\documentclass[reqno]{amsart} \usepackage{hyperref} \usepackage{amssymb} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2013 (2013), No. 41, pp. 1--9.\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-2013/41\hfil Oscillation of fixed points of solutions] {Oscillation of fixed points of solutions to complex linear differential equations} \author[A. El Farissi, M. Benbachir \hfil EJDE-2013/41\hfilneg] {Abdallah El Farissi, Maamar Benbachir} \address{Abdallah El Farissi \newline Faculty of Sciences and Technology, Bechar University \\ Bechar, Algeria} \email{elfarissi.abdallah@yahoo.fr, elfarissi.a@gmail.com} \address{Maamar Benbachir \newline Sciences and Technology Faculty \\ Khemis Miliana University, Ain Defla, Algeria} \email{mbenbachir2001@gmail.com, mbenbachir2001@yahoo.fr} \thanks{Submitted June 13, 2012. Published February 6, 2013.} \subjclass[2000]{34M10, 30D35} \keywords{Linear differential equation; entire solution; hyper order; \hfill\break\indent exponent of convergence; hyper exponent of convergence} \begin{abstract} In this article, we study the relationship between the derivatives of the solutions to the differential equation $f^{(k)}+A_{k-1}f^{(k-1)}+\dots +A_0f=0$ and entire functions of finite order. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{definition}[theorem]{Definition} \allowdisplaybreaks \section{Introduction and statement of results} Throughout this article, we assume that the reader is familiar with the fundamental results and the standard notations of the Nevanlinna's value distribution theory \cite{Wang,Nevanlina}. In addition, we will use $\lambda(f)$ and $\lambda(1/f)$ to denote respectively the exponents of convergence of the zero-sequence and the pole-sequence of a meromorphic function $f$, $\rho(f)$ to denote the order of growth of $f$, $\overline{\lambda}(f)$ and $\overline{\lambda}(1/f)$ to denote respectively the exponents of convergence of the sequence of distinct zeros and distinct poles of $f$. A meromorphic function $\varphi(z)$ is called a small function of a meromorphic function $f(z)$ if $T(r,\varphi) =o(T(r,f))$ as $r\to+\infty$, where $T(r,f)$ is the Nevanlinna characteristic function of $f$. In order to express the rate of growth of meromorphic solutions of infinite order, we recall the following definitions. \begin{definition}[\cite{Liu},\cite{Wang}] \label{def1} \rm Let $f$ be a meromorphic function and let $z_1,z_2,\dots $ such that $(| z_j| =r_j$, $00:\text{ }\sum _{j=1}^{+\infty}| z_j| ^{-\tau}<+\infty\Big\} . \] \end{definition} Clearly, \begin{equation} \overline{\tau}(f)=\limsup_{r\to+\infty} \frac{\log\overline{N}(r,\frac{1}{f-z})}{\log r}, \label{e1.1} \end{equation} where $\overline{N}(r,\frac{1}{f-z})$ is the counting function of distinct fixed points of $f(z)$ in $\{ |z| 0$. Many important results have been obtained on the fixed points of general transcendental meromorphic functions