\documentclass[twoside]{article} \usepackage{amsfonts,amsmath} \pagestyle{myheadings} \markboth{\hfil Orders of solutions \hfil EJDE--2001/61} {EJDE--2001/61\hfil Benharrat Bela\"\i di \& Saada Hamouda \hfil} \begin{document} \title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent {\sc Electronic Journal of Differential Equations}, Vol. {\bf 2001}(2001), No. 61, pp. 1--5. \newline ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp ejde.math.swt.edu (login: ftp)} \vspace{\bigskipamount} \\ % Orders of solutions of an n-th order linear differential equation with entire coefficients % \thanks{ {\em Mathematics Subject Classifications:} 30D35, 34M10, 34C10, 34C11. \hfil\break\indent {\em Key words:} Linear differential equations, entire functions, order of growth. \hfil\break\indent \copyright 2001 Southwest Texas State University. \hfil\break\indent Submitted July 23, 2001. Published September 17, 2001.} } \date{} % \author{Benharrat Bela\"\i di \& Saada Hamouda} \maketitle \begin{abstract} We study the solutions of the differential equation $$f^{(n)}+A_{n-1}(z) f^{(n-1) }+\dots+A_{1}(z)f'+A_{0}(z) f=0,$$ where the coefficients are entire functions. We find conditions on the coefficients so that every solution that is not identically zero has infinite order. \end{abstract} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \section{Introduction} For $n\geq 2$, we consider the linear differential equation $$f^{(n) }+A_{n-1}(z) f^{(n-1)}+ \dots +A_{1}(z) f'+A_{0}(z) f=0\,, \tag{1.1}$$ where $A_{0}(z) ,\dots,A_{n-1}(z)$ are entire functions with $A_{0}(z)\not\equiv 0$. Let $\rho (f)$ denote the order of the growth of an entire function $f$ as defined in \cite{h1}: $$\rho(f)=\limsup_{r\to \infty}\frac{\log \big(\log\big(\max_{|z|=r}|f(z)| \big)\big)}{\log r}\,.$$ The value $T(r,f)=\log(\max_{|z|=r}|f(z)|)$ is known as the Nevanlinna characteristic of $f$ \cite{h1}. It is well known that all solutions of (1.1) are entire functions and when some of the coefficients of (1.1) are transcendental, (1.1) has at least one solution with order $\rho (f) =\infty$. The question which arises is: \begin{quote} What conditions on $A_{0}(z) ,\dots,A_{n-1}(z)$ will guarantee that every solution $f\not\equiv 0$ has infinite order? \end{quote} In this paper we prove two results concerning this question. When $A_{0}(z) ,\dots, A_{n-1}(z)$ are polynomials with $A_{0}(z)\not\equiv 0$, every solution of (1.1) is an entire function with finite rational order; see for example \cite{g3}, \cite[pp. 199-209]{j1}, \cite[pp. 106-108]{v1}, and \cite[pp. 65-67]{w1}. In the study of the differential equation $$f''+A(z) f'+B(z) f=0 \tag{1.2}$$ where $A(z)$ and $B(z)\not\equiv 0$ are entire functions, Gundersen proved the following results. \begin{theorem}[{\cite[p. 418]{g1}}] \label{thmA} Let $A(z)$ and $B(z)\not\equiv 0$ be