\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2013 (2013), No. 183, pp. 1--8.\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-2013/183\hfil Growth of solutions] {Growth of solutions to linear complex differential equations in an angular region} \author[N. Wu\hfil EJDE-2013/183\hfilneg] {Nan Wu} % in alphabetical order \address{Nan Wu \newline Department of Mathematics, School of Science, China University of Mining and Technology (Beijing), Beijing 100083, China} \email{wunan2007@163.com} \thanks{Submitted April 26, 2013. Published August 10, 2013.} \subjclass[2000]{30D10, 30D20, 30B10, 34M05} \keywords{Meromorphic solutions; order of a function; angular region} \begin{abstract} In this article, we consider the growth of solutions of higher-order linear differential equations in an angular region instead of the complex plane. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction and statement of results} We assume that the reader is familiar with the fundamental results and standard notations of the Nevanlinna theory in the unit disk $\Delta=\{z:|z|<1\}$ and in the complex plane $\mathbb{C}$ (see \cite{Hayman,Tsuji,Yang}), such as $T(r,f), N(r,f), m(r,f),\delta(a,f)$. The order and lower order of $f$ in $\mathbb{C}$ or in $\Delta$ are defined as follows: \begin{gather*} \rho_\mathbb{C}(f)=\limsup_{r\to\infty}\frac{\log T(r,f)}{\log r}, \quad \rho_\Delta(f)=\limsup_{r\to1-}\frac{\log T(r,f)}{-\log(1-r)},\\ \mu_\mathbb{C}(f)=\liminf_{r\to\infty}\frac{\log T(r,f)}{\log r}, \quad \mu_\Delta(f)=\liminf_{r\to1-}\frac{\log T(r,f)}{-\log(1-r)}. \end{gather*} The meromorphic functions in the unit disk can be divided into the following three classes: \begin{itemize} \item[(1)] bounded type: $T(r,f)=O(1)$ as $r\to 1-$; \item[(2)] rational type: $T(r,f)=O(\log (1-r)^{-1})$ and $f(z)$ does not belong to (1); \item[(3)] admissible in $\Delta$: $$\limsup_{r\to 1-}\frac{T(r,f)}{-\log(1-r)}=\infty.$$ \end{itemize} Meromorphic functions in the complex plane can also be divided into the following three classes: \begin{itemize} \item[(1)] bounded type: $T(r,f)=O(1)$ as $r\to \infty$; \item[(2)] rational type: $T(r,f)=O(\log r)$ and $f(z)$ does not belong to (1); \item[(3)] admissible in $\mathbb{C}$: $$\limsup_{r\to \infty}\frac{T(r,f)}{\log r}=\infty.$$ \end{itemize} The growth of solutions to higher-order linear differential equations in $\mathbb{C}$ and in $\Delta$ has been investigated by many authors. Gundersen \cite{G} and Heittokangas \cite{Heitiokangas} considered the growth of solutions of the second-order linear differential equations and obtained a theorem in $\mathbb{C}$ and in $\Delta$ respectively as follows. \begin{theorem}[\cite{G,Heitiokangas}] \label{thm1.1} Let $B(z)$ and $C(z)$ be the analytic coefficients of the equation $$\label{1.1} g''+B(z)g'+C(z)g=0$$ in $\mathbb{C}$ (or in $\Delta$). If either (i) $\rho(B)<\rho(C)$ or (ii) $B(z)$ is non-admissible while $C(z)$ is admissible, then all solutions $g\not\equiv0$ of \eqref{1.1} are of infinite order of growth. \end{theorem} Chen \cite{Chen} generalized Theorem \ref{thm1.1} as follows. \begin{theorem}[\cite{Chen}] \label{thm1.4} Let $A_0(z),\dots, A_k(z)$ be the analytic coefficients of the equation $$\label{1.3} A_k(z)f^{(k)}+A_{k-1}(z)f^{(k-1)}+\dots+A_0(z)f=0$$ in $\mathbb{C}$ (or in $\Delta$). If either (i) $\max_{1\leq j\leq k}\rho(A_j)<\rho(A_0)$, or (ii) $A_j(z)$ $(j=1,2,\dots,k)$ are non-admissible while $A_0(z)$ is admissible, then all solutions $f\not\equiv0$ of \eqref{1.3} are of infinite order of growth. \end{theorem} In 1994, Wu \cite{Wu,Wu1} used the Nevanlinna theory in an angle to study the growth of solutions of the second-order linear differential equation in an angular region and obtained the following two theorems. \begin{theorem}[\cite{Wu}] \label{thm1.6} Let $A(z)$ and $B(z)$ be meromorphic in $\mathbb{C}$ with $\rho(A)<\rho(B)$ and $\delta(\infty,B)>0$. Then every nontrivial meromorphic solution $f$ of the equation $$\label{1.4} f''+A(z)f'+B(z)f=0$$ has infinite order. Furthermore, if $\rho(B)\leq1/2$ and $\delta(\infty,B)=1$, then $\rho_{\alpha,\beta}(f)=+\infty$ for every angular region $\Omega(\alpha,\beta)$. \end{theorem} \begin{theorem}[\cite{Wu1}]\label{thm1.5} Let $A(z)$ and $B(z)$ be analytic on $\overline{\Omega}(\alpha,\beta)$. If for any $K>0$, the measure of $$\Big\{ \theta: \alpha<\theta<\beta, \liminf_{r\to\infty}\frac{(|A(re^{i\theta})|+1)r^K}{|B(re^{i\theta})|}=0\Big\}$$ is larger than zero, then any solution $f\not\equiv0$ of \eqref{1.4} has $\varrho_{\alpha,\beta}(f)=+\infty$. \end{theorem} In 2009, Xu and Yi \cite{Xu} generalized Theorem \ref{thm1.5} to the case of linear higher order differential equation and obtained the following theorem. \begin{theorem}\cite{Xu} \label{thm1.7} Let $A_j(z)(j=0,1,\dots,k-1)$ be analytic on $\Omega(\alpha,\beta)(0<\beta-\alpha\leq2\pi)$, if for any $K>0$ the $\theta$'s which satisfy $\alpha\leq\theta\leq\beta$ and $$\label{1.6} \liminf_{r\to\infty}\frac{(|A_1(re^{i\theta})|+\dots +|A_{k-1}(re^{i\theta})|+1)r^K}{|A_0(re^{i\theta})|}=0$$ form a set of positive measure. Then for every solution $f\not\equiv0$ of \eqref{1.6} we have $\varrho_{\alpha,\beta}(f)=+\infty$. \end{theorem} \begin{remark} \rm The order $\varrho_{\alpha,\beta}(f)$ in Theorems \ref{thm1.5} and \ref{thm1.7} is defined by $$\varrho_{\alpha,\beta}(f)=\limsup_{r\to\infty} \frac{\log\log M(r,\overline{\Omega},f)}{\log r},$$ where $M(r,\overline{\Omega},f)=\max_{\alpha\leq\theta\leq\beta}|f(re^{i\theta})|$ and $f\not\equiv0$ is a function analytic on the set $\overline{\Omega}(\alpha,\beta)=\{z:\alpha\leq\arg z\leq\beta\}(0<\beta-\alpha\leq2\pi)$. The order $\rho_{\alpha,\beta}(f)$ in this paper is different from $\varrho_{\alpha,\beta}(f)$. \end{remark} It is natural to pose the following question: \begin{quote} How does the solutions of linear differential equations with analytic or meromorphic coefficients grow in an angular region? \end{quote} Before stating our