for almost four decades (see \cite{Zhang}). However, there are a few studies on the fixed points of solutions of differential equations. In \cite{Wang}, Wang and L\"u investigated the fixed points and hyper-order of solutions of second order linear differential equations with meromorphic coefficients and their derivatives, they obtained the following result. \begin{theorem}[\cite{Wang}] \label{thm4} Suppose that $A(z)$ is a transcendental meromorphic function satisfying $\delta(\infty,A)=\liminf_{r\to+\infty} \frac{m(r,A)}{T(r,A)}=\delta>0$, $\rho(A)=\rho<+\infty$. Then every meromorphic solution $f\not\equiv 0$ of the equation \begin{equation} f''+A(z)f=0, \label{e1.5} \end{equation} satisfies that $f,f',f''$ all have infinitely many fixed points and \begin{gather} \overline{\tau}(f)=\overline{\tau}(f') =\overline{\tau}(f'')=\rho(f)=+\infty, \label{e1.6} \\ \overline{\tau}_2(f)=\overline{\tau}_2(f')=\overline{\tau}_2(f'')=\rho _2(f)=\rho. \label{e1.7} \end{gather} \end{theorem} The above theorem has been generalized to higher order differential equations by Liu Ming-Sheng and Zhang Xiao-Mei as follows. \begin{theorem}[\cite{Liu}] \label{th5} Suppose that $k\geq2$ and $A(z)$ is a transcendental meromorphic function satisfying \[ \delta(\infty,A)=\liminf_{r\to+\infty} \frac{m(r,A)}{T(r,A)}=\delta >0, \] and $\rho(A)=\rho<+\infty$. Then every meromorphic solution $f\not \equiv 0$ of \eqref{e1.4} satisfies that $f$ and $f',f'',\dots ,f^{(k)}$ all have infinitely many fixed points and \begin{gather} \overline{\tau}(f)=\overline{\tau}(f') =\overline{\tau}(f'')=\dots =\overline{\tau}(f^{(k)})=\rho(f)=+\infty, \label{e1.8} \\ \overline{\tau}_2(f)=\overline{\tau}_2(f^{' })=\overline{\tau}_2(f'') =\dots =\overline{\tau}_2(f^{(k)})=\rho _2(f)=\rho. \label{e1.9} \end{gather} \end{theorem} In \cite{EL Farissi}, El Farissi and Belaidi extended the result of Theorem \ref{th5} and gave the following theorem. \begin{theorem} \label{thm6} Suppose that $k\geq2$ and $A(z)$ is a transcendental meromorphic function satisfying $\delta(\infty,A)=\liminf_{r\to+\infty} \frac{m(r,A)}{T(r,A)}=\delta>0$, and $0<\rho(A)=\rho<+\infty$. If $\varphi\not \equiv 0$ is a meromorphic function with finite order $\rho(\varphi)<+\infty $, then every meromorphic solution $f\not \equiv 0$ of \eqref{e1.4} satisfies \begin{gather} \overline{\lambda}(f-\varphi)=\overline{\lambda}( f'-\varphi)=\dots =\overline{\lambda}(f^{( k)}-\varphi)=\rho(f)=+\infty, \label{e1.10} \\ \overline{\lambda}_2(f-\varphi)=\overline{\lambda}_2( f'-\varphi)=\dots =\overline{\lambda}_2(f^{( k)}-\varphi)=\rho_2(f)=\rho. \label{e1.11} \end{gather} \end{theorem} \section{Our contribution} The main purpose of this article is to study the relationship between the derivatives of the solutions to the differential equation \begin{equation} f^{(k)}+A_{k-1}f^{(k-1)}+\dots +A_0f=0,\quad k\geq2, \label{e1.12} \end{equation} and entire functions