\documentclass[reqno]{amsart} \usepackage{hyperref} \usepackage{amssymb} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2010(2010), No. 65, pp. 1--12.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2010 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2010/65\hfil Growth of solutions] {Growth of solutions of higher-order linear differential equations} \author[K. Hamani\hfil EJDE-2010/65\hfilneg] {Karima Hamani} \address{Karima Hamani \newline Department of Mathematics\\ Laboratory of Pure and Applied Mathematics\\ University of Mostaganem, B. P. 227 Mostaganem, Algeria} \email{hamanikarima@yahoo.fr} \thanks{Submitted February 3, 2010. Published May 8, 2010.} \subjclass[2000]{34A20, 30D35} \keywords{Linear differential equation; entire function; hyper-order} \begin{abstract} In this article, we study the growth of solutions of the linear differential equation $f^{(k)}+(A_{k-1}(z)e^{P_{k-1}(z)}+B_{k-1}(z)) f^{(k-1)}+\dots +(A_0(z)e^{P_0(z)}+B_0(z))f=0,$ where $k\geq 2$ is an integer, $P_j(z)$ are nonconstant polynomials and $A_j(z), B_j(z)$ are entire functions, not identically zero. We determine the hyper-order of these solutions, under certain conditions. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \section{Introduction and statement of results} In this article, we assume that the reader is familiar with the fundamental results and standard notation of the Nevanlinna value distribution theory of meromorphic functions \cite{h1}. Let $\sigma (f)$ denote the order of growth of an entire function $f(z)$ and $\sigma _2(f)$ the hyper-order of $f(z)$, which as in \cite{k1,y1} is defined by $$\sigma _2(f)=\limsup_{r\to +\infty}\frac{\log \log T(r,f)}{\log r}=\limsup_{r\to +\infty}\frac{\log \log \log M(r,f)}{\log r}, \label{e1.1}$$ where $M(r,f)=\max_{|z|=r} | f( z)|$. We define the linear measure of a set $E\subset [ 0,+\infty )$ by $m(E)=\int_0^{+\infty }\chi _E(t)dt$ and the logarithmic measure of a set $H\subset [ 1,+\infty )$ by $lm(H)=\int_1^{+\infty }\frac{\chi _{H}(t)}{t}dt$, where $\chi _E$ is the characteristic function of a set $E$ . Several authors \cite{c1,g2,k1} have studied the second-order linear differential equation $$f''+h_1(z)e^{P(z)}f'+h_0(z)e^{Q(z)}f=0, \label{e1.2}$$ where $P(z)$ and $Q(z)$ are nonconstant polynomials, $h_1(z)$ and $h_0(z)\not\equiv 0$ are entire functions satisfying $\sigma (h_1)<\deg P$ and $\sigma (h_0)<\deg Q$. Gundersen showed in \cite[p. 419]{g2} that if $\deg P\neq \deg Q$, then every nonconstant solution of \eqref{e1.2} is of infinite order. If $\deg P=\deg Q$, then \eqref{e1.2} can have nonconstant solutions of finite order. Indeed, $f(z)=z$ satisfies $f''-z^{3}e^{z}f'+z^{2}e^{z}f=0$. Kwon \cite{k1} studied the case where $\deg P=\deg Q$ and proved the following result: \begin{theorem}[\cite{k1}] \label{thmA} Let $P(z)$ and $Q(z)$ be nonconstant polynomials such that \begin{gather} P(z)=a_{n}z^{n}+\dots +a_1z+a_0, \label{e1.3} \\ Q(z)=b_{n}z^{n}+\dots +b_1z+b_0, \label{e1.4} \end{gather} where $a_{i},b_{i}$ $(i=0,1,\dots ,n)$ are complex numbers, $a_{n}\neq 0$ and $b_{n}\neq 0$. Let $h_j(z)$ $(j=0,1)$ be entire functions with $\sigma ( h_j)1$ and $\epsilon >0$\ be given constants. Then there exist a set $E_1\subset [ 1,+\infty )$ having finite logarithmic measure and a constant $B>0$ that depends only on $\alpha$ and $(i,j)$ ($i,j$ positive integers with $i>j$) such that for all $z$\ satisfying $|z|=r\notin [ 0,1] \cup E_1$, we have $$\big|\frac{f^{(i)}(z)}{f^{(j)}(z)} \big| \leq B\big[ \frac{T(\alpha r,f)}{r}(\log ^{\alpha }r)\log T(\alpha r,f)\big] ^{i-j}. \label{e2.1}$$ \end{lemma} \begin{lemma}[\cite{c3}] \label{lem2.2} Let $f(z)$ be a transcendental entire function Then there exists a set $E_2\subset [ 1,+\infty )$ that has finite logarithmic measure such that for all $z$ satisfying $|z|=r\notin [ 0,1] \cup E_2$ and $|f(z)|=M(r,f)$, we have $$\big|\frac{f(z)}{f^{(s)}(z)} \big| \leq 2r^{s} ,\label{e2.2}$$ where $s\geq 1$ is an integer. \end{lemma} \begin{lemma}[\cite{m1}] \label{lem2.3} Let $P(z)=(\alpha +i\beta )z^{n}+\dots$ ($\alpha$, $\beta$ are real numbers, $|\alpha |+|\beta |\neq 0$) be a polynomial with degree $n\geq 1$ and $A(z)$ be an entire function with $\sigma (A)0$, there exists a set $E_3\subset [ 1,+\infty)$ having finite logarithmic measure such that for any $\theta \in [ 0,2\pi )\setminus H$ $(H=\{ \theta \in [ 0,2\pi ):\delta ( P,\theta ) =0\} )$ and for $|z|=r\notin [ 0,1] \cup E_3$, we have \begin{itemize} \item[(i)] if $\delta (P,\theta )>0$, then $$\exp \{ (1-\epsilon )\delta (P,\theta ) r^{n}\} \leq |g(re^{i\theta })|\leq \exp \{ (1+\epsilon )\delta (P,\theta )r^{n}\} , \label{e2.3}$$ \item[(ii)] if $\delta (P,\theta )<0$, then $$\exp \{ (1+\epsilon )\delta (P,\theta ) r^{n}\} \leq |g(re^{i\theta })|\leq \exp \{ (1-\epsilon )\delta (P,\theta )r^{n}\} . \label{e2.4}$$ \end{itemize} \end{lemma} \begin{lemma}[\cite{c3}] \label{lem2.4} Let $k\geq 2$ be an integer and let $A_j(z)$ $(j=0,1,\dots ,k-1)$ be entire functions of finite order. If $f$ is a solution of the differential equation $$f^{(k)}+A_{k-1}(z)f^{(k-1)}+\dots +A_1( z)f'+A_0(z)f=0, \label{e2.5}$$ then $\sigma _2(f)\leq \max \{\sigma (A_j)$ $(j=0,1,\dots ,k-1)\}$. \end{lemma} \section{Proof of main results} \subsection{Proof of Theorem \ref{thm1.1}} Assume $f$ is a transcendental solution of \eqref{e1.6}. By Lemma \ref{lem2.1}, there exist a constant $B>0$ and