results, we give some notation and definitions of a meromorphic function in an angular region $\Omega(\alpha,\beta)=\{z:\alpha<\arg z<\beta\}$. In this article, $\Omega$ usually denotes the angular region $\Omega(\alpha,\beta)$ and $\Omega_\varepsilon=\{z:\alpha+\varepsilon<\arg z<\beta-\varepsilon\}$, where $0<\varepsilon<(\beta-\alpha)/2$. Let $f(z)$ be a meromorphic function on $\overline{\Omega}(\alpha,\beta)=\{z:\alpha\leq\arg z\leq\beta\}$. Recall the definition of Ahlfors-Shimizu characteristic in an angular region (see \cite{Tsuji}). Set $\Omega(r)=\Omega(\alpha,\beta)\cap\{z:0<|z|0$, write $\omega=\frac{\pi}{\beta-\alpha}, \eta=\frac{\varepsilon}{\beta-\alpha}$. Then the following inequalities hold: \begin{gather}\label{2.2} \mathcal {T}(r, \mathbb{C}, f(z(\zeta)))\leq 2\mathcal{T}\Big(\big(\frac{2}{1-r}\big)^{1/\omega},\Omega,f(z)\Big)+O(1), \\ \label{2.3} \mathcal {T}(r,\Omega_\varepsilon,f(z)) \leq \frac{r^\omega}{\omega\eta}\mathcal{T}(1-\eta r^{-\omega},\mathbb{C}, f(z(\zeta)))+O(1), \end{gather} where $z=z(\zeta)$ is the inverse transformation of \eqref{2.1}. Consequently, $$\label{2.4} \rho_\Delta(f(z(\zeta)))\leq\frac{1}{\omega}\rho_\Omega(f(z)),\quad \rho_{\Omega_\varepsilon}(f(z))\leq(\rho_\Delta(f(z(\zeta)))+1)\omega.$$ \end{lemma} \begin{proof} By Lemma \ref{lem2.1}, for the inverse of the transformation \eqref{2.1} it follows that $$z(\Delta_h)\subset \Omega\cap\Big\{z: |z|\leq \big(\frac{2}{1-h}\big) ^{1/\omega}\Big\}, \quad \text{where } \Delta_h=\{z:|z|1,\alpha<\arg z theta<\beta\}(0<\beta-\alpha<2\pi) onto the unit disk \{\zeta:|\zeta|<1\} while the transformation \eqref{2.1} maps the angular region \{z:\alpha<\arg z <\beta\}(0<\beta-\alpha<2\pi) onto the unit disk \{\zeta:|\zeta|<1\}. For completeness, we give the proof of Lemma \ref{lem2.3} using the method of \cite[Lemma 1]{Edrei1}. \begin{proof} Put$$ V(\zeta)=\frac{1}{z'(\zeta)}. $$By a simple calculation, we have$$ f'(z(\zeta))=V(\zeta)F'(\zeta). $$An obvious induction shows that$$ \psi(\zeta)=f^{(l)}(z(\zeta))=\sum_{j=1}^l\alpha_jF^{(j)}(\zeta) $$where the coefficients \alpha_j are polynomials (with numerical coefficients) in the variables V, V', V'', \dots. Taking the derivative on both side of \eqref{2.6}, we obtain that$$ \frac{dz}{d\zeta}=\frac{e^{i\theta_0}}{\omega} \Big(\frac{1+\zeta}{1-\zeta}\Big)^{\frac{1}{\omega}-1} \frac{2}{(1-\zeta)^2},\quad \omega=\frac{\pi}{\beta-\alpha},\theta_0=\frac{\alpha+\beta}{2}. $$Then$$ \big|\frac{dz}{d\zeta}\big|\leq\frac{1}{\omega}\frac{2^{1/\omega}}{(1-|\zeta|) ^{\frac{1}{\omega}+1}}. $$Therefore,$$ T(r,z'(\zeta))\leq\log M(r,z'(\zeta))\leq\Big(\frac{1}{\omega}+1\Big) \log\frac{2}{1-r}+\log\frac{1}{\omega}, $$By the first fundamental theorem,$$ T\Big(r,\frac{1}{z'(\zeta)}\Big) =T(r,z'(\zeta))+\log\frac{1}{|z'(0)|} \leq\Big(\frac{1}{\omega}+1\Big)\log\frac{2}{1-r}+\log\frac{1}{\omega} +\log2\omega. $$Thus,$$ T(r,V(\zeta))=T\Big(r,\frac{1}{z'(\zeta)}\Big) \leq\Big(\frac{1}{\omega}+1\Big)\log\frac{2}{1-r}+\log\frac{1}{\omega} +\log2\omega, \begin{align*} T(r,V^{(k)})&=m(r,V^{(k)})+N(r,V^{(k)}) \leq