho (A) $. \end{corollary} \begin{corollary} \label{coro1.5} Under the hypotheses of Corollary \ref{coro1.4}, let $\varphi (z) \not\equiv 0$ be a meromorphic function with finite order such that $\psi (z) \not\equiv 0$. Then the differential polynomial \eqref{e1.2} satisfies \begin{gather*} \overline{\lambda }(g_{f}-\varphi ) =\lambda (g_{f}-\varphi ) =\rho (f) =\infty,\\ \overline{\lambda }_2(g_{f}-\varphi ) =\lambda _2( g_{f}-\varphi ) =\rho _2(f) =\rho (A) . \end{gather*} \end{corollary} \section{Auxiliary lemmas} \begin{lemma}[\cite{b1,c2}] \label{lem2.1} Let $A_0,A_1,\dots ,A_{k-1},F\not\equiv 0$ be finite order meromorphic functions. If $f$ is a meromorphic solution of the equation \begin{equation} f^{(k) }+A_{k-1}f^{(k-1) }+\dots +A_1f'+A_0f=F \label{e2.1} \end{equation} with $\rho (f) =+\infty $ and $\rho _2(f) =\rho $, then $f$ satisfies \begin{gather*} \overline{\lambda }(f) =\lambda (f) =\rho (f) =+\infty ,\\ \overline{\lambda }_2(f) =\lambda _2(f) =\rho_2(f)=\rho . \end{gather*} \end{lemma} The following lemma is a special case of the result due to Cao, Chen, Zheng and Tu \cite{c1}. \begin{lemma} \label{lem2.2} Let $A_0,A_1,\dots ,A_{k-1},F\not\equiv 0$ be finite order meromorphic functions. If $f$ is a meromorphic solution of \eqref{e2.1} with \[ \max \{ \rho (A_{j}) \text{ }(j=0,1,\dots ,k-1) ,\rho (F) \} <\rho (f) <+\infty , \] then \[ \overline{\lambda }(f) =\lambda (f) =\rho (f) . \] \end{lemma} By using similar proofs as in \cite[Propositions 5.1 and 5.5]{l1}, we easily obtain the following lemma. \begin{lemma} \label{lem2.3} For all non-trivial solutions $f$ of \eqref{e1.1} the following statements hold: \begin{itemize} \item[(i)] If $A$ is a polynomial with $\deg A=n\geqslant 1$, then \begin{equation} \rho (f) =\frac{n+k}{k}. \label{e2.2} \end{equation} \item[(ii)] If $A$ is transcendental and $\rho (A) <\infty $, then \begin{equation} \rho (f) =\infty \quad \text{and}\quad \rho _2(f) =\rho (A) . \label{e2.3} \end{equation} \end{itemize} \end{lemma} \begin{lemma}[\cite{b1}] \label{lem2.4} Let $A(z)$ be a transcendental meromorphic function of finite order $\rho (A) >0$ such that $\delta (\infty ,A)=\delta >0$, and let $f\not\equiv 0$ be a meromorphic solution of \eqref{e1.1}. Suppose, moreover, that either: \begin{itemize} \item[(i)] all poles of $f$ are uniformly bounded multiplicity, or \item[(ii)] $\delta (\infty ,f) >0$. \end{itemize} Then $\rho (f) =\infty $ and $\rho_2(f) =\rho (A) $. \end{lemma} We remark that for $k=2$, Lemma \ref{lem2.4} was obtained by Laine and Rieppo in \cite{l2}. Using the properties of the order of growth and the definition of the type, we easily obtain the following result which we omit the proof. \begin{lemma} \label{lem2.5} Let $f$ and $g$ be meromorphic functions such that $0<\rho (f),\rho (g) <\infty $ and $0<\tau (f) ,\tau (g) <\infty $. Then \begin{itemize} \item[(i)] If $\rho (f) >\rho(g) $, then \begin{equation} \tau (f+g) =\tau (fg) =\tau (f) . \label{e2.4} \end{equation} \item[(ii)] If $\rho (f) =\rho (g) $ and $\tau (f) \neq \tau (g) $, then \begin{equation} \rho (f+g) =\rho (fg) =\rho (f) =\rho(g) . \label{e2.5} \end{equation} \end{itemize} \end{lemma} \begin{lemma}[\cite{h1}] \label{lem2.6} Let $f$ be a meromorphic function and let $k\geqslant 1$ be an integer. Then \[ m(r,\frac{f^{(k) }}{f}) =S(r,f) , \] where $S(r,f) =O(\log T(r,f) +\log r) $, possibly outside of an exceptional set $E\subset (0,+\infty ) $ with finite linear measure. If $f$ is of finite order of growth, then \[ m(r,\frac{f^{(k) }}{f}) =O(\log r) . \] \end{lemma} \section{Proofs of main results} \begin{proof}[Proof of Theorem \ref{thm1.1}] Suppose that $f$ is an infinite order meromorphic solution of \eqref{e1.1} with $\rho _2(f) =\rho $. By \eqref{e1.1}, we have \begin{equation} f^{(k) }=-Af \label{e3.1} \end{equation} which implies \begin{equation} g_{f}=d_{k}f^{(k) }+d_{k-1}f^{(k-1) }+\dots +d_0f =d_{k-1}f^{(k-1) }+\dots +(d_0-d_{k}A) f. \label{e3.2} \end{equation} We can rewrite this euqality as \begin{equation} g_{f}=\overset{k-1}{\underset{i=0}{\sum }}\alpha _{i,0}f^{(i) }, \label{e3.3} \end{equation} where $\alpha _{i,0}$ are defined in \eqref{e1.6}. Differentiating both sides of equation \eqref{e3.3} and replacing $f^{(k) }$\ with $f^{(k) }=-Af$, we obtain \begin{equation} \begin{aligned} g_{f}' &=\overset{k-1}{\underset{i=0}{\sum }}\alpha _{i,0}'f^{(i) } +\overset{k-1}{\underset{i=0}{\sum }}\alpha_{i,0}f^{(i+1) } =\overset{k-1}{\underset{i=0}{\sum }}\alpha _{i,0}'f^{(i) }+\overset{k}{\underset{i=1}{\sum }} \alpha _{i-1,0}f^{(i) } \\ &=\alpha _{0,0}'f+\overset{k-1}{\underset{i=1}{\sum }}\alpha _{i,0}'f^{(i) }+\overset{k-1}{\underset{i=1}{\sum }} \alpha _{i-1,0}f^{(i) }+\alpha _{k-1,0}f^{(k) } \\ &=\alpha _{0,0}'f+\overset{k-1}{\underset{i=1}{\sum }}(\alpha _{i,0}'+\alpha _{i-1,0}) f^{(i) }-\alpha _{k-1,0}Af \\ &=\overset{k-1}{\underset{i=1}{\sum }}(\alpha _{i,0}'+\alpha _{i-1,0}) f^{(i) }+(\alpha _{0,0}'-\alpha _{k-1,0}A) f. \end{aligned}\label{e3.4} \end{equation} We can rewrite the above equality as \begin{equation} g_{f}'=\overset{k-1}{\underset{i=0}{\sum }}\alpha _{i,1}f^{(i) }, \label{e3.5} \end{equation} where \begin{equation} \alpha _{i,1}=\begin{cases} \alpha _{i,0}'+\alpha _{i-1,0},&\text{for }i=1,\dots ,k-1, \\ \alpha _{0,0}'-A\alpha _{k-1,0},&\text{for }i=0. \end{cases} \label{e3.6} \end{equation} Differentiating both sides of \eqref{e3.5} and replacing $f^{(k) }$ with $f^{(k) }=-Af$, we obtain \begin{equation} \begin{aligned} g_{f}'' &=\overset{k-1}{\underset{i=0}{\sum }}\alpha _{i,1}'f^{(i) }+\overset{k-1}{\underset{i=0}{\sum }} \alpha _{i,1}f^{(i+1) } =\overset{k-1}{\underset{i=0}{\sum }} \alpha _{i,1}'f^{(i) }+\overset{k}{\underset{i=1}{\sum } }\alpha _{i-1,1}f^{(i) } \\ &=\alpha _{0,1}'f+\overset{k-1}{\underset{i=1}{\sum }}\alpha _{i,1}'f^{(i) }+\overset{k-1}{\underset{i=1}{\sum }} \alpha _{i-1,1}f^{(i) }+\alpha _{k-1,1}f^{(k) } \\ &=\alpha _{0,1}'f+\overset{k-1}{\underset{i=1}{\sum }}(\alpha _{i,1}'+\alpha _{i-1,1}) f^{(i) }-\alpha _{k-1,1}Af \\ &=\overset{k-1}{\underset{i=1}{\sum }}(\alpha _{i,1}'+\alpha _{i-1,1}) f^{(i) }+(\alpha _{0,1}'-\alpha _{k-1,1}A) f \end{aligned} \label{e3.7} \end{equation} which implies that \begin{equation} g_{f}''=\overset{k-1}{\underset{i=0}{\sum }}\alpha _{i,2}f^{(i) }, \label{e3.8} \end{equation} where \begin{equation} \alpha _{i,2}=\begin{cases} \alpha _{i,1}'+\alpha _{i-1,1}, &\text{for }i=1,\dots ,k-1, \\ \alpha _{0,1}'-A\alpha _{k-1,1}, &\text{for }i=0. \end{cases} \label{e3.9} \end{equation} By using the same method as above we can easily deduce that \begin{equation} g_{f}^{(j) }=\overset{k-1}{\underset{i=0}{\sum }}\alpha _{i,j}f^{(i) },\quad j=0,1,\dots ,k-1, \label{e3.10} \end{equation} where \begin{equation} \alpha _{i,j}=\begin{cases} \alpha _{i,j-1}'+\alpha _{i-1,j-1},&\text{for }i=1,\dots ,k-1, \\ \alpha _{0,j-1}'-A\alpha _{k-1,j-1},&\text{for }i=0 \end{cases} \label{e3.11} \end{equation} and \begin{equation} \alpha _{i,0}=\begin{cases} d_{i},&\text{for }i=1,\dots ,k-1, \\ d_0-d_{k}A,&\text{for }i=0. \end{cases} \label{e3.12} \end{equation} By \eqref{e3.3}--\eqref{e3.12}, we obtain the system of equations \begin{equation} \begin{gathered} g_{f}=\alpha _{0,0}f+\alpha _{1,0}f'+\dots +\alpha _{k-1,0}f^{(k-1) }, \\ g_{f}'=\alpha _{0,1}f+\alpha _{1,1}f'+\dots +\alpha _{k-1,1}f^{(k-1) }, \\ g_{f}''=\alpha _{0,2}f+\alpha _{1,2}f'+\dots +\alpha _{k-1,2}f^{(k-1) }, \\ \dots \\ g_{f}^{(k-1) }=\alpha _{0,k-1}f+\alpha _{1,k-1}f'+\dots +\alpha _{k-1,k-1}f^{(k-1) }. \end{gathered} \label{e3.13} \end{equation} By Cramer's rule, and since $h\not\equiv 0$ we have \begin{equation} f=\frac{1}{h} \begin{vmatrix} g_{f} & \alpha _{1,0} & \dots & \alpha _{k-1,0} \\ g_{f}' & \alpha _{1,1} & \dots & \alpha _{k-1,1} \\ \vdots & & & \vdots \\ g_{f}^{(k-1) } & \alpha _{1,k-1} & \dots & \alpha _{k-1,k-1} \end{vmatrix}. \label{e3.14} \end{equation} Then \begin{equation} f=C_0g_{f}+C_1g_{f}'+\dots +C_{k-1}g_{f}^{(k-1) }, \label{e3.15} \end{equation} where $C_{j}$ are finite order meromorphic functions depending on $\alpha _{i,j}$, where $\alpha _{i,j}$ are defined in \eqref{e3.11}. If $\rho (g_{f}) <+\infty $, then by \eqref{e3.15}, we obtain $\rho (f) <+\infty $, which is a contradiction. Hence $\rho (g_{f}) =\rho (f)=+\infty $. Now, we prove that $\rho _2(g_{f}) =\rho _2(f) =\rho $. By \eqref{e3.2} , we have $\rho_2(g_{f}) \leqslant \rho _2(f) $ and by \eqref{e3.15}, we have $\rho _2(f) \leqslant \rho _2(g_{f}) $. This yield $\rho _2(g_{f}) =\rho _2(f) =\rho $. Furthermore, if $f$ is a finite order meromorphic solution of equation \eqref{e1.1} such that \begin{equation} \rho (f) >\max \{ \rho (A) ,\rho ( d_{j}) \text{ }(j=0,1,\dots ,k) \} , \label{e3.16} \end{equation} then \begin{equation} \rho (f) >\max \{ \rho (\alpha _{i,j}) :i=0,\dots ,k-1,j=0,\dots ,k-1\} . \label{e3.17} \end{equation} By \eqref{e3.2} and \eqref{e3.16} we have $\rho (g_{f})\leqslant \rho (f) $. Now, we prove $\rho (g_{f})=\rho (f) $. If $\rho (g_{f}) <\rho (f)$, then by \eqref{e3.15} and \eqref{e3.17}, we obtain \[ \rho (f) \leqslant \max \{ \rho (C_{j}) \text{ } (j=0,\dots ,k-1) ,\rho (g_{f}) \} <\rho(f) \] and this is a contradiction. Hence $\rho (g_{f}) =\rho (f) $. \end{proof} \begin{remark} \label{rmk3.1}\rm From \eqref{e3.15}, it follows that the condition $h\not\equiv 0$ is equivalent to the condition $ g_{f},g_{f}',g_{f}'',\dots ,g_{f}^{(k-1) }$ are linearly independent over the field of meromorphic functions of finite order. \end{remark} \begin{proof}[Proof of Theorem \ref{thm1.2}] Suppose that $f$ is an infinite order meromorphic solution of equation \eqref{e1.1} with $\rho _2(f) =\rho $. Set $w(z) =g_{f}-\varphi $. Since $\rho (\varphi ) <\infty $, then by Theorem \ref{thm1.1} we have $\rho (w) =\rho (g_{f}) =\infty $ and $\rho _2(w) =\rho _2(g_{f}) =\rho $. To prove $\overline{\lambda } (g_{f}-\varphi ) =\lambda (g_{f}-\varphi ) =\infty $ and $\overline{\lambda }_2(g_{f}-\varphi ) =\lambda _2( g_{f}-\varphi ) =\rho $ we need to prove $\overline{\lambda }(w) =\lambda (w) =\infty $ and $\overline{\lambda }_2(w) =\lambda _2(w) =\rho $. By $g_{f}=w+\varphi $ and \eqref{e3.15}, we obtain \begin{equation} f=C_0w+C_1w'+\dots +C_{k-1}w^{(k-1) }+\psi (z) , \label{e3.18} \end{equation} where \begin{equation} \psi (z) =C_0\varphi +C_1\varphi '+\dots +C_{k-1}\varphi ^{(k-1) }. \label{e3.19} \end{equation} Substituting \eqref{e3.18} in \eqref{e1.1}, we obtain \begin{equation} C_{k-1}w^{(2k-1) }+\underset{i=0}{\overset{2k-2}{\sum }}\phi _{i}w^{(i) }=-\big(\psi ^{(k) }+A(z) \psi \big) =H, \label{e3.20} \end{equation} where $\phi _{i}$ $(i=0,\dots ,2k-2) $ are meromorphic functions with finite order. Since $\psi (z) $ is not a solution of \eqref{e1.1}, it follows that $H\not\equiv 0$. Then by Lemma \ref{lem2.1}, we obtain $\overline{\lambda }(w) =\lambda (w)=\infty $ and $\overline{\lambda }_2(w) =\lambda _2(w) =\rho $; i. e., $\overline{\lambda }(g_{f}-\varphi ) =\lambda (g_{f}-\varphi ) =\infty $ and $\overline{\lambda }_2(g_{f}-\varphi ) =\lambda _2(g_{f}-\varphi ) =\rho $. Suppose that $f$ is a finite order meromorphic solution of \eqref{e1.1} such that \eqref{e1.10} holds. Set $w(z) =g_{f}-\varphi $. Since $\rho (\varphi ) <\rho (f) $, then by Theorem \ref{thm1.1} we have $\rho (w) =\rho (g_{f}) =\rho (f) $. To prove $ \overline{\lambda }(g_{f}-\varphi ) =\lambda (g_{f}-\varphi ) =\rho (f) $ we need to prove $\overline{\lambda }(w) =\lambda (w) =\rho (f) $. Using the same reasoning as above, we obtain \[ C_{k-1}w^{(2k-1) }+\underset{i=0}{\overset{2k-2}{\sum }}\phi _{i}w^{(i) }=-\big(\psi ^{(k) }+A(z) \psi \big) =F, \] where $C_{k-1}$, $\phi _{i}$ $(i=0,\dots ,2k-2) $ are meromorphic functions with finite order $\rho (C_{k-1}) <\rho (w) $, $\rho (\phi _{i}) <\rho (w) $ $ (i=0,\dots ,2k-2) $ and \[ \psi (z) =C_0\varphi +C_1\varphi '+\dots +C_{k-1}\varphi ^{(k-1) },\text{ }\rho (F) <\rho (w) . \] Since $\psi (z) $ is not a solution of \eqref{e1.1}, it follows that $F\not\equiv 0$. Then by Lemma \ref{lem2.2}, we obtain $\overline{\lambda }(w) =\lambda (w) =$ $\rho (f) $; i. e., $\overline{\lambda }(g_{f}-\varphi ) =\lambda ( g_{f}-\varphi ) =\rho (f) $. \end{proof} \begin{proof}[Proof of Corollary \ref{coro1.3}] Suppose that $f\not\equiv 0$ is a solution of \eqref{e1.1}. Since $A$ is a nonconstant polynomial, then by Lemma \ref{lem2.3}, we have $\rho (f) =\frac{\deg (A)+k}{k}$, which implies that \[ \rho (f) >\max \{ \rho (A) ,\rho (d_{j}) \quad (j=0,1,\dots ,k) \} =0. \] Thus, by Theorem \ref{thm1.1}, we obtain $\rho (g_{f}) =\rho (f) =\frac{\deg (A) +k}{k}$. \end{proof} \begin{proof}[Proof of Corollary \ref{coro1.4}] Suppose that $f\not\equiv 0$ is a meromorphic solution of \eqref{e1.1} such that: (i) all poles of $f$ are uniformly bounded multiplicity, or that (ii) $\delta (\infty ,f) >0$. Then by Lemma \ref{lem2.4}, we have $\rho (f) =\infty $ and $\rho _2(f) =\rho (A) $. Now, by using Theorem \ref{thm1.1}, we obtain $\rho (g_{f}) =\rho (f) =\infty $ and $\rho _2(g_{f}) =\rho _2(f) =\rho (A) $. \end{proof} \section{Discussion and applications} In this section, we consider the differential equation \begin{equation} f'''+A(z) f=0, \label{e4.1} \end{equation} where $A(z) $ is a meromorphic function of finite order. It is clear that the difficulty of the study of the differential polynomial generated by solutions lies in the calculation of the coefficients $\alpha _{i,j}$. We explain here that by using our method, the calculation of the coefficients $\alpha _{i,j}$ can be deduced easily. We study for example the growth of the differential polynomial \begin{equation} g_{f}=f'''+f''+f'+f. \label{e4.2} \end{equation} We have \begin{equation} \begin{gathered} g_{f}=\alpha _{0,0}f+\alpha _{1,0}f'+\alpha _{2,0}f'', \\ g_{f}'=\alpha _{0,1}f+\alpha _{1,1}f'+\alpha _{2,1}f'', \\ g_{f}''=\alpha _{0,2}f+\alpha _{1,2}f'+\alpha _{2,2}f''. \end{gathered} \label{e4.3} \end{equation} By \eqref{e1.6} we obtain \begin{equation} \alpha _{i,0}=\begin{cases} 1,&\text{for }i=1,2, \\ 1-A,&\text{for }i=0. \end{cases} \label{e4.4} \end{equation} Now, by \eqref{e3.6} we obtain \[ \alpha _{i,1}=\begin{cases} \alpha _{i,0}'+\alpha _{i-1,0},&\text{for }i=1,2 \\ \alpha _{0,0}'-A\alpha _{2,0},&\text{for }i=0. \end{cases} \] Hence \begin{equation} \begin{gathered} \alpha _{0,1}=\alpha _{0,0}'-A\alpha _{2,0}=-A'-A, \\ \alpha _{1,1}=\alpha _{1,0}'+\alpha _{0,0}=1-A, \\ \alpha _{2,1}=\alpha _{2,0}'+\alpha _{1,0}=1. \end{gathered} \label{e4.5} \end{equation} Finally, by \eqref{e3.9} we have \[ \alpha _{i,2}=\begin{cases} \alpha _{i,1}'+\alpha _{i-1,1},&\text{for}i=1,2, \\ \alpha _{0,1}'-A\alpha _{2,1},&\text{for }i=0. \end{cases} \] So, we obtain \begin{equation} \begin{gathered} \alpha _{0,2}=\alpha _{0,1}'-A\alpha _{2,1}=-A''-A'-A, \\ \alpha _{1,2}=\alpha _{1,1}'+\alpha _{0,1}=-2A'-A, \\ \alpha _{2,2}=\alpha _{2,1}'+\alpha _{1,1}=1-A. \end{gathered} \label{e4.6} \end{equation} Hence \begin{equation} \begin{gathered} g_{f}=(1-A) f+f'+f'', \\ g_{f}'=(-A'-A) f+(1-A) f'+f'', \\ g_{f}''=(-A''-A'-A) f+(-2A'-A) f'+(1-A) f'' \end{gathered} \label{e4.7} \end{equation} and \begin{equation} \begin{aligned} h&=\begin{vmatrix} 1-A & 1 & 1 \\ -A'-A & 1-A & 1 \\ -A''-A'-A & -2A'-A & 