a set $E_1\subset [ 1,+\infty )$ having finite logarithmic measure such that for all $z$ satisfying $|z|=r\notin [ 0,1]\cup E_1$, we have \begin{gather} \big|\frac{f^{(j)}(z)}{f^{(s)}(z)}\big|\leq Br [ T(2r,f)] ^{j-s+1}\quad (j=s+1,\dots,k), \label{e3.1} \\ |\frac{f^{(j)}(z)}{f(z)}|\leq Br[ T(2r,f)] ^{j+1}\quad (j=1,2,\dots,s-1). \label{e3.2} \end{gather} By Lemma \ref{lem2.2}, there exists a set $E_2\subset [ 1,+\infty )$ that has finite logarithmic measure such that for all $z$ satisfying $|z|=r\notin [ 0,1] \cup E_2$ and $|f(z)| =M(r,f)$, we have $$\big|\frac{f(z)}{f^{(s)}(z)} \big|\leq 2r^{s}. \label{e3.3}$$ Since $\arg a_{n,j}\neq \arg a_{n,s}$ $(j\neq s)$, there is a ray $\arg z=\theta \in [ 0,2\pi )\setminus H$, where $H=\{\theta \in [ 0,2\pi ):\delta (P_0,\theta )=0$ or $\dots$ or $\delta (P_{k-1},\theta )=0$\}, such that $\delta ( P_s,\theta )>0$, $\delta (P_j,\theta )<0$ $(j\neq s)$. Set $\beta =\max \{ \sigma ( B_j)\; (j=0,\dots ,k-1)\}$. By Lemma \ref{lem2.3}, for any given $\epsilon$ ($0<2\epsilon <\min \{ 1,n-\beta\}$), there exists a set $E_3\subset [ 1,+\infty )$ having finite logarithmic measure such that for all $z$ with $\arg z=\theta$, $|z|=r\notin [ 0,1] \cup E_3$ and a sufficiently large $r$, we have $$|A_s(z)e^{P_s(z)}+B_s(z)|\geq (1-o(1))\exp \{ (1-\epsilon )\delta (P_s,\theta ) r^{n}\} \label{e3.4}$$ and \begin{aligned} |A_j(z)e^{P_j(z)}+B_j(z)| &\leq \exp \{ (1-\epsilon)\delta (P_j,\theta )r^{n}\} +\exp \{ r^{\sigma (B_j)+\frac{\epsilon }{2}}\} \\ &\leq \exp \{ r^{\sigma (B_j)+\epsilon }\}\\ &\leq \exp \{ r^{\beta +\epsilon }\} \quad (j\neq s). \end{aligned}\label{e3.5} We can rewrite \eqref{e1.6} as \begin{aligned} &A_s(z)e^{P_s(z)}+B_s(z)\\ &=\frac{f^{(k)}}{f^{( s) }}+(A_{k-1}(z)e^{P_{k-1}(z)} +B_{k-1}(z))\frac{f^{(k-1)}}{f^{(s)}} +\dots \\ &\quad+ (A_{s+1}(z)e^{P_{s+1}(z)}+B_{s+1}(z)) \frac{ f^{(s+1)}}{f^{(s)}}+(A_{s-1}(z)e^{P_{s-1}(z)} +B_{s-1}(z))\frac{f^{(s-1)}}{f} \frac{f}{f^{(s)}} \\ &\quad +\dots +(A_1(z)e^{P_1(z)}+B_1(z))\frac{ f'}{f}\frac{f}{f^{(s)}} +(A_0(z)e^{P_0(z)}+B_0(z))\frac{f}{f^{( s)}}. \end{aligned} \label{e3.6} Hence from \eqref{e3.1}-\eqref{e3.6}, for all $z$ with $\arg z=\theta$, $|z|=r\notin [ 0,1] \cup E_1\cup E_2\cup E_3$, $|f( z) |=M(r,f)$ and a sufficiently large $r$, we have $$(1-o(1))\exp \{ (1-\epsilon )\delta (P_s,\theta )r^{n}\} \leq M_1r^{s+1}\exp \{ r^{\beta +\epsilon }\} [ T(2r,f)] ^{k}, \label{e3.7}$$ where $M_1$ is some positive constant. Thus $0<2\epsilon <\min \{ 1,n-\beta \}$ implies $\sigma (f)=+\infty$ and $\sigma _2(f)\geq n$. By Lemma \ref{lem2.4}, we have $\sigma _2(f)=n$. \subsection{Proof of Theorem \ref{thm1.2}} Assume $f$ is a transcendental solution of \eqref{e1.6}. Since $a_{n,j}=c_ja_{n,s}$ $(00$. By Lemma \ref{lem2.3}, for any given $\epsilon$ $(0<2\epsilon <\frac{1-c}{1+c})$, there exists a set $E_3\subset [ 1,+\infty )$ having finite logarithmic measure such that for all $z$ with $\arg z=\theta$, $|z|=r\notin [ 0,1] \cup E_3$ and a sufficiently large $r$, we have $$|A_s(z)e^{P_s(z)}+B_s(z)|\geq (1-o(1))\exp \{ (1-\epsilon )\delta (P_s,\theta ) r^{n}\} \label{e4.1}$$ and $$|A_j(z)e^{P_j(z)}+B_j(z)|\leq (1+o(1))\exp \{ (1+\epsilon )c\delta (P_s,\theta ) r^{n}\} \quad (j\neq s).\label{e4.2}$$ Thus by \eqref{e3.1}-\eqref{e3.3}, \eqref{e3.6}, \eqref{e4.1} and \eqref{e4.2}, we obtain that for all $z$ with $\arg z=\theta$, $|z|=r\notin [ 0,1] \cup E_1\cup E_2\cup E_3$, $|f(z)|=M(r,f)$ and a sufficiently large $r$, \begin{aligned} &(1-o(1))\exp \{ (1-\epsilon )\delta (P_s,\theta )r^{n}\} \\ &\leq M_2r^{s+1}(1+o(1))\exp \{ ( 1+\epsilon ) c\delta (P_s,\theta )r^{n}\} [ T(2r,f)] ^{k}, \end{aligned} \label{e4.3} where $M_2$ is a positive constant. By $0<2\epsilon <\frac{1-c}{1+c}$ and \eqref{e4.3}, we have $$\exp \{ \frac{(1-c)}{2}\delta (P_s,\theta ) r^{n}\} \leq M_3r^{s+1}[ T(2r,f)] ^{k}, \label{e4.4}$$ where $M_3$ is a positive constant. Hence \eqref{e4.4} implies $\sigma (f)=+\infty$ and $\sigma _2( f)\geq n$. By Lemma \ref{lem2.4}, we have $\sigma _2(f)=n$. Now we prove that if $\max \{ c_1,\dots c_{s-1}\} 0$. By Lemma \ref{lem2.3}, for any given $\epsilon$ $(0<2\epsilon <\min \{ \frac{1-c}{1+c} ,\frac{c_0-c'}{c_0+c'}\} )$, there exists a set $E_3\subset [ 1,+\infty )$ having finite logarithmic measure such that for all $z$ with $\arg z=\theta$, $|z|=r\notin [ 0,1] \cup E_3$ and a sufficiently large $r$, we have \eqref{e4.1} and \eqref{e4.2}. If $m\geq s$, by \eqref{e1.6}, \eqref{e4.1} and \eqref{e4.2}, we obtain for all $z$ with $\arg z=\theta$, $|z|=r\notin [ 0,1] \cup E_3$ and a sufficiently large $r$, \begin{aligned} &d_1r^{m-s}(1-o(1))\exp \{ ( 1-\epsilon )\delta (P_s,\theta )r^{n}\}\\ &\leq |A_s(z)e^{P_s(z)}+B_s(z)||f^{(s) }(z)| \\ &\leq d_2r^{m}(1+o(1))\exp \{ ( 1+\epsilon ) c\delta (P_s,\theta )r^{n}\} , \end{aligned}\label{e4.5} where $d_1$, $d_2$ are positive constants. By \eqref{e4.5}, $$\exp \{ \frac{(1-c)}{2}\delta (P_s,\theta ) r^{n}\} \leq d_3r^{s}, \label{e4.6}$$ where $d_3$ is a positive constant. Hence \eqref{e4.6} is not possible. If $m0$ and a set $E_1\subset [ 1,+\infty )$ having finite logarithmic measure such that for all $z$ satisfying $|z|=r\notin [ 0,1] \cup E_1$, we have $$\big|\frac{f^{(j)}(z)}{f(z)}\big|\leq Br[ T(2r,f)] ^{k+1} \quad (j=1,2,\dots,k). \label{e5.1}$$ Set $\beta =\max \{ \sigma (B_j)\; (j=0,\dots ,k-1)\}$. Suppose that $a_{n,j_1},\dots ,a_{n,j_{m}}$ satisfy $a_{n,j_{\alpha }}=c_{j_{\alpha }}a_{n,0}$, $j_{\alpha }\in \{ 1,\dots ,s-1,s+1,\dots k-1\}$, $\alpha \in \{ 1,\dots ,m\}$, $1\leq m\leq k-2$ and $\arg a_{n,j}=\theta _s$ for $j\in \{ 1,\dots ,s-1,s+1,\dots ,k-1\} \setminus \{ j_1,\dots ,j_{m} \}$. Choose a constant $c$ satisfying $\max \{ c_{j_1},\dots ,c_{j_{m}}\}=c<1$. We divide the proof into two cases: $c<0$ and $0\leq c<1$. \textbf{Case (a): $c<0$.