of finite order, where $A_j$ are entire functions of finite order. We prove the following result. \begin{theorem}\label{Theorem main} Let $k\geq2$ and $A_j$ be entire functions of finite order such that $\max\{ \rho(A_j),j=1,\dots ,k-1\} <\rho(A_0)<+\infty$. If $\varphi\not \equiv 0$ is an entire function with finite order, $\rho(\varphi)<+\infty$, then every solution $f\not \equiv 0$ of \eqref{e1.12} satisfies \begin{equation} \overline{\lambda}(f^{(i)}-\varphi) =\lambda(f^{(i)}-\varphi)=\rho(f) =+\infty,\quad i\in \mathbb{N}\label{e1.13} \\ \end{equation} and \begin{equation} \overline{\lambda}_2(f^{(i)}-\varphi) =\lambda_2(f^{(i)}-\varphi)=\rho_2(f) =\rho(A_0)=\rho,\text{ }i\in \mathbb{N}. \label{e1.14} \end{equation} \end{theorem} For $\varphi(z)=z$ in Theorem \ref{Theorem main}, we obtain the following result. \begin{corollary} \label{coro8} Let $k\geq2$ and $A_j$ be entire functions of finite order such that $\max\{ \rho(A_j),j=1,\dots ,k-1\} <\rho(A_0)<+\infty$. Then every solution $f\not \equiv 0$ of \eqref{e1.12}, its derivatives $f^{(i)}$ $(i\in\mathbb{N})$ have infinitely many fixed points and \begin{gather} \overline{\tau}(f^{(i)})=\tau(f^{(i)})=\rho(f)=+\infty,\quad i\in\mathbb{N}, \label{e1.15}\\ \overline{\tau}_2(f^{(i)})=\tau_2(f^{(i)})=\rho_2(f)=\rho(A_0) =\rho,\quad i\in\mathbb{N}. \label{e1.16} \end{gather} \end{corollary} \begin{corollary} \label{coro9} Suppose that $k\geq2$ and $A(z)$ is a transcendental entire function such that $0<\rho(A) =\rho<+\infty$. If $\varphi\not \equiv 0$ is an entire function with finite order, $\rho(\varphi)<+\infty$, then every solution $f\not \equiv 0$ of \eqref{e1.4} satisfies \eqref{e1.13} and \eqref{e1.14}. \end{corollary} \section{Auxiliary Lemmas} The following lemmas will be used in the proof of Theorem \ref{Theorem main}. \begin{lemma}[\cite{Gundersen}]\label{Lemme21} Let $f$ be a transcendental meromorphic function of finite order $\rho$, let $\Gamma=\{ (k_1,j_1),(k_2,j_2),\dots ,(k_m,j_m)\} $ denote a finite set of distinct pairs of integers that satisfy $k_{i}>j_{i}\geq0$ for $i=1,\dots ,m$ and let $\varepsilon>0$ be a given constant. Then the following estimations hold: (i) There exists a set $E_1\subset [0,2\pi)$ that has linear measure zero, such that if $\psi \in[0,2\pi)-E_1$, then there is a constant $R_1 =R_1(\psi)>1$ such that for all $z$ satisfying $\arg z=\psi$ and $| z|\geqslant R_1$ and for all $(k,j)\in\Gamma$, we have \begin{equation} \big| \frac{f^{(k)}(z)}{f^{(j)}(z)}\big| \leqslant| z| ^{(k-j)(\rho-1+\varepsilon)}. \label{e2.1} \end{equation} (ii) There exists a set $E_2\subset(1,\infty)$ that has finite logarithmic measure $lm(E_2)=\int_1^{+\infty}\frac{\chi_{_{E_2}}(t)}{t}dt$, where $\chi_{_{E_2}}$ is the characteristic function of $E_2$, such that for all $z$ satisfying $| z| \notin E_2\cup[0,1]$ and for all $(k,j)\in\Gamma$, we have \begin{equation} | \frac{f^{(k)}(z)}{f^{(j)}(z)}| \leqslant| z| ^{(k-j)(\rho-1+\varepsilon)}. \label{e2.2} \end{equation} \end{lemma} To avoid some problems caused by the exceptional set we recall the following Lemmas. \begin{lemma}[{\cite[p. 68]{Bank}}] \label{Lemma2.2} Let $g:[0,+\infty)\to\mathbb{R}$ and $h:[ 0,+\infty)\to\mathbb{R}$ be monotone non-decreasing functions such that $g(r) \leq h(r)$ outside of an exceptional set $E$ of finite linear measure. Then for any $\alpha>1$, there exists $r_0>0$ such that $g(r)\leq h(\alpha r)$ for all $r>r_0$. \end{lemma} \begin{lemma}[\cite{Chen}]\label{Lemma2.3} Let $A_0$, $A_1,\dots $, $A_{k-1}$, $F\not \equiv 0$ be finite order meromorphic functions. If $f$ is a meromorphic solution with $\rho(f)=+\infty$ of the equation \begin{equation} f^{(k)}+A_{k-1}f^{(k-1)}+\dots +A_1f'+A_0f=F, \label{e2.3} \end{equation} then $\overline{\lambda}(f)=\lambda(f)=\rho(f)=+\infty$. \end{lemma} \begin{lemma}[\cite{EL Farissi}] \label{Lemma2.4} 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 \eqref{e2.3} with $\rho(f)=+\infty$ and $\rho_2(f) =\rho$, then $f$ satisfies $\overline{\lambda}_2(f)=\lambda_2(f)=\rho_2(f)=\rho$. \end{lemma} \begin{lemma}[\cite{ChenYang}] \label{Lemma2.5} Let $A_j$ be entire functions of finite order such that $$ \max\{\rho(A_j),j=1,\dots ,k-1\} <\rho(A_0)=\rho<+\infty, $$ then every solution $f\not \equiv 0$, of \eqref{e1.12}, satisfies $\rho_2(f)=\rho$. \end{lemma} Let $A_j$ $(j=0,1,\dots ,k-1)$ be entire functions. We define a sequences of functions as follows: \begin{equation} \begin{gathered} A_j^{0}=A_j\quad j=0,1,\dots ,k-1\\ A_{k-1}^{i}=A_{k-1}^{i-1}-\frac{(A_0^{i-1})'} {A_0^{i-1}}\quad i\in\mathbb{N}\\ A_j^{i}=A_j^{i-1}+A_{j+1}^{i-1}\frac{(\Psi_{j+1}^{i-1})'}{\Psi_{j+1}^{i-1}} \quad j=0,1,\dots ,k-2;\; i\in\mathbb{N}, \end{gathered} \label{e2.4} \end{equation} where $\Psi_{j+1}^{i-1}=\frac{A_{j+1}^{i-1}}{A_0^{i-1}}$. \begin{lemma}\label{Lemma2.6} Let $A_j$ be entire functions of finite order such that $$ \beta=\max\{ \rho(A_j),j=1,\dots ,k-1\} <\rho(A_0)=\alpha<+\infty. $$ Then (1) There exists a set $E_{i}\subset(1,\infty)$ that has finite logarithmic measure such that for all $z$ satisfying $| z| \notin E_{i}\cup[0,1]$, we have \[ | A_j^{i}| \leqslant M_{i}r^{\mu_{i}}\exp\{ \gamma_{i}r^{\beta}\} ,\quad \text{for } j=1,\dots ,k-1, \] where $M_{i},\mu_{i},\gamma_{i}$ are positive real numbers. (2) There exists a set $E_{i}\subset(1,\infty)$ that has finite logarithmic measure such that for all $z$ satisfying $| z| \notin E_{i}\cup[ 0,1]$, we have \[ | A_0^{i}-A_0| \leqslant M_{i}r^{\mu_{i}}\exp\{ \gamma_{i}r^{\beta}\} \quad \text{for } i\in\mathbb{N}, \] where $M_{i},\mu_{i},\gamma_{i}$ are positive numbers (3) For all $i\in\mathbb{N}$, $A_0^{i}\not \equiv 0$. \end{lemma} \begin{proof} We use the induction on $i$: If $i=1$, then by \eqref{e2.4}, we have \begin{equation} | A_j^{1}| =| A_j^{0}+A_{j+1}^{0}\frac{( \Psi_{j+1}^{0})'}{\Psi_{j+1}^{0}}| \leqslant | A_j^{0}| +| A_{j+1}^{0}\frac{( \Psi_{j+1}^{0})'}{\Psi_{j+1}^{0}}| .