\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, $$\overline{\tau}(f)=\limsup_{r\to+\infty} \frac{\log\overline{N}(r,\frac{1}{f-z})}{\log r}, \label{e1.1}$$ 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 $$f''+A(z)f=0, \label{e1.5}$$ 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 $$f^{(k)}+A_{k-1}f^{(k-1)}+\dots +A_0f=0,\quad k\geq2, \label{e1.12}$$ 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 $$\overline{\lambda}(f^{(i)}-\varphi) =\lambda(f^{(i)}-\varphi)=\rho(f) =+\infty,\quad i\in \mathbb{N}\label{e1.13} \$/extract_tex] and $$\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}$$ \end{theorem} For \varphi(z)=z in Theorem \ref{Theorem main}, we obtain the following result. \begin{corollary} \label{coro8} Let k\geq2 and A_j be entire functions of finite order such that \max\{ \rho(A_j),j=1,\dots ,k-1\} <\rho(A_0)<+\infty. Then every solution f\not \equiv 0 of \eqref{e1.12}, its derivatives f^{(i)} (i\in\mathbb{N}) have infinitely many fixed points and \begin{gather} \overline{\tau}(f^{(i)})=\tau(f^{(i)})=\rho(f)=+\infty,\quad i\in\mathbb{N}, \label{e1.15}\\ \overline{\tau}_2(f^{(i)})=\tau_2(f^{(i)})=\rho_2(f)=\rho(A_0) =\rho,\quad i\in\mathbb{N}. \label{e1.16} \end{gather} \end{corollary} \begin{corollary} \label{coro9} Suppose that k\geq2 and A(z) is a transcendental entire function such that 0<\rho(A) =\rho<+\infty. If \varphi\not \equiv 0 is an entire function with finite order, \rho(\varphi)<+\infty, then every solution f\not \equiv 0 of \eqref{e1.4} satisfies \eqref{e1.13} and \eqref{e1.14}. \end{corollary} \section{Auxiliary Lemmas} The following lemmas will be used in the proof of Theorem \ref{Theorem main}. \begin{lemma}[\cite{Gundersen}]\label{Lemme21} Let f be a transcendental meromorphic function of finite order \rho, let \Gamma=\{ (k_1,j_1),(k_2,j_2),\dots ,(k_m,j_m)\} denote a finite set of distinct pairs of integers that satisfy k_{i}>j_{i}\geq0 for i=1,\dots ,m and let \varepsilon>0 be a given constant. Then the following estimations hold: (i) There exists a set E_1\subset [0,2\pi) that has linear measure zero, such that if \psi \in[0,2\pi)-E_1, then there is a constant R_1 =R_1(\psi)>1 such that for all z satisfying \arg z=\psi and | z|\geqslant R_1 and for all (k,j)\in\Gamma, we have $$\big| \frac{f^{(k)}(z)}{f^{(j)}(z)}\big| \leqslant| z| ^{(k-j)(\rho-1+\varepsilon)}. \label{e2.1}$$ (ii) There exists a set E_2\subset(1,\infty) that has finite logarithmic measure lm(E_2)=\int_1^{+\infty}\frac{\chi_{_{E_2}}(t)}{t}dt, where \chi_{_{E_2}} is the characteristic function of E_2, such that for all z satisfying | z| \notin E_2\cup[0,1] and for all (k,j)\in\Gamma, we have $$| \frac{f^{(k)}(z)}{f^{(j)}(z)}| \leqslant| z| ^{(k-j)(\rho-1+\varepsilon)}. \label{e2.2}$$ \end{lemma} To avoid some problems caused by the exceptional set we recall the following Lemmas. \begin{lemma}[{\cite[p. 68]{Bank}}] \label{Lemma2.2} Let g:[0,+\infty)\to\mathbb{R} and h:[ 0,+\infty)\to\mathbb{R} be monotone non-decreasing functions such that g(r) \leq h(r) outside of an exceptional set E of finite linear measure. Then for any \alpha>1, there exists r_0>0 such that g(r)\leq h(\alpha r) for all r>r_0. \end{lemma} \begin{lemma}[\cite{Chen}]\label{Lemma2.3} Let A_0, A_1,\dots , A_{k-1}, F\not \equiv 0 be finite order meromorphic functions. If f is a meromorphic solution with \rho(f)=+\infty of the equation $$f^{(k)}+A_{k-1}f^{(k-1)}+\dots +A_1f'+A_0f=F, \label{e2.3}$$ then \overline{\lambda}(f)=\lambda(f)=\rho(f)=+\infty. \end{lemma} \begin{lemma}[\cite{EL Farissi}] \label{Lemma2.4} Let A_0, A_1,\dots ,A_{k-1},F\not \equiv 0 be finite order meromorphic functions. If f is a meromorphic solution of the \eqref{e2.3} with \rho(f)=+\infty and \rho_2(f) =\rho, then f satisfies \overline{\lambda}_2(f)=\lambda_2(f)=\rho_2(f)=\rho. \end{lemma} \begin{lemma}[\cite{ChenYang}] \label{Lemma2.5} Let A_j be entire functions of finite order such that \max\{\rho(A_j),j=1,\dots ,k-1\} <\rho(A_0)=\rho<+\infty, then every solution f\not \equiv 0, of \eqref{e1.12}, satisfies \rho_2(f)=\rho. \end{lemma} Let A_j (j=0,1,\dots ,k-1) be entire functions. We define a sequences of functions as follows: $$\begin{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}$$ 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 $$| A_j^{1}| =| A_j^{0}+A_{j+1}^{0}\frac{( \Psi_{j+1}^{0})'}{\Psi_{j+1}^{0}}| \leqslant | A_j^{0}| +| A_{j+1}^{0}\frac{( \Psi_{j+1}^{0})'}{\Psi_{j+1}^{0}}| .