} Since $\theta _0\neq \theta _s$, there is a ray $\arg z=\theta \in [ 0,2\pi )\setminus H$, where $H=\{ \theta \in [ 0,2\pi ):\delta (P_0,\theta )=0\text{ or } \delta (P_s,\theta )=0\}$ such that $\delta ( P_0,\theta )>0$ and $\delta (P_s,\theta )<0$. Hence \begin{gather} \delta (P_{j_{\alpha }},\theta )=c_{j_{\alpha }}\delta (P_0,\theta )<0 (\alpha =1,\dots ,m), \label{e5.2} \\ \delta (P_j,\theta )=|a_{n,j}|\cos (\theta _s+n\theta )<0, \label{e5.3} \end{gather} where $j\in \{ 1,\dots ,s-1,s+1,\dots ,k-1\} \setminus \{ j_1,\dots ,j_{m}\}$. By Lemma \ref{lem2.3}, for any given $\epsilon$ $(0<2\epsilon <\min \{ 1,n-\beta \})$, there exists a set $E_3\subset [ 1,+\infty )$ having finite logarithmic measure such that for all $z$ with $\arg z=\theta$, $|z|=r\notin [ 0,1] \cup E_3$ and a sufficiently large $r$, we have $$|A_0(z)e^{P_0(z)}+B_0(z)|\geq (1-o(1))\exp \{ (1-\epsilon )\delta (P_0,\theta ) r^{n}\} \label{e5.4}$$ and \begin{aligned} |A_j(z)e^{P_j(z)}+B_j(z)| &\leq \exp \{ (1-\epsilon )\delta (P_j,\theta )r^{n}\} +\exp \{ r^{\sigma (B_j)+\frac{\epsilon}{2}}\} \\ &\leq \exp \{ r^{\sigma (B_j)+\epsilon }\} \\ &\leq \exp \{ r^{\beta +\epsilon }\} ( j=1,\dots ,k-1). \end{aligned} \label{e5.5} We rewrite \eqref{e1.6} as \begin{aligned} &A_0(z)e^{P_0(z)}+B_0(z)\\ &=\frac{f^{(k)}}{f}+(A_{k-1}(z)e^{P_{k-1}(z)}+B_{k-1}(z)) \frac{f^{(k-1)}}{f} +\dots \\ &\quad + (A_s(z)e^{P_s(z)}+B_s(z))\frac{f^{(s)}}{f} +\dots +(A_1(z)e^{P_1(z)}+B_1(z))\frac{ f'}{f}. \end{aligned} \label{e5.6} Hence by \eqref{e5.1} and \eqref{e5.4}-\eqref{e5.6}, we obtain for all $z$ with $\arg z=\theta$, $|z|=r\notin [ 0,1]\cup E_1\cup E_3$ and a sufficiently large $r$, \begin{aligned} &(1-o(1))\exp \{ (1-\epsilon )\delta (P_0,\theta ) r^{n}\} \\ &\leq (1+(k-1)\exp \{r^{\beta +\epsilon }\})Br[ T(2r,f)] ^{k+1} \\ &\leq kBr\exp \{r^{\beta +\epsilon }\}[ T(2r,f)] ^{k+1}. \end{aligned} \label{e5.7} Thus $0<2\epsilon <\min \{ 1,n-\beta \}$ implies $\sigma (f)=+\infty$ and $\sigma _2(f)\geq n$. By Lemma \ref{lem2.4}, we have $\sigma _2(f)=n$. \textbf{Case (b): $0\leq c<1$.