1-A \end{vmatrix} \\ &=3A'-A-AA'-AA''+A^2-A^{3}+2(A')^2+1. \end{aligned} \label{e4.8} \end{equation} Suppose that $h\not\equiv 0$, by simple calculations we have \begin{equation} f=\frac{Ag_{f}''+(-1-2A') g_{f}'+(1-A+2A'+A^2) g_{f}}{h} \label{e4.9} \end{equation} and by different conditions on the solution $f$ we can ensure that \[ \rho (g_{f}) =\rho (f'''+f''+f'+f) =\rho (f) . \] Turning now to the problem of oscillation, for that we consider a meromorphic function $\varphi (z) \not\equiv 0$ of finite order. From \eqref{e4.9}, we obtain \begin{equation} f=\frac{Aw''+(-1-2A') w'+(1-A+2A'+A^2) w}{h}+\psi (z) , \label{e4.10} \end{equation} where $w=g_{f}-\varphi $ and \begin{equation} \psi (z) =\frac{A\varphi ''+(-1-2A') \varphi '+(1-A+2A'+A^2) \varphi }{h} . \label{e4.11} \end{equation} Hence \begin{equation} f=\frac{A}{h}w''+C_1w'+C_0w+\psi , \label{e4.12} \end{equation} where \[ C_1=-\frac{1+2A'}{h},\quad C_0=\frac{1-A+2A'+A^2}{h}. \] Substituting \eqref{e4.12} into \eqref{e4.1} , we obtain \[ \frac{A}{h}w^{(5) }+\underset{i=0}{\overset{4}{\sum }}\phi _{i}w^{(i) }=-\big(\psi ^{(3) }+A(z) \psi \big) , \] where $\phi _{i}$ $(i=0,\dots ,4) $ are meromorphic functions with finite order. Suppose that all meromorphic solutions $f\not\equiv 0$ of \eqref{e4.1} are of infinite order and $\rho _2(f)=\rho $. If $\psi \not\equiv 0$, then by Lemma \ref{lem2.1}, we obtain \begin{gather} \overline{\lambda }(g_{f}-\varphi ) =\lambda (g_{f}-\varphi ) =\rho (f) =+\infty, \label{e4.13} \\ \overline{\lambda }_2(g_{f}-\varphi ) =\lambda _2( g_{f}-\varphi ) =\rho _2(f) =\rho . \label{e4.14} \end{gather} Suppose that $f$ is a meromorphic solution of \eqref{e4.1} of finite order such that \[ \rho (f) >\max \{ \rho (A) ,\rho (\varphi ) \} . \] If $\psi ^{(3) }+A(z) \psi \not\equiv 0$, then by Lemma \ref{lem2.2}, we obtain \[ \overline{\lambda }(g_{f}-\varphi ) =\lambda ( g_{f}-\varphi ) =\rho (f) . \] Finally, we can state the following two results. \begin{theorem} \label{thm4.1} Let $A(z) $ be a transcendental entire function of finite order satisfying $0<\rho (A) <\infty $ and $0<\tau (A) <\infty $, and let $d_{j}(z) $ $(j=0,1,2,3) $ be finite order entire functions that are not all vanishing identically such that \[ \max \{ \rho (d_{j}) \text{ }(j=0,1,2,3) \} <\rho (A) . \] If $f$ is a nontrivial solution of \eqref{e4.1} , then the differential polynomial \begin{equation} g_{f}=d_3f^{(3) }+d_2f''+d_1f'+d_0f \label{e4.15} \end{equation} satisfies \begin{gather*} \rho (g_{f}) =\rho (f) =\infty,\\ \rho _2(g_{f}) =\rho _2(f) =\rho (A). \end{gather*} \end{theorem} \begin{theorem} \label{thm4.2} Under the hypotheses of Theorem \ref{thm4.1}, let $\varphi (z) \not\equiv 0$ be an entire function with finite order. If $f$ is a nontrivial solution of \eqref{e4.1}, then the differential polynomial $g_{f}=d_3f^{(3) }+d_2f''+d_1f'+d_0f$ with $d_3\not\equiv 0$ satisfies \begin{gather} \overline{\lambda }(g_{f}-\varphi ) =\lambda ( g_{f}-\varphi ) =\rho (f) =\infty , \label{e4.16}\\ \overline{\lambda }_2(g_{f}-\varphi ) =\lambda _2( g_{f}-\varphi ) =\rho _2(f) =\rho (A) . \label{e4.17} \end{gather} \end{theorem} \begin{proof}[Proof of Theorem \ref{thm4.1}] Suppose that $f$ is a nontrivial solution of \eqref{e4.1}. Then by Lemma \ref{lem2.3}, \[ \rho (f) =\infty ,\text{ }\rho _2(f) =\rho (A) . \] First, we suppose that $d_3\not\equiv 0$. By the same reasoning as before we obtain that \[ h=\begin{vmatrix} H_0 & H_1 & H_2 \\ H_3 & H_4 & H_5 \\ H_6 & H_7 & H_8 \end{vmatrix}, \] where $H_0=d_0-d_3A$, $H_1=d_1$, $H_2=d_2$, $H_3=d_0'-(d_2+d_3') A-d_3A'$, $H_4=d_0+d_1'-d_3A$, $H_5=d_1+d_2'$, $H_6=d_0''-(d_1+2d_2'+d_3'') A-(d_2+d_3') A'-d_3A''$, $H_7=2d_0'+d_1''-(d_2+2d_3') A-2d_3A'$, $H_8=d_0+2d_1'+d_2''-d_3A$. Then \begin{align*} h&=\Big( 3d_0d_1d_2+3d_0d_1d_3'+3d_0d_2d_2'-6d_0d_3d_1'+3d_1d_2d_1' +3d_1d_3d_0'\\ &\quad +d_0d_2d_3''-2d_0d_3d_2''+d_1d_2d_2''+d_1d_3d_1'' +d_2d_3d_0'' +2d_0d_2'd_3'+2d_1d_1'd_3'-4d_2d_0'd_3'\\ &\quad +2d_2d_1'd_2'+2d_3d_0'd_2'-d_1d_2'd_3''+d_1d_3'd_2'' +d_2d_1'd_3''-d_2d_1''d_3'-d_3d_1'd_2'' \\ &\quad +d_3d_2'd_1''-d_1^{3}-3d_0^2d_3-2d_1(d_2')^2-3d_1^2d_2' -2d_3(d_1')^2-d_2^2d_1''-d_1^2d_3''-3d_2^2d_0'\Big) A\\ &\quad +\Big( 2d_0d_2d_3'+2d_0d_3d_2'-d_1d_2d_2'+2d_1d_3d_1' -4d_2d_3d_0'+d_1d_3d_2''\\ &\quad -d_2d_3d_1''-2d_1d_2'd_3'+2d_2d_1'd_3'+3d_0d_1d_3 +d_0d_2^2-d_1^2d_2+d_2^2d_1'-2d_1^2d_3'\Big) A' \\ &\quad +\Big(d_2d_3d_1'+d_0d_2d_3-d_1d_3d_2'-d_1^2d_3\Big)A'' +\Big(2d_2d_3d_3'-3d_1d_3^2+2d_2^2d_3-2d_3^2d_2'\Big)AA' \\ &\quad+\Big( d_2^{3}-3d_1d_2d_3-3d_1d_3d_3'-3d_2d_3d_2'-d_2d_3d_3'' -2d_3d_2'd_3' \\ &\quad +3d_0d_3^2+3d_3^2d_1'+2d_2(d_3')^2+3d_2^2d_3' +d_3^2d_2''\Big) A^2 \\ &\quad -d_3^{3}A^{3}+2d_2d_3^2(A')^2-d_2d_3^2AA'' -3d_0d_1d_0'-d_0d_1d_1''-d_0d_2d_0''-2d_0d_0'd_2' \\ &\quad +d_1d_0''d_2'+d_2d_0'd_1''-d_2d_1'd_0''+d_0^{3} +2d_0(d_1')^2+3d_0^2d_1'+2d_2(d_0')^2 \\ &\quad +d_1^2d_0''+d_0^2d_2''-2d_1d_0'd_1'+d_0d_1'd_2''-d_0d_2'd_1'' -d_1d_0'd_2''. \end{align*} Finally, if $d_3 \equiv 0$, $d_2 \equiv 0$, $d_1 \equiv 0$ and $d_0 \not\equiv 0$, we have $h = d_3 \not\equiv 0$. Hence $h \not\equiv 0$. By $d_3\not\equiv 0$, $A\not\equiv 0$ and Lemma \ref{lem2.5}, we have $\rho (h) =\rho (A) $, hence $h$ $\not\equiv 0$. For the cases (i) $d_3\equiv 0$, $d_2\not\equiv 0$; (ii) $d_3\equiv 0,d_2\equiv 0$ and $d_1\not\equiv 0$, by using a similar reasoning as above we obtain $h\not\equiv 0$. By $h\not\equiv 0$, we obtain \[ f=\frac{1}{h}\begin{vmatrix} g_{f} & d_1 & d_2 \\ g_{f}' & d_0+d_1'-d_3A & d_1+d_2' \\ g_{f}'' & 2d_0'+d_1''-( d_2+2d_3') A-2d_3A' & d_0+2d_1'+d_2''-d_3A \end{vmatrix}, \] which we can write \begin{equation} f=\frac{1}{h}(D_0g_{f}+D_1g_{f}'+D_2g_{f}'') , \label{e4.18} \end{equation} where \begin{gather*} \begin{aligned} D_0&=\Big( d_1d_2-2d_0d_3+2d_1d_3'+d_2d_2'-3d_3d_1'-d_3d_2'' +2d_2'd_3'\Big) A \\ &\quad +\Big( 2d_1d_3+2d_3d_2'\Big) A' +A^2d_3^2+3d_0d_1'-2d_1d_0'+d_0d_2''-d_1d_1'' \\ &\quad -2d_0'd_2'+d_1'd_2''-d_2'd_1''+d_0^2+2(d_1')^2, \end{aligned}\\ D_1=\Big( d_1d_3-2d_2d_3'-d_2^2\Big) A+d_2d_1''-d_0d_1-2d_1d_1'+2d_2d_0'-d_1d_2'', \\ D_2=d_2d_3A+d_1^2-d_2d_1'+d_1d_2'-d_0d_2. \end{gather*} If $\rho (g_{f}) <+\infty $, then by \eqref{e4.18}, we obtain $\rho (f) <+\infty $, and this is a contradiction. Hence $\rho (g_{f}) =\rho (f) =+\infty $. Now, we prove that $\rho _2(g_{f}) =\rho_2(f) =\rho (A) $. By \eqref{e4.15}, we obtain $\rho _2(g_{f}) \leqslant \rho _2(f) $ and by \eqref{e4.18} we have $\rho _2(f) \leqslant \rho _2(g_{f}) $. This yield $\rho _2(g_{f}) =\rho _2(f) =\rho (A) $. \end{proof} \begin{proof}[Proof of Theorem \ref{thm4.2}] Suppose that $f$ is a nontrivial solution of \eqref{e4.1}. By setting $w=g_{f}-\varphi $ in \eqref{e4.18}, we have \begin{equation} f=\frac{1}{h}(D_0w+D_1w'+D_2w'')+\psi , \label{e4.19} \end{equation} where \begin{equation} \psi =\frac{D_2\varphi ''+D_1\varphi '+D_0\varphi }{h}. \label{e4.20} \end{equation} Since $d_3\not\equiv 0$, then $h\not\equiv 0$. It follows from Theorem \ref{thm4.1} that $g_{f}$ is of infinite order and $\rho _2(g_{f}) =\rho(A) $. Substituting \eqref{e4.19} into \eqref{e4.1}, we obtain \[ \frac{D_2}{h}w^{(5) }+\underset{i=0}{\overset{4}{\sum }}\phi _{i}w^{(i) }=-\big(\psi ^{(3) }+A(z)\psi \big) , \] where $\phi _{i}$ $(i=0,\dots ,4) $ are meromorphic functions with finite order. First, we prove that $\psi \not\equiv 0$. Suppose that $\psi \equiv 0$, then by \eqref{e4.20} we obtain \begin{equation} D_2\varphi ''+D_1\varphi '+D_0\varphi =0 \label{e4.21} \end{equation} and by Lemma \ref{lem2.5}, we have \begin{equation} \rho (D_0) >\max \{ \rho (D_1) ,\rho (D_2) \} . \label{e4.22} \end{equation} By \eqref{e4.21}, we can write \[ D_0=-\big(D_2\frac{\varphi ''}{\varphi }+D_1\frac{ \varphi '}{\varphi }\big) . \] Since $\rho (\varphi ) <\infty $, by Lemma \ref{lem2.6} we obtain \[ T(r,D_0) \leqslant T(r,D_1) +T(r,D_2) +O(\log r) . \] Then \[ \rho (D_0) \leqslant \max \{ \rho (D_1),\rho (D_2) \} , \] which is a contradiction with \eqref{e4.22}. It is clear now that $\psi \not\equiv 0$ cannot be a solution of \eqref{e4.1} because $\rho (\psi ) <\infty $. Then, by Lemma \ref{lem2.1} we \begin{gather*} \overline{\lambda }(w) =\lambda (w) =\overline{ \lambda }(g_{f}-\varphi ) =\lambda (g_{f}-\varphi ) =\rho (f) =\infty,\\ \overline{\lambda }_2(w) =\lambda _2(w) = \overline{\lambda }_2(g_{f}-\varphi ) =\lambda _2( g_{f}-\varphi ) =\rho _2(f) =\rho (A) . \end{gather*} \end{proof} \subsection*{Acknowledgments} The authors would like to thank the anonymous referees for their valuable suggestions and helpful remarks. This research is supported by ATRST (Agence Th\'ematique de Recherche en Sciences et Technologies) and University of Mostaganem (UMAB), (PNR Project Code 8/u27/3144). \begin{thebibliography}{00} \bibitem{b1} B. Bela\"{\i}di; \emph{Growth and oscillation theory of solutions of some linear differential equations}, Mat. Vesnik 60 (2008), no. 4, 233--246. \bibitem{c1} T. B. Cao, Z. X. Chen, X. M. Zheng, J. Tu; \emph{On the iterated order of meromorphic solutions of higher order linear differential equations}, Ann. 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