entire functions such that for real constants $\alpha$, $\beta$, $\theta _{1}$, $\theta _{2}$ with $\alpha>0$, $\beta >0$, and $\theta _{1}<\theta _{2}$, we have $$|B(z)| \geq \exp \{ (1+o(1)) \alpha | z| ^{\beta }\} \tag{1.3}$$ and $$| A(z) | \leq \exp \{ o(1)| z| ^{\beta }\} \tag{1.4}$$ as $z\to \infty$ with $\theta _{1}\leq \arg z\leq \theta _{2}$. Then every solution $f\not\equiv 0$ of (1.2) has infinite order. \end{theorem} \begin{theorem}[{\cite[p. 419]{g1}}] \label{thmB} Let $\{ \Phi _{k}\}$ and $\{ \theta _{k}\}$ be two finite collections of real numbers satisfying $\Phi _{1}<\theta _{1} <\Phi _{2}<\theta _{2}<\dots< \Phi _{n}<\theta _{n}< \Phi _{n+1}$, where $\Phi _{n+1}=\Phi _{1}+2\pi$, and set $$\mu =\max_{1\leq k\leq n} (\Phi _{k+1}-\theta _{k}). \tag{1.5}$$ Suppose that $A(z)$ and $B(z)$ are entire functions such that for some constant $\alpha \geq 0$, $$| A(z) | =O(| z| ^{\alpha }) \tag{1.6}$$ as $z\to \infty$ with $\Phi _{k}\leq \arg z\leq \theta _{k}$ for $k=1,\dots,n$ and where $B(z)$ is transcendental with $\rho (B) <\frac{\pi }{\mu }$. Then every solution $f\not\equiv 0$ of (1.2) has infinite order. \end{theorem} \section{Statement and proof of results} In this paper we prove the following two theorems: \begin{theorem} \label{thm1} Let $A_{0}(z),\dots, A_{n-1}(z)$, $A_{0}(z)\not\equiv 0$ be entire functions such that for real constants $\alpha$, $\beta$, $\mu$, $\theta _{1}$, $\theta _{2}$, where $0\leq \beta <\alpha$, $\mu >0$ and $\theta _{1}<\theta _{2}$ we have $$| A_{0}(z) | \geq e^{\alpha | z| ^{\mu }} \tag{2.1}$$ and $$| A_{k}(z) | \leq e^{\beta | z|^{\mu }},\quad k=1,\dots,n-1 \tag{2.2}$$ as $z\to \infty$ with $\theta _{1}\leq \arg z\leq \theta _{2}$. Then every solution $f\not\equiv 0$ of (1.1) has infinite order. \end{theorem} \begin{theorem} \label{thm2} Let $\{ \Phi _{k}\}$ and $\{ \theta _{k}\}$ be two finite collections of real numbers satisfying $\Phi _{1}<\theta _{1}< \Phi _{2}<\theta _{2}<\dots< \Phi _{m}<\theta _{m}< \Phi _{m+1}$ where $\Phi _{m+1}=\Phi _{1}+2\pi$, and set $$\mu =\max_{1\leq k\leq m}(\Phi _{k+1}-\theta _{k}) .\tag{2.3}$$ Suppose that $A_{0}(z) ,\dots,A_{n-1}(z)$ are entire functions such that for some constant $\alpha \geq 0$, $$| A_{j}(z) | =O(| z| ^{\alpha}),\quad j=1,\dots,n-1 \tag{2.4}$$ as $z\to \infty$ with $\Phi _{k}\leq \arg z\leq \theta _{k}$ for $k=1,\dots,m$ and where $A_{0}(z)$ is transcendental with $\rho (A_{0}) <\pi/\mu$. Then every solution $f\not\equiv 0$ of (1.1) has infinite order. \end{theorem} Next, we provide a lemma that is used in the proofs of our theorems. \begin{lemma}[{\cite[p. 89]{g2}}] \label{lm1} Let $w$ be a transcendental entire 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 satisfying $k_{i}>j_{i}\geq 0$ for $i=1,\dots,m$, and let $\varepsilon >0$ be a given constant. Then there exists a set $E\subset [ 0,2\pi )$ that has linear measure zero, such that if $\psi _{0}\in [ 0,2\pi )-E$, then there is a constant $R_{0}=R_{0}(\psi _{0}) >0$ such that for all $z$ satisfying $\arg z=\psi _{0}$ and $|z| \geq R_{0}$ and for all $(k,j) \in \Gamma$, we have $$\Big| \frac{w^{(k) }(z) }{w^{(j)}(z) }\Big| \leq | z| ^{(k-j) (\rho -1+\varepsilon ) }.