\label{e2.5} \end{equation} By Lemma \ref{Lemme21} (ii), there exists a set $E_1\subset( 1,\infty)$ that has finite logarithmic measure such that for all $z $ satisfying $| z| \notin E_1\cup[0,1]$, we have \begin{gather} | \frac{(\Psi_{j+1}^{0})'}{\Psi_{j+1}^{0}}| \leqslant r^{\mu_1},\label{e2.6} \\ | A_j^{0}| \leqslant\exp\{ \gamma r^{\beta}\} ,\quad j=0,\dots ,k-2;\label{e2.7} \\ | A_{j+1}^{0}| \leqslant\exp\{ \gamma r^{\beta}\} ,\quad j=0,\dots ,k-2.\label{e2.8} \end{gather} Combining \eqref{e2.5}, \eqref{e2.6}, \eqref{e2.7} and \eqref{e2.8}, yields \begin{equation} | A_j^{1}| \leqslant M_1r^{\mu_1}\exp\{ \gamma_1r^{\beta}\} .\label{e2.9} \end{equation} Then (1) is true for $i=1$. Now suppose that the assertion (1) is true for the values which are strictly smaller than a certain $i$. Let \begin{equation} | A_j^{i}| =\big| A_j^{i-1}+A_{j+1}^{i-1} \frac{(\Psi_{j+1}^{i-1})'}{\Psi_{j+1}^{i-1}}\big| \leqslant| A_j^{i-1}| +\big| A_{j+1}^{i-1} \frac{(\Psi_{j+1}^{i-1})'}{\Psi_{j+1}^{i-1}}\big|.\label{e2.10} \end{equation} By the induction hypothesis, there exists a set $E_{i-1}\subset(1,\infty)$ that has finite logarithmic measure such that for all $z$ satisfying $| z| \notin E_{i-1}\cup[0,1]$ we have \begin{equation} | A_{k}^{i-1}| \leqslant M_{i-1}r^{\mu_{i-1}}\exp\{ \gamma_{i-1}r^{\beta}\} ,\quad (k=j,j+1)\label{e2.11} \end{equation} and by Lemma \ref{Lemme21} there exist a set $E_{i-1}'\subset(1,\infty)$ that has finite logarithmic measure such that for all $z$ satisfying $| z| \notin E_{i-1}'\cup[ 0,1] $ and we have \begin{equation} | \frac{(\Psi_{j+1}^{i-1})'}{\Psi_{j+1}^{i-1} }| \leqslant r^{\mu_{i-1}}.\label{e2.12} \end{equation} Hence, thanks to \eqref{e2.10},\eqref{e2.11} and \eqref{e2.12}, there exist a set $E_{i}=E_{i-1}\cup E_{i-1}'\subset(1,\infty)$ that has finite logarithmic measure such that for all $z$ satisfying $| z| \notin E_{i}\cup[ 0,1] $ and we have \begin{equation} | A_j^{i}| \leqslant M_{i}r^{\mu_{i}}\exp\{ \gamma_{i}r^{\beta}\} .\label{e2.13} \end{equation} Where $M_{i},\mu_{i},\gamma_{i}$ are positive numbers; the proof of part (1) is complete. (2) We use the same arguments as before. For $i=1$ by \eqref{e2.4} we have \[ | A_0^{1}-A_0| =\Big| A_0^{0}+A_1^{0} \frac{(\Psi_1^{0})'}{\Psi_1^{0}}-A_0\Big| =| A_1^{0}\frac{(\Psi_1^{0})'}{\Psi_1^{0}}| \] Using Lemma \ref{Lemme21} (ii), we can state that there exists a set $E_1\subset(1,\infty)$ that has finite logarithmic measure such that for all $z$ satisfying $ z| \notin E_1\cup[ 0,1] $ and we have \begin{gather} \big| \frac{(\Psi_1^{0})'}{\Psi_1^{0} }\big| \leqslant r^{\mu_1}, \label{e2.14} \\ | A_1^{0}| \leqslant\exp\{ \gamma_1r^{\beta}\} . \label{e2.15} \end{gather} From these two inequalities, we find that \begin{equation} | A_0^{1}-A_0| \leqslant M_1r^{\mu_1}\exp\{\gamma_1r^{\beta}\} . \label{e2.16} \end{equation} Consequently (2) is true for $i=1$. Now suppose that the assertion is true for the values which are strictly smaller than a certain $i$, then using \eqref{e2.4} we obtain \begin{equation} | A_0^{i}-A_0| =\big| A_0^{i-1}+A_1^{i-1} \frac{(\Psi_1^{i-1})'}{\Psi_1^{i-1}}-A_0\big| \leqslant| A_0^{i-1}-A_0| + \big|A_1^{i-1}\frac{(\Psi_1^{i-1})'}{\Psi_1^{i-1}}\big| \label{e2.17} \end{equation} By induction hypothesis, there exists a set $E_{i-1}\subset(1,\infty)$ that has finite logarithmic measure such that for all $z$ satisfying $|z| \notin E_{i-1}\cup[0,1] $ and we have \begin{equation} | A_0^{i-1}-A_0| \leqslant M_{i-1}r^{\mu_{i-1}} \exp\{ \gamma_{i-1}r^{\beta}\} . \label{e2.18} \end{equation} Using Lemma \ref{Lemme21}, we deduce that there exist a set $E_{i-1}'\subset(1,\infty)$ that has finite logarithmic measure such that for all $z$ satisfying $| z| \notin E_{i-1}'\cup[ 0,1]$ and we can write \begin{equation} \big| \frac{(\Psi_1^{i-1})'}{\Psi_1^{i-1} }\big| \leqslant r^{\mu_{i-1}}. \label{e2.19} \end{equation} Using assertion (1), we obtain \begin{equation} | A_{k}^{i-1}| \leqslant M_{i-1}r^{\mu_{i-1}}\exp\{ \gamma_{i-1}r^{\beta}\} . \label{e2.20} \end{equation} By \eqref{e2.17}, \eqref{e2.18}, \eqref{e2.19} and \eqref{e2.20} there exists a set $E_{i}=E_{i-1}\cup E_{i-1} '\subset(1,\infty)$ that has finite logarithmic measure such that for all $z$ satisfying $|z| \notin E_{i}\cup[ 0,1] $ we have \begin{equation} | A_0^{i}-A_0| \leqslant M_{i}r^{\mu_{i}}\exp\{ \gamma_{i}r^{\beta}\} , \label{e2.21} \end{equation} where $M_{i},\mu_{i},\gamma_{i}$ are positive real numbers. The proof of part (2) is complete. Now we prove part (3) Suppose that there exists $i_0\in\mathbb{N}$, such that. $A_0^{i_0}\equiv0$, this implies that $-A_0 =A_0^{i_0}-A_0$. By (2), there exists a set $E_{i_0}\subset(1,\infty)$ that has finite logarithmic measure such that for all $z$ satisfying $| z| \notin E_{i_0}\cup[ 0,1]$, we have \[ | A_0| =| A_0^{i_0}-A_0| \leqslant M_{i_0}r^{\mu_{i_0}}\exp\{ \gamma_{i_0}r^{\beta}\} , \] which contradicts $\rho(A_0)>\beta$. \end{proof} \begin{lemma} \label{Lemma2.7} Let $A_j$ be entire functions of finite order such that $\max\{\rho(A_j), j=1,\dots ,k-1\} =\beta<\rho(A_0)=\alpha<+\infty$. Then every non trivial meromorphic solution of the equation \begin{equation} g^{(k)}+A_{k-1}^{i}g^{(k-1)}+\dots +A_0^{i}g=0,\quad k\geq2 \label{e2.22} \end{equation} has infinite order, where $A_j^{i}$, $j=0,1,\dots ,k-1$ are defined as in \eqref{e2.4}. \end{lemma} \begin{proof} Assume that \eqref{e2.22} has a meromorphic solution $g$ with $\rho(g)<\infty$. We rewrite \eqref{e2.22} as \begin{equation} \frac{g^{(k)}}{g}+A_{k-1}^{i}\frac{g^{(k-1)}} {g}+\dots +A_0^{i}-A_0=-A_0,\quad k\geq2. \label{e2.23} \end{equation} By Lemma \ref{Lemme21} (ii), there exists a set $E\subset(1,\infty)$ of finite logarithmic measure such that for all $z$, $|z| \notin E\cup[ 