} Using the same reasoning as above, there exists a ray $\arg z=\theta \in [ 0,2\pi )\setminus H$, where $H$ is defined as above, such that $\delta (P_0,\theta )>0$, and $\delta (P_s,\theta )<0$. Hence \begin{gather} \delta (-cP_0,\theta )=-c\delta (P_0,\theta )<0, \delta ((1-c)P_0,\theta )=( 1-c)\delta (P_0,\theta ) >0, \label{e5.8} \\ \delta (P_j,\theta )=|a_{n,j}|\cos (\theta _s+n\theta)<0, \label{e5.9} \end{gather} where $j\in \{ 1,\dots ,s-1,s+1,\dots ,k-1\} \setminus \{ j_1,\dots ,j_{m}\}$, \begin{gather} \delta (P_j-cP_0,\theta )<0, j\in \{ 1,\dots ,k-1\} \setminus \{ j_1,\dots ,j_{m}\}, \label{e5.10} \\ \delta (P_{j_{\alpha }}-cP_0,\theta )=(c_{j_{\alpha }}-c) \delta (P_0,\theta )<0 (\alpha =1,\dots ,m). \label{e5.11} \end{gather} By Lemma \ref{lem2.3}, for any given $\epsilon$ $(0<2\epsilon <1)$, there exists a set $E_3\subset [ 1,+\infty )$ having finite logarithmic measure such that for all $z$ with $\arg z=\theta$, $|z|=r\notin [ 0,1] \cup E_3$ and a sufficiently large $r$, we have \begin{gather} |A_0(z)e^{(1-c)P_0(z)}|\geq \exp \{ (1-\epsilon )(1-c)\delta ( P_0,\theta )r^{n}\} , \label{e5.12} \\ |e^{-cP_0(z)}|\leq \exp \{ -(1-\epsilon )c\delta (P_0,\theta )r^{n}\} 0$, and$\delta (P_s,\theta )<0$. By Lemma \ref{lem2.3}, for any given$\epsilon ( 0<2\epsilon <\min \{1,n-\beta \} )$, there exists a set$E_3\subset [1,+\infty )$having finite logarithmic measure such that for all$z$with$\arg z=\theta $,$|z|=r\notin [ 0,1 ] \cup E_3$and a sufficiently large$r$, we have \eqref{e5.4} and \eqref{e5.5}. By \eqref{e1.6}, \eqref{e5.4} and \eqref{e5.5}, for all$z$with$\arg z=\theta $,$|z|=r\notin [ 0,1 ] \cup E_3$and a sufficiently large$r, we have \begin{aligned} \gamma _1r^{q}(1-o(1))\exp \{ ( 1-\epsilon ) \delta (P_0,\theta )r^{n}\} &\leq | A_0(z)e^{P_0(z)}+B_0(z)||f( z) | \\ &\leq k\gamma _2r^{q-1}\exp \{r^{\beta +\epsilon }\}. \end{aligned}\label{e5.18} where\gamma _1$and$\gamma _2$are positive constants. From \eqref{e5.18}, $$\exp \{ (1-\epsilon )\delta (P_0,\theta ) r^{n}\} \leq \frac{\gamma _3}{r}, \label{e5.19}$$ where$\gamma _3$is a positive constant. This is a contradiction. Suppose now that$0\leq c<1$. Using the same reasoning as above, there is a ray$\arg z=\theta \in [ 0,2\pi )\setminus H$, where$H$is defined as above, such that$\delta (P_0,\theta )>0$, and$\delta (P_s,\theta )<0$. By Lemma \ref{lem2.3}, for any$\epsilon (0<2\epsilon <1)$, there exists a set$E_3\subset [ 1,+\infty )$having finite logarithmic measure such that for all$z$with$\arg z=\theta $,$|z|=r\notin [ 0,1 ] \cup E_3$and a sufficiently large$r$, we have \eqref{e5.12}-\eqref{e5.15}. By \eqref{e1.6}, \eqref{e5.12}-\eqref{e5.15}, for all$z$with$\arg z=\theta $,$|z|=r\notin [ 0,1] \cup E_3$and a sufficiently large$r, we have \begin{aligned} &\gamma _{4}r^{q}\exp \{ (1-\epsilon )( 1-c)\delta (P_0,\theta )r^{n}\} \\ &\leq | A_0(z)e^{(1-c)P_0(z)}||f( z) |\\ &\leq |B_0(z)e^{-cP_0(z)}||f(z)|+|e^{-cP_0( z)}| |f^{(k)}(z)|\\ &\quad + |A_{k-1}(z)e^{P_{k-1}(z)-cP_0(z) }+B_{k-1}(z) e^{-cP_0(z)}||f^{(k-1)}(z) |\\ &\quad +\dots +|A_1(z)e^{P_1(z)-cP_0(z) }+B_1(z) e^{-cP_0(z)}||f'(z) |\\ &\leq \gamma _{5}r^{q}, \end{aligned} \label{e5.20} where\gamma _{4}$and$\gamma _{5}$are positive constants. From \eqref{e5.20}, we obtain for$|z|=r\notin [ 0,1] \cup E_3$and a sufficiently large$r$, $$\exp \{ (1-\epsilon )(1-c)\delta ( P_0,\theta ) r^{n}\} \leq \frac{\gamma _{5}}{\gamma _{4}}\,. \label{e5.21}$$ This is a contradiction; hence \eqref{e1.6} cannot have a nonzero polynomial solution. If$\arg a_{n,j}=\theta _s(j=1,\dots ,s-1,s+1,\dots ,k-1)$, then$\arg a_{n,j}\neq \arg a_{n,0}(j=1,\dots ,k-1)$and by Theorem \ref{thmC}, it follows that every solution$f(\not\equiv 0)$of \eqref{e1.6} is of infinite order and satisfies$\sigma_2(f) =n$. \subsection*{Proof of Theorem \ref{thm1.4}} Assume$f$is a transcendental solution of \eqref{e1.6}. By Lemma \ref{lem2.1}, there exist a constant$B>0$and a set$E_1\subset [ 1,+\infty )$having finite logarithmic measure such that for all$z$satisfying$|z|=r\notin [ 0,1]\cup E_1$, we have \begin{gather} |\frac{f^{(j)}(z)}{f^{(d)}(z)}|\leq Br [ T(2r,f)] ^{j-d+1}\quad (j=d+1,\dots,k) \label{e6.1} \\ |\frac{f^{(j)}(z)}{f(z)}|\leq Br[ T(2r,f)] ^{j+1}\quad (j=1,2,\dots,d-1). \label{e6.2} \end{gather} By Lemma \ref{lem2.2}, there exists a set$E_2\subset [ 1,+\infty )$having finite logarithmic measure such that for all$z$satisfying$|z|=r\notin [ 0,1] \cup E_2$and$|f(z)| =M(r,f)$, we have $$|\frac{f(z)}{f^{(d)}(z)} |\leq 2r^{d}. \label{e6.3}$$ Set$\beta =\max \{ \sigma (B_j)\; (j=0,\dots ,k-1)\}$. Suppose that$a_{n,j_1},\dots ,a_{n,j_{m}}$satisfy$a_{n,j_{\alpha }}=c_{j_{\alpha }}a_{n,d}$,$j_{\alpha }\in \{ 0,\dots ,k-1\} \setminus \{ d,s\} $,$\alpha \in \{ 1,\dots ,m\} $,$1\leq m\leq k-2$and$\arg a_{n,j}=\theta _s$for$j\in \{ 0,\dots ,k-1\} \setminus \{ d,s,j_1,\dots ,j_{m}\} $. Choose a constant$c$satisfying$\max \{ c_{j_1},\dots ,c_{j_{m}}\} =c<1$. We divide the proof into two cases:$c<0$and$0\leq c<1$. \textbf{Case (a):$c<0$.} Since$\theta _d\neq \theta _s$, there is a ray$\arg z=\theta \in [ 0,2\pi )\setminus H$, where$H=\{ \theta \in [ 0,2\pi ):\delta (P_d,\theta )=0\text{ or } \delta (P_s,\theta )=0\} $such that$\delta ( P_d,\theta )>0$and$\delta (P_s,\theta )<0$. Hence \begin{gather} \delta (P_{j_{\alpha }},\theta )=c_{j_{\alpha }}\delta (P_d,\theta )<0 \quad (\alpha =1,\dots ,m), \label{e6.4} \\ \delta (P_j,\theta )=|a_{n,j}|\cos (\theta _s+n\theta )<0, \quad j\in \{ 0,\dots ,k-1\} \setminus \{ d,s,j_1,\dots ,j_{m}\} . \label{e6.5} \end{gather} By Lemma \ref{lem2.3}, for any$\epsilon (0<2\epsilon <\min \{ 1,n-\beta \} )$, there exists a set$E_3\subset [ 1,+\infty )$having