\label{e2.5}$$ 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 $$| A_j^{1}| \leqslant M_1r^{\mu_1}\exp\{ \gamma_1r^{\beta}\} .\label{e2.9}$$ 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 $$| A_j^{i}| =\big| A_j^{i-1}+A_{j+1}^{i-1} \frac{(\Psi_{j+1}^{i-1})'}{\Psi_{j+1}^{i-1}}\big| \leqslant| A_j^{i-1}| +\big| A_{j+1}^{i-1} \frac{(\Psi_{j+1}^{i-1})'}{\Psi_{j+1}^{i-1}}\big|.\label{e2.10}$$ 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 $$| A_{k}^{i-1}| \leqslant M_{i-1}r^{\mu_{i-1}}\exp\{ \gamma_{i-1}r^{\beta}\} ,\quad (k=j,j+1)\label{e2.11}$$ 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 $$| \frac{(\Psi_{j+1}^{i-1})'}{\Psi_{j+1}^{i-1} }| \leqslant r^{\mu_{i-1}}.\label{e2.12}$$ 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 $$| A_j^{i}| \leqslant M_{i}r^{\mu_{i}}\exp\{ \gamma_{i}r^{\beta}\} .\label{e2.13}$$ 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 $$| A_0^{1}-A_0| \leqslant M_1r^{\mu_1}\exp\{\gamma_1r^{\beta}\} . \label{e2.16}$$ 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 $$| A_0^{i}-A_0| =\big| A_0^{i-1}+A_1^{i-1} \frac{(\Psi_1^{i-1})'}{\Psi_1^{i-1}}-A_0\big| \leqslant| A_0^{i-1}-A_0| + \big|A_1^{i-1}\frac{(\Psi_1^{i-1})'}{\Psi_1^{i-1}}\big| \label{e2.17}$$ 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 $$| A_0^{i-1}-A_0| \leqslant M_{i-1}r^{\mu_{i-1}} \exp\{ \gamma_{i-1}r^{\beta}\} . \label{e2.18}$$ 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 $$\big| \frac{(\Psi_1^{i-1})'}{\Psi_1^{i-1} }\big| \leqslant r^{\mu_{i-1}}. \label{e2.19}$$ Using assertion (1), we obtain $$| A_{k}^{i-1}| \leqslant M_{i-1}r^{\mu_{i-1}}\exp\{ \gamma_{i-1}r^{\beta}\} . \label{e2.20}$$ 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 $$| A_0^{i}-A_0| \leqslant M_{i}r^{\mu_{i}}\exp\{ \gamma_{i}r^{\beta}\} , \label{e2.21}$$ 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 $$g^{(k)}+A_{k-1}^{i}g^{(k-1)}+\dots +A_0^{i}g=0,\quad k\geq2 \label{e2.22}$$ 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 $$\frac{g^{(k)}}{g}+A_{k-1}^{i}\frac{g^{(k-1)}} {g}+\dots +A_0^{i}-A_0=-A_0,\quad k\geq2. \label{e2.23}$$ 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 $$| \frac{g^{(j)}}{g}| \leqslant r^{\alpha}, \quad j=k,k-1,\dots ,1. \label{e2.24}$$ 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 $$| A_j^{i}| \leqslant M_{i}r^{\mu_{i}}\exp\{ \gamma_{i}r^{\beta}\} . \label{e2.25}$$ 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 $$| A_0^{i}-A_0| \leqslant M_{i}r^{\mu_{i}}\exp\{\gamma_{i}r^{\beta}\}. \label{e2.26}$$ 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$z$,$| z| \notin E_{i}'\cup[ 0,1] , we have \begin{align*} | A_0| & \leqslant| \frac{g^{(k)}}{g}| +| A_{k-1}^{i}| | \frac{g^{(k-1)}}{g}| +\dots +| A_0^{i}-A_0| \\ & \leqslant r^{\alpha}+M_{i}r^{\mu_{i}}\exp\{ \gamma_{i}r^{\beta }\} r^{\alpha}+M_{i}r^{\mu_{i}}\exp\{ \gamma_{i}r^{\beta }\} r^{\alpha}\\ & \leqslant Mr^{\mu}\exp\{ \gamma r^{\beta}\} r^{\alpha}, \end{align*} whereM,\mu,\gamma$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 $$g_{i}^{(k)}+A_{k-1}^{i}g_{i}^{(k-1)} +\dots +A_0^{i}g_{i}=0, \label{e2.27}$$ 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 fori=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 $$g_{i-1}^{(k)}+A_{k-1}^{i-1}g_{i-1}^{(k-1) }+A_{k-2}^{i-1}g_{i-1}^{(k-2)}\dots +A_0^{i-1}g_{i-1}=0, \label{e2.29}$$ 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 thatf$is a solution of \eqref{e1.12}. By Lemma \ref{Lemma2.5}, we have$\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 $$w^{(k)}+A_{k-1}^{i}w^{(k-1)}+\dots +A_0 ^{i}w=-\Big(\varphi^{(k)}+A_{k-1}^{i}\varphi^{( k-1)}+\dots +A_0^{i}\varphi\Big)=F. \label{e3.1}$$ By$\rho(\varphi)<\infty$and Lemma \ref{Lemma2.7}, we obtain$F\not \equiv 0$and$\rho(F)<\infty$. 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