$$ \end{lemma} \subsection*{Proof of Theorem \ref{thm1}} Suppose that $f\not\equiv 0$ is a solution of (1.1) with $\rho (f) <\infty$. Set $\delta =\rho (f)$. Then from Lemma 1, there exists a real constant $\psi_{0}$ where $\theta _{1}\leq \psi _{0}\leq \theta _{2}$, such that $$\big| \frac{f^{(k) }(z) }{f(z) }\big|=o(1) | z| ^{k\,\delta },\quad k=1,\dots,n \tag{2.5} % 3.1$$ as $z\to \infty$ with $\arg z=\psi _{0}$. Then from (2.5) and (1.1), we obtain that $$| A_{0}(z) | \leq o(1) | z|^{\delta }| A_{1}(z) | +\dots+o(1) | z| ^{(n-1) \,\delta }| A_{n-1}(z) |+o(1) | z| ^{n\,\delta } \tag{2.6} %3.2$$ as $z\to \infty$ with $\arg z=\psi _{0}$. However this contradicts (2.1) and (2.2). Therefore, every solution $f\not\equiv 0$ of (1.1) has infinite order. Next we give an example that illustrates Theorem \ref{thm1}. \paragraph{Example 1.} Consider the differential equation $$f''-(3+6e^{z}) f''+(2+6e^{z}+11e^{2z}) f'-6e^{3z}f=0 \tag{2.7}$$ In this equation, for $z=re^{i\theta}$, $r\to +\infty$, $\frac{\pi }{6}\leq \theta \leq \frac{\pi }{4}$ we have \begin{gather*} | A_{0}(z) | =\ | -6\,e^{3z}| =6e^{3r\cos \theta }>e^{3\frac{\sqrt{2}}{2}r}, \\ | A_{1}(z) | = 2+6e^{z}+11e^{2z}| \leq 19e^{2r\cos \theta }\leq 19e^{\sqrt{3}\,r}0$,$\beta \in \mathbb{C}$,$| \beta | \geq 1$, and$P_{1},\dots, P_{n-1}$are polynomials. If we take the sector$\theta _{1}\leq \arg z\leq \theta_{2}$,$\theta _{1}$,$\theta _{2}\in ] 0,\frac{\pi }{2}[ $with$\theta _{1}$near enough to$\theta _{2}$such that$\max_{1\leq k\leq n-1} \deg (P_{k}) <\alpha \frac{\cos \theta _{2}} {\cos\theta _{1}}$, then conditions (2.1) and (2.2) of Theorem \ref{thm1} are satisfied as$z\to\infty $with$\theta _{1}\leq \arg z\leq \theta _{2}$. From Theorem \ref{thm1}, it follows that every solution$f\not\equiv 0$of (2.8) has infinite order. \subsection*{Proof of Theorem \ref{thm2}} Suppose that$f\not\equiv 0$is a solution of (1.1) where$\rho (f) <\infty $and we set$\beta =\rho (f)$. From Lemma 1, there exists a set$E\subset [ 0,2\pi )$that has linear measure zero, such that if$\psi _{0}\in [ \Phi _{k},\theta _{k})-E$for some$k$, then $$| \frac{f^{(l) }(z) }{f(z) }|=O(| z| ^{l\beta }) ,\quad l=1,\dots,n \tag{2.9} %5.1$$ as$z\to \infty $with$\arg z=\psi _{0}$. From (2.9) ,(2.4) and (1.1), we obtain that $$| A_{0}(z) | \leq | \frac{f^{(n) }}{f}| +| A_{n-1}(z) | | \frac{f^{( n-1) }}{f}| +\dots+| A_{1\,}(z) | |\frac{f^{^{/}}}{f}| =O(| z| ^{\sigma }) \tag{2.10} %5.2}$$ as$z\to \infty $with$\arg z=\psi _{0}$, where$\sigma=\alpha +n\beta $. Let$\varepsilon >0$be a small constant that satisfies$\rho (A_{0}) <\frac{\pi }{\mu +2\varepsilon }$(this is possible since$\rho (A_{0}) <\frac{\pi }{\mu }$). By using the Phragm\'{e}n-Lindel\"of theorem on (2.10), it can be deduced that for some integer$s>0$$$| A_{0}(z) | =O(| z| ^{s}) \tag{2.11} %5.3}$$ as$z\to \infty $with$\Phi _{k}+\varepsilon \leq \arg z\leq \theta _{k}-\varepsilon $for$k=1,\dots,m$. Now for each$k$, we have from (2.3) that$\Phi _{k+1}+\varepsilon -(\theta _{k}-\varepsilon ) \leq \mu +2\varepsilon $, and so$\rho (A_{0}) <\frac{\pi }{\Phi _{k+1}-\theta _{k}+2\varepsilon }$. Hence using the Phragm\'{e}n-Lindel\"of theorem on (2.11) we can deduce that$|A_{0}(z) |=O(| z|^{s})$as$z\to \infty $in the whole complex plane. This means that$A_{0}(z)$is a polynomial which contradicts our hypothesis and completes the proof of Theorem \ref{thm2}. Next we give an example that illustrates Theorem \ref{thm2}. \paragraph{Example 3.} If$A_{0}(z)$is transcendental with$\rho (A_{0}) <2$, then from Theorem \ref{thm2}, every solution$f\not\equiv 0$of the equation $$f^{(n) }+P_{n-1}(z) f^{(n-1)}+\dots+P_{2}(z) f^{^{\prime \prime }} +(e^{z^{3}}+e^{i\,z^{3}}) f'+A_{0}(z) f=0\,, %\tag{5.4}$$ where$P_{n-1},\dots,P_{2}\$ are polynomials, is of infinite order. \paragraph{Acknowledgement.} The authors would like to thank the referee for his/her helpful remarks and suggestions. \begin{thebibliography}{0} \frenchspacing \bibitem{g1} G. Gundersen, \textit{Finite order solutions of second order linear differential equations}, Trans. Amer. Math. Soc. 305 (1988), pp. 415-429. \bibitem{g2} G. Gundersen, \textit{Estimates for the logarithmic derivative of a meromorphic functions, plus similar estimates}, J. London Math. Soc. (2) 37 (1988), pp. 88-104. \bibitem{g3} G. Gundersen, M. Steinbart M., and S. Wang, \textit{The possible orders of solutions of linear differential equations with polynomial coefficients}, Trans. Amer. Math. Soc. 350 (1998), pp. 1225-1247. \bibitem{h1} W. K. Hayman W. K, \textit{Meromorphic functions}, Clarendon Press, Oxford, 1964. \bibitem{j1} G. Jank and L. Volkmann, \textit{Einf\"uhrung in die Theorie der ganzen und meromorphen Funktionen mit Anwendungen auf Differentialgleichungen}, Birkh\"auser, Basel-Boston-Stuttgart, 1985. \bibitem{v1} G. Valiron, \textit{Lectures on the general theory of integral functions}, translated by E. F. Collingwood, Chelsea, New York, 1949. \bibitem{w1} H. Wittich, \textit{Neuere Untersuchungen \"uber eindeutige analytishe Funktionen}, 2nd Edition, Springer-Verlag, Berlin-Heidelberg-New York, 1968. \end{thebibliography} \noindent\textsc{Benharrat Bela\"idi} (e-mail:belaidi.benharrat@caramail.com)\\ \textsc{Saada Hamouda } (e-mail: hamouda.saada@caramail.com)\\[3pt] Department of Mathematics, University of Mostaganem \\ B. P. 227 Mostaganem, Algeria \end{document}