0,1]$, we have \begin{equation} | \frac{g^{(j)}}{g}| \leqslant r^{\alpha}, \quad j=k,k-1,\dots ,1. \label{e2.24} \end{equation} On the other hand, Lemma \ref{Lemma2.6} (1) implies that there exists a set $E_{i}\subset(1,\infty)$ of finite logarithmic measure such that for all $z$, $| z| \notin E_{i}\cup[0,1] $, we have \begin{equation} | A_j^{i}| \leqslant M_{i}r^{\mu_{i}}\exp\{ \gamma_{i}r^{\beta}\} . \label{e2.25} \end{equation} And by \ref{Lemma2.6} (2) there exists a set $E_{i}' \subset(1,\infty)$ of finite logarithmic measure such that for all $z$, $| z| \notin E_{i}'\cup[0,1] $, we have \begin{equation} | A_0^{i}-A_0| \leqslant M_{i}r^{\mu_{i}}\exp\{\gamma_{i}r^{\beta}\}. \label{e2.26} \end{equation} By \eqref{e2.23}, \eqref{e2.24}, \eqref{e2.25}, and \eqref{e2.26}, we can find a set $E':=E\cup E_{i}'\cup E_{i}'$ of finite logarithmic measure such that for all $z$, $| z| \notin E_{i}'\cup[ 0,1] $, we have \begin{align*} | A_0| & \leqslant| \frac{g^{(k)}}{g}| +| A_{k-1}^{i}| | \frac{g^{(k-1)}}{g}| +\dots +| A_0^{i}-A_0| \\ & \leqslant r^{\alpha}+M_{i}r^{\mu_{i}}\exp\{ \gamma_{i}r^{\beta }\} r^{\alpha}+M_{i}r^{\mu_{i}}\exp\{ \gamma_{i}r^{\beta }\} r^{\alpha}\\ & \leqslant Mr^{\mu}\exp\{ \gamma r^{\beta}\} r^{\alpha}, \end{align*} where $M,\mu,\gamma$ are positive real numbers. This leads to a contradiction with $\beta<\rho(A_0)$, hence $\rho(g)=\infty$. \end{proof} \begin{lemma} \label{Lemma2.8} Let $A_j$ $(j=0,1,\dots,k-1)$ be entire functions. If $f$ is a solution of an equation of the form \eqref{e1.12} then $g_{i}=f^{(i)}$ $(i\in\mathbb{N})$ is an entire solution of the equation \begin{equation} g_{i}^{(k)}+A_{k-1}^{i}g_{i}^{(k-1)} +\dots +A_0^{i}g_{i}=0, \label{e2.27} \end{equation} where $A_j^{i}$, $j=0,1,\dots ,k-1$ are defined in \eqref{e2.4}. \end{lemma} \begin{proof} Assume that $f$ is a solution of \eqref{e1.12} and let $g_{i}:=f^{(i)}$ $(i\in\mathbb{N})$. We shall prove that $g_{i}$ is an entire solution of \eqref{e2.27}. To do this, we use induction. For $i=1$, differentiating both sides of \eqref{e1.12}, we write \begin{equation*} f^{(k+1)}+A_{k-1}f^{(k)}+(A_{k-1} '+A_{k-2})f^{(k-1)}+\dots +( A_1'+A_0)f'+A_0'f=0. %\label{e2.28} \end{equation*} Taking \[ f=-\frac{(f^{(k)}+A_{k-1}f^{(k-1)}+\dots +A_1f')}{A_0}, \] we obtain \begin{align*} &f^{(k+1)}+(A_{k-1}-\frac{A_0'}{A_0})f^{(k)}+(A_{k-1}'+A_{k-2} -A_{k-1}\frac{A_0'}{A_0})f^{(k-1)}\\ &+\dots +(A_1'+A_0-A_1\frac{A_0'}{A_0})f'=0; \end{align*} that is, \[ g_1^{(k)}+A_{k-1}^{1}g_1^{(k-1)}+A_{k-2} ^{1}g_1^{(k-2)}\dots +A_0^{1}g_1=0. \] Hence \eqref{e2.27} is true for $i=1$. Now suppose that \eqref{e2.27} is true for the values which are strictly smaller than a certain $i$. If $g_{i-1}$ is a solution of the equation \begin{equation} g_{i-1}^{(k)}+A_{k-1}^{i-1}g_{i-1}^{(k-1) }+A_{k-2}^{i-1}g_{i-1}^{(k-2)}\dots +A_0^{i-1}g_{i-1}=0, \label{e2.29} \end{equation} then by differentiation both sides of \eqref{e2.29}, we obtain \begin{align*} &g_{i-1}^{(k+1)}+A_{k-1}^{i-1}g_{i-1}^{(k) }+((A_{k-1}^{i-1})'+A_{k-2}^{i-1}) g_{i-1}^{(k-1)}+\dots\\ &+((A_1^{i-1})'+A_0^{i-1}) g_{i-1}'+A_0'g_{i-1}=0. %\label{e2.30} \end{align*} Taking \[ g_{i-1}=-\frac{(g_{i-1}^{(k)}+A_{k-1}^{i-1}g_{i-1}^{( k-1)}+A_{k-2}^{i-1}g_{i-1}^{(k-2)}\dots +A( g_{i-1})')}{A_0^{i-1}}, \] we obtain \begin{align*} &g_{i-1}^{(k+1)}+\Big(A_{k-1}^{i-1}-\frac{(A_0 ^{i-1})'}{A_0^{i-1}}\Big)g_{i-1}^{(k) }+\Big((A_{k-1}^{i-1})'+A_{k-2}^{i-1} -A_{k-1}^{i-1}\frac{(A_0^{i-1})'}{A_0^{i-1} }\Big)g_{i-1}^{(k-1)}\\ &+\dots +\Big((A_1^{i-1})'+A_0^{i-1}-A_1 ^{i-1}\frac{(A_0^{i-1})'}{A_0^{i-1}}\Big) g_{i-1}'=0; \end{align*} %\label{e2.31} that is, \[ g_i^{(k)}+A_{k-1}^{i-1}g_i^{(k-1)} +A_{k-2}^{i-1}g_i^{(k-2)}\dots +A_0^{i-1}g_i=0. \] Lemma \ref{Lemma2.8} is thus proved. \end{proof} \subsection{Proof of Theorem \ref{Theorem main}} Assume that $f$ is a solution of \eqref{e1.12}. By Lemma \ref{Lemma2.5}, we have $\rho_2(f)=\rho(A_0)$. Using Lemma \ref{Lemma2.8}, we can state that $g_{i}:=f^{(i)}$ $(i\in\mathbb{N})$ is a solution of \eqref{e2.27}. Let $w(z):=g_{i}( z)-\varphi(z)$; $\varphi$ is an entire finite order function. Then $\rho(w)=\rho(g_{i})=\rho(f)=\infty$ and $\rho_2(w)=\rho_2(g_{i})=\rho_2(f)=\rho(A_0)$. To prove $\overline{\lambda}(g_{i}-\varphi) =\lambda(g_{i}-\varphi)=\infty$ and $\overline{\lambda} _2(g_{i}-\varphi)=\lambda_2(g_{i}-\varphi) =\rho(A_0)$, we need to prove only that $\overline{\lambda}( w)=\infty$ and $\overline{\lambda}_2(w)=\rho(A_0)$. Using the fact that $g_{i}=w+\varphi$, and by Lemma \ref{Lemma2.8}, we can write \begin{equation} w^{(k)}+A_{k-1}^{i}w^{(k-1)}+\dots +A_0 ^{i}w=-\Big(\varphi^{(k)}+A_{k-1}^{i}\varphi^{( k-1)}+\dots +A_0^{i}\varphi\Big)=F. \label{e3.1} \end{equation} By $\rho(\varphi)<\infty$ and Lemma \ref{Lemma2.7}, we obtain $F\not \equiv 0$ and $\rho(F)<\infty$. By Lemma \ref{Lemma2.4} $\overline{\lambda}(w)=\lambda(w)=\rho( w)=\infty$ and $\overline{\lambda}_2(w)=\lambda _2(w)=\rho_2(w)=\rho(A_0)$. The proof of Theorem \ref{Theorem main} is complete. \subsection*{Acknowledgments} This work was supported by l’ANDRU, Agence Nationale pour le Developpement de la Recherche Universitaire (PNR Projet 2011-2013). \begin{thebibliography}{99} \bibitem{Bank} S. Bank; \emph{A general theorem concerning the growth of solutions of first-order algebraic differential equations}, Compositio Math. \textbf{25} (1972), 61-70. \bibitem{EL Farissi} B. Bela\"{\i}di, A. El Farissi; \emph{Oscillation theory to some complex linear large differential equations}, Annals of Differential Equations, 2009, no. 1, 1-7. \bibitem{Belaidi} B. Bela\"{\i}di, A. El Farissi; \emph{Differential polynomials generated by some complex linear differential equations with meromorphic coefficients,} Glasnik Matematicki, Vol. 43(63) 2008, 363-373. \bibitem{Chen} Z. X. 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