m\Big(r,\frac{V^{(k) }}{V}\Big)+m(r,V)+kN(r,V)\\ &\leq m\Big(r,\frac{V^{(k)}}{V}\Big)+(k+1)T(r,V) \leq O\Big(\log\frac{2}{1-r}\Big), \quad k=1,2,\dots. \end{align*} In view of the coefficients \alpha_j are polynomials (with numerical coefficients) in the variables V, V', V'', \dots, we have T(r,\alpha_j)\leq O\Big(\log\frac{2}{1-r}\Big), \quad j=1,2,\dots, l. The proof is complete. \end{proof} \section{Proof of Theorem \ref{thm1.3}} Suppose that f\not\equiv0 is a solution of f^{(k)}+A(z)f=0 in \Omega. Then F(\zeta)=f(z(\zeta)) is a solution of the differential equation $$\alpha_kF^{(k)}(\zeta)+\alpha_{k-1}F^{(k-1)}(\zeta)+\dots +\alpha_1F'(\zeta)+B(\zeta)F(\zeta)=0$$ in \Delta, where B(\zeta)=A(z(\zeta)), \alpha_j(j=1,2,\dots,k) are described in Lemma \ref{lem2.3}. From the condition \eqref{1.5} and the inequality \eqref{2.3}, we obtain that B is admissible in \Delta. By Lemma \ref{lem2.3}, we get that T(r,\alpha_j)=O(\log(1-r)^{-1})(j=1,2,\dots,k), so \alpha_j(j=1,2,\dots,k) are non-admissible in \Delta. By Theorem \ref{thm1.4}, we have \rho_\Delta(F)=\infty. Combining this with \eqref{2.4} leads to \rho_\Omega(f)=\infty. Theorem \ref{thm1.3} follows. \section{Proof of Theorem \ref{thm1.2}} Suppose that f\not\equiv0 is a solution of \eqref{1.3} in \Omega. In view of \eqref{2.5}, we have \begin{align*} &\sum_{i=1}^kA_i(z(\zeta))f^{(i)}(z(\zeta))+A_0(z(\zeta))f(z(\zeta))\\ &=\sum_{i=1}^kA_i(z(\zeta))\sum_{j=1}^i\alpha_jF^{(j)}(\zeta) +A_0(z(\zeta))f(z(\zeta))\\ &=\sum_{j=1}^k\alpha_j\sum_{i=j}^kA_i(z(\zeta))F^{(j)}(\zeta) +A_0(z(\zeta))f(z(\zeta)). \end{align*} Then F(\zeta)=f(z(\zeta)) is a solution of the differential equation $$B_k(\zeta)F^{(k)}(\zeta)+B_{k-1}(\zeta)F^{(k-1)}(\zeta)+\dots +B_1(\zeta)F'(\zeta)+B_0(\zeta)F(\zeta)=0$$ in \Delta, where B_0(\zeta)=A_0(z(\zeta)), B_j(\zeta)=\alpha_j\sum_{i=j}^kA_i(z(\zeta))(j=1,2,\dots,k). Since T(r,\alpha_j)=O(\log(1-r)^{-1})(j=1,2,\dots,k), it follows that \begin{align*} T(r, B_j) &\leq T(r,\alpha_j)+\sum_{i=j}^kT(r,A_i(z(\zeta)))+O(1)\\ &=\sum_{i=j}^kT(r,A_i(z(\zeta)))+O(\log(1-r)^{-1}),\quad j=1,2,\dots,k. \end{align*} By Lemma \ref{lem2.2}, we have \begin{gather*} \frac{1}{\omega}\rho_{\Omega_\varepsilon}(A_0)\leq\rho_\Delta(B_0)+1,\\ \rho_\Delta(B_j)\leq \frac{1}{\omega}\max_{1\leq j\leq k}\rho_\Omega(A_j),\quad \text{for } j=1,2,\dots,k. \end{gather*} Combining the above with the condition (i) gives \rho_\Delta(B_j)<\rho_\Delta(B_0), \quad\text{for } j=1,2,\dots,k.  By Theorem \ref{thm1.4}, we have $\rho_\Delta(F)=\infty$. Combining this with \eqref{2.4} leads to $\rho_\Omega(f)=\infty$. In view of $\mathcal{T}(r,\Omega, A_j)=O(\log r)$ it follows that $T(r,B_j)=O(\log(1-r)^{-1})$. From the condition \eqref{1.2} and the inequality \eqref{2.3}, we obtain that $B_0$ is admissible in $\Delta$. By Theorem \ref{thm1.4}, we have that $\rho_\Delta(F)=\infty$. 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