finite logarithmic measure such that for all$z$with$\arg z=\theta $,$|z|=r\notin [ 0,1] \cup E_3$and a sufficiently large$r, we have $$|A_d(z)e^{P_d(z)}+B_d(z)|\geq (1-o(1))\exp \{ (1-\epsilon )\delta (P_d,\theta ) r^{n}\} \label{e6.6}$$ and \begin{aligned} |A_j(z)e^{P_j(z)}+B_j(z)| &\leq \exp \{ (1-\epsilon )\delta (P_j,\theta )r^{n}\} +\exp \{ r^{\sigma (B_j)+\frac{\epsilon }{2}}\} \\ &\leq \exp \{ r^{\sigma (B_j)+\epsilon }\} \\ &\leq \exp \{ r^{\beta +\epsilon }\} (j\neq d). \end{aligned}\label{e6.7} By \eqref{e1.6}, we have \begin{aligned} &A_d(z)e^{P_d(z)}+B_d(z)\\ &=\frac{f^{(k)}}{f^{( d) }}+\big(A_{k-1}(z)e^{P_{k-1}(z)}+B_{k-1}(z)\big)\frac{ f^{(k-1)}}{f^{(d)}}+\dots \\ &\quad + \big(A_{d+1}(z)e^{P_{d+1}(z)}+B_{d+1}(z)\big) \frac{ f^{(d+1)}}{f^{(d)}}\\ &\quad +\big(A_{d-1}(z)e^{P_{d-1}(z)} +B_{d-1}(z)\big)\frac{f^{(d-1)}}{f}\frac{f}{f^{(d)}} +\dots \\ &\quad +(A_1(z)e^{P_1(z)}+B_1(z))\frac{ f'}{f}\frac{f}{f^{(d)}}+( A_0(z)e^{P_0(z)}+B_0(z)) \frac{f}{f^{(d)}}\, . \end{aligned} \label{e6.8} Hence by \eqref{e6.1}-\eqref{e6.3} and \eqref{e6.6}-\eqref{e6.8}, we get for allz$with$\arg z=\theta $,$|z|=r\notin [ 0,1]\cup E_1\cup E_2\cup E_3$,$|f(z)|=M(r,f)$and a sufficiently large$r$, $$(1-o(1))\exp \{ (1-\epsilon )\delta (P_d,\theta )r^{n}\} \leq M_1r^{d+1}\exp \{r^{\beta +\epsilon }\}[ T(2r,f)] ^{k+1}, \label{e6.9}$$ where$M_1$is a positive constant. Thus$0<2\epsilon <\min \{ 1,n-\beta \} $implies$\sigma (f)=+\infty $and$\sigma _2(f)\geq n$. By Lemma \ref{lem2.4}, we have$\sigma _2(f) =n$. \textbf{Case (b):$0\leq c<1$.} Using the same reasoning as above, there exists a ray$\arg z=\theta \in [ 0,2\pi )\setminus H$, where$H$is defined as above, such that$\delta (P_d,\theta )>0$, and$\delta (P_s,\theta )<0$. Hence \begin{gather} \delta (-cP_d,\theta )=-c\delta (P_d,\theta )<0, \delta ((1-c)P_d,\theta )=( 1-c)\delta (P_d,\theta ) >0, \label{e6.10} \\ \delta (P_j,\theta )=|a_{n,j}|\cos (\theta _s+n\theta )<0,\quad j\in \{ 0,\dots ,k-1\} \setminus \{ d,s,j_1,\dots ,j_{m}\} , \label{e6.11} \\ \delta (P_j-cP_d,\theta )<0 \quad j\in \{ 0,\dots ,k-1\} \setminus \{ d,j_1,\dots ,j_{m}\}, \label{e6.12} \\ \delta (P_{j_{\alpha }}-cP_d,\theta )=(c_{j_{\alpha }}-c) \delta (P_d,\theta )<0 \quad (\alpha =1,\dots ,m). \label{e6.13} \end{gather} By Lemma \ref{lem2.3}, for any given$\epsilon (0<2\epsilon <1)$, there exists a set$E_3\subset [ 1,+\infty )$having finite logarithmic measure such that for all$z$with$\arg z=\theta $,$|z|=r\notin [ 0,1] \cup E_3$and a sufficiently large$r\$, we have \begin{gather} |A_d(z)e^{(1-c)P_d(z)}|\geq \exp \{ (1-\epsilon )(1-c)\delta ( P_d,\theta )r^{n}\} , \label{e6.14} \\ |e^{-cP_d(z)}|\leq \exp \{ -(1-\epsilon )c\delta (P_d,\theta )r^{n}\}