\documentclass[reqno]{amsart} \usepackage{amsfonts} \usepackage{hyperref} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2007(2007), No. 83, pp. 1--7.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2007 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2007/83\hfil Iterated order of solutions] {Iterated order of solutions of certain linear differential equations with entire coefficients} \author[S. Hamouda\hfil EJDE-2007/83\hfilneg] {Saada Hamouda} \address{Saada Hamouda \newline Department of Mathematics, University of Mostaganem, B. P. 227 Mostaganem, Algeria} \email{hamouda\_saada@yahoo.fr} \thanks{Submitted March 8, 2007. Published June 6, 2007.} \subjclass[2000]{30D35, 34M10} \keywords{Linear differential equations; growth of solutions; iterated order} \begin{abstract} In this paper, we investigate the iterated order of solutions of homogeneous and nonhomogeneous linear differential equations where the coefficients are entire functions and satisfy certain growth conditions. \end{abstract} \maketitle \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \numberwithin{equation}{section} \section{Introduction and statement of results} In this paper, we assume that the reader is familiar with the fundamental results and the standard notation of the Nevanlinna value distribution theory \cite{haym1}. For $n\geq 2$, we consider the linear differential equation $$w^{(n)}+A_{n-1}(z)w^{^{(n-1)}}+\dots +A_{0}(z)w=B(z), \label{e1}$$ where $A_{0}(z),\dots ,A_{n-1},B(z)$ are entire functions with $A_{0}(z)\not\equiv 0$ or $B(z)\not\equiv 0$. It is well known that all solutions of \eqref{e1} are entire functions, and if some of the coefficients are transcendental, then \eqref{e1} has at least one solution of infinite order. Thus, the question which arises is: What conditions on $A_{0}(z),\dots ,A_{n-1},B(z)$ will guarantee that every solution $w\not\equiv 0$ has infinite order? For the above question, there are many results, (see for example \cite{gund1,gund3,bel.ham1}). Gundersen and Steinbart \cite{gund3} proved the following results. \begin{theorem}[\cite{gund3}] \label{thm1.1} Let $\mu$, $\theta_{1}$, $\theta _{2}$ be real constants satisfying $\mu >0$ and $\theta _{1}<\theta _{2}$. Suppose that, in the differential equation $$w^{(n)}+A_{n-1}(z)w^{^{(n-1)}}+\dots +A_{0}(z)w=B(z), \label{g1}$$ there exists a unique coefficient $A_{q}(z)$ such that for any $\theta \in (\theta _{1}, \theta_{2})$ there exist real constants $\alpha =\alpha (\theta )$ and $\beta =\beta (\theta )$ satisfying $0\leq \beta <\alpha$, so that the following conditions hold as $z\to \infty$ along $\arg z=\theta$: \begin{gather*} | A_{q}(z)| \geq \exp \{ (\alpha +o(1))| z| ^{\mu }\} , \\ | A_{k}(z)| \leq \exp \{ (\beta +o(1))| z| ^{\mu }\} \quad \text{for all }k\neq q, \\ | B(z)| \leq \exp \{ (\beta +o(1))| z| ^{\mu }\} . \end{gather*} Assume that $\alpha (\theta )$ and $\beta(\theta )$ are continuous functions on $\theta_{1}<\theta <\theta _{2}$. Suppose that $w$ is transcendental solution of \eqref{g1} with $\sigma (w)<\infty$. If $l\geq q$ is an integer, then for any $\theta \in (\theta_{1},\theta _{2})$ we have \begin{equation*} | w^{(l)}(z)| \leq \exp \{-(\alpha -\beta +o(1))| z|^{\mu }\} \end{equation*} as $z\to \infty$ along $\arg z=\theta$. \end{theorem} \begin{corollary}[\cite{gund3}] \label{coro1.1} Let $\theta _{1},\theta_{2},\dots ,\theta _{m}$ be a finite set of real numbers that satisfy $\theta _{1}<\theta _{2}<\dots <\theta _{m}+2\pi$. Suppose that for each $i=1,2,\dots ,m-1$, there exists in \eqref{g1} one particular coefficient $A_{q_{i}}(z)$ and a corresponding constant $\mu _{i}>0$, such that for any $\theta \in (\theta _{i},\theta _{i+1})$ there exist constants $\alpha _{i}=\alpha _{i}(\theta )$ and $\beta _{i}=\beta _{i}(\theta )$ satisfying $0\leq \beta _{i}<\alpha _{i}$, so that the following conditions hold as $z\to \infty$ along $\arg z=\theta$: \begin{gather*} | A_{q_{i}}(z)| \geq \exp \{ (\alpha _{i}+o(1))| z| ^{\mu_{i}}\} , \\ | A_{k}(z)| \leq \exp \{ (\beta_{i}+o(1))| z| ^{\mu _{i}}\} \quad \text{for all }k\neq q_{i},\\ | B(z)| \leq \exp \{ (\beta_{i}+o(1))| z| ^{\mu _{i}}\} . \end{gather*} For each $i=1,2,\dots ,m-1$, assume that $\alpha _{i}(\theta )$ and $\beta _{i}(\theta )$ are continuous functions on $\theta _{i}<\theta <\theta_{i+1}$. Then every transcendental solution $w$ of \eqref{g1} satisfies $\sigma (w)=\infty$. \end{corollary} In 2001, Bela\"{\i}di and Hamouda proved the following result. \begin{theorem}[\cite{bel.ham1}] \label{thm1.2} Let $A_{0}(z),\dots ,A_{n-1}$ be entire functions with $A_{0}(z) \not\equiv 0$, such that for some real constants $\alpha ,\beta ,\mu ,\theta _{1},\theta_{2}$, with $0\leq \beta <\alpha$, $\mu >0$, $\theta _{1}<\theta _{2}$, we have \begin{gather*} | A_{0}(z)| \geq e^{\alpha |z| ^{\mu }},\\ | A_{k}(z)| \leq e^{\beta |z| ^{\mu }}\quad (k=1,\dots ,n-1), \end{gather*} as $z\to \infty$ with $\theta _{1}\leq \arg z\leq \theta _{2}$. Then every solution $w\not\equiv 0$ of the differential equation $$w^{(n)}+A_{n-1}(z)w^{^{(n-1)}}+\dots +A_{0}(z)w=0 \label{homg}$$ has infinite order. \end{theorem} Another question is: For solutions of infinite order, how can we describe precisely their growth? For $r\in [ 0,\infty )$, we define $\exp _{0}r=r$, $\exp_{1}r=e^{r}$ and $\exp _{i+1}r=\exp (\exp _{i}r)$ $(i\in\mathbb{N})$. For $r$ sufficiently large, we define $\log _{0}r=r$, $\log_{1}r=\log r$, $\log _{i+1}r=\log (\log _{i}r)$ $(i\in\mathbb{N})$. Also, we can define $\exp _{-i}r=\log _{i}r$ and $\log_{-i}r=\exp _{i}r$. To express the rate of growth of entire function of infinite order, we introduce the notion of iterated order \cite{sato,lain,jan.vol}. \begin{def} \label{def1.1} \rm The iterated $i$-order of entire function $w$ is defined by $$\sigma _{i}(w)=\limsup_{r\to +\infty } \frac{\log _{i}T(r,w)}{\log r}=\limsup_{r\to +\infty } \frac{\log _{i+1}M(r,w)}{\log r} \quad (i\in \mathbb{N}). \label{3}$$ \end{def} Recently, Tu, Chen and Zheng proved the following result. \begin{theorem}[\cite{jimtu}] \label{thm1.3} Let $A_{0}(z),\dots ,A_{n-1}$ be entire functions with $A_{0}(z)\not\equiv 0$, such that for real constants $\alpha$, $\beta$, $\mu$, $\theta _{1}$, $\theta _{2}$, and positive integer $p$ with $0\leq \beta <\alpha$, $\mu >0$, $\theta _{1}<\theta _{2}$, we have \begin{gather*} | A_{0}(z)| \geq \exp _{p}\{ \alpha| z| ^{\mu }\}, \\ | A_{k}(z)| \leq \exp _{p}\{ \beta| z| ^{\mu }\} ,\quad (k=1,\dots ,n-1), \end{gather*} as $z\to \infty$ with $\theta _{1}\leq \arg z\leq \theta _{2}$. Then $\sigma _{p+1}(w)\geq \mu$ holds for all non-trivial solutions of \eqref{homg}. \end{theorem} In this paper, we will prove the following results. \begin{theorem} \label{thm1.4} Let $A_{0}(z),\dots ,A_{n-1}$ be entire functions with $A_{0}(z)\not\equiv 0$, and let $\theta _{1},\theta _{2},\dots ,\theta _{m}$ be a finite set of real numbers that satisfy $0=\theta _{1}<\theta_{2}<\dots <\theta _{m}=2\pi$ such that for each $i\in \{1,2,\dots ,m-1\}$. Then there exists in \eqref{homg} one particular coefficient $A_{s_{i}}(z)$ and a corresponding constant $\mu _{i}>0$, such that for any $\theta \in (\theta _{i}$, $\theta _{i+1})$ there exist constants $\alpha _{i}$ and $\beta _{i}$ satisfying $0\leq \beta _{i}<\alpha _{i}$, we have \begin{gather} | A_{s_{i}}(z)| \geq \exp _{p}\{ \alpha _{i}| z| ^{_{\mu _{i}}}\}, \label{4} \\ | A_{k}(z)| \leq \exp _{p}\{ \beta _{i}| z| ^{\mu _{i}}\} \quad \text{for all } k\neq s_{i}), \label{5} \end{gather} as $z\to \infty$ along $\arg z=\theta$, where $p\geq 1$ is an integer. Then, every transcendental solution $w$ of \eqref{homg} satisfies \begin{equation*} \sigma _{p+1}(w)\geq \min_{1\leq i\leq m-1} \{\mu _{i}\} . \end{equation*} \end{theorem} \begin{theorem} \label{thm1.5} Suppose that $A_{0}(z),\dots ,A_{n-1},B(z)$ are entire functions such that for real constants $\alpha$, $\beta$, $\mu$, $\theta _{1}$, $\theta _{2}$, with $0\leq \beta <\alpha$, $\mu >0$, $0\leq \theta _{1}<\theta_{2}\leq 2\pi$, we have, for some integer $s$ $(1\leq s\leq n-1)$, \begin{gather} | A_{s}(z)| \geq \exp _{p}\{ \alpha| z| ^{\mu }\} , \label{6s} \\ | A_{k}(z)| \leq \exp _{p}\{ \beta | z| ^{\mu }\} \quad \text{for all } k\neq s, \label{7s}\\ | B(z)| \leq \exp _{p}\{ \beta | z| ^{\mu }\} \label{8s} \end{gather} as $z\to \infty$ with $\theta _{1}\leq \arg z\leq \theta _{2}$, where $p\geq 1$ is an integer. Given $\varepsilon >0$ small enough. If $w$ is a transcendental solution with $\sigma_{p}(w)<\infty$ of the differential equation $$w^{(n)}+A_{n-1}(z)w^{^{(n-1) }}+\dots +A_{0}(z)w=B(z), \label{9s}$$ then there exists a constant $M>0$ such that, for each integer $l\geq s$, we have $$| w^{(l)}(z)| \leq M \label{10s}$$ along any ray $\arg z=\theta \in S(\varepsilon ) =\{ \theta :\theta _{1}+\varepsilon \leq \arg z\leq \theta _{2}-\varepsilon \}$. \end{theorem} \begin{corollary} \label{coro1.2} Let $A_{0}(z),\dots ,A_{n-1}$, $B(z)$ be entire functions with $B(z)\not\equiv 0$, let $\theta _{1},\theta _{2},\dots ,\theta_{m}$ be a finite set of real numbers that satisfy $0=\theta_{1}<\theta _{2}<\dots <\theta _{m}=2\pi$ such that for each $i\in\{ 1,2,\dots ,m-1\}$ there exists in \eqref{9s} one particular coefficient $A_{s_{i}}(z)$ and a corresponding constants $\mu _{i}$, $\alpha _{i}$, $\beta _{i}$ satisfying $0\leq \beta _{i}<\alpha _{i}$, $\mu _{i}>0$, we have \begin{gather} | A_{s_{i}}(z)| \geq \exp _{p}\{ \alpha _{i}| z| ^{_{\mu _{i}}}\} , \label{11s} \\ | A_{k}(z)| \leq \exp _{p}\{ \beta _{i}| z| ^{\mu _{i}}\} \quad \text{for all } k\neq s_{i}, \label{12s}\\ | B(z)| \leq \exp _{p}\{ \beta _{i}| z| ^{\mu _{i}}\} \label{13s} \end{gather} as $z\to \infty$ with $\theta _{i}<\arg z<\theta _{i+1}$, where $p\geq 1$ is an integer. Then, every transcendental solution $w$ of \eqref{9s} satisfies $\sigma _{p}(w)=\infty$. \end{corollary} In the above corollary, the differential equation \eqref{9s} may have a polynomial solution. For example, $w(z)=z$ is a solution of \begin{equation*} w^{(4)}+e^{z}w^{(3)}+e^{-z}w^{\prime \prime }+w^{\prime }+w=z+1. \end{equation*}% In general, $w(z)=z$ is a solution of the differential equation \begin{equation*} w^{(n)}+A_{n-1}(z)w^{(n-1)}+\dots +A_{2}(z)w^{\prime \prime }+w^{\prime }+w=z+1, \end{equation*} where the coefficients $A_{2}(z),\dots ,A_{n-1}(z)$ satisfy the conditions \eqref{11s} and \eqref{12s}. Our proofs depend mainly on the following Lemmas. \section{Auxiliary Lemmas} \begin{lemma}[\cite{gund2}] \label{lem2.1} Let $w(z)$ be a transcendental entire function. Let $$\Gamma =\{ (k_{1},q_{1}),(k_{2},q_{2}),\dots ,(k_{m},q_{m})\}$$ denote a finite set of distinct pairs of integers that satisfy $k_{i}>q_{i}\geq 0$ $(i=1,\dots ,m)$ and let $\alpha >0$ be a given real constant. Then there exists a set $E\subset [ 0,2\pi )$ that has linear measure zero and a constant $c>0$ that depend only on $\alpha$, such that if $\psi _{0}\in [ 0,2\pi )-E$, then there is a constant $R_{0}=R_{0}(\psi _{0})>1$ such that for all $z$ satisfying $\arg z=\psi _{0}$ and $| z| =r\geq R_{0}$, and for all $(k,q)\in \Gamma$, we have \begin{equation*} \big| \frac{w^{(k)}(z)}{w^{(q)}(z)}\big| \leq c[ T(\alpha r,w)\frac{1}{r}\log ^{\alpha }(r)\log T(\alpha r,w)] ^{k-q}. \end{equation*} \end{lemma} From the above Lemma, which is also \cite[Theorem 2]{gund2}, we obtain the following result. \begin{lemma} \label{lem2.2} Let $w$ be a transcendental entire function with $\sigma _{p}(w)=\sigma <\infty$ $(p\geq 1)$. Let $\Gamma =\{ (k_{1},q_{1}),(k_{2},q_{2}),\dots ,(k_{m},q_{m})\}$ denote a finite set of distinct pairs of integers that satisfy $k_{i}>q_{i}\geq 0$ $(i=1,\dots ,m)$. 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 $R_{0}=R_{0}(\psi _{0})>1$ such that for all $z$ satisfying $\arg z=\psi _{0}$ and $|z| =r\geq R_{0}$, and for all $(k,q)\in \Gamma$, we have \begin{equation*} \big| \frac{w^{(k)}(z)}{w^{(q)}(z)}\big| \leq [ \exp _{p-1}\{ r^{\sigma+\varepsilon }\} ] ^{k-q}, \end{equation*} where $\varepsilon >0$. \end{lemma} \begin{proof} The definition \begin{equation*} \sigma _{p}(w)=\limsup_{r\to +\infty } \frac{\log _{p}T(r,w)}{\log r}=\sigma \end{equation*} implies that for any given $\varepsilon ^{\prime}>0$ there exists $r_{0}>0$ such that for all $r\geq r_{0}$ we have \begin{equation*} \frac{\log _{p}T(r,w)}{\log r}<\sigma +\varepsilon ^{\prime}; \end{equation*} which implies $$T(r,w)<\exp _{p-1}\{ r^{\sigma +\varepsilon ^{\prime}}\} . \label{tr}$$ Combining \eqref{tr} with Lemma \ref{lem2.1}, for $\alpha >0$, there exists a set $E\subset [ 0,2\pi )$ that has linear measure zero and a constant $c>0$, such that if $\psi _{0}\in [ 0,2\pi )-E$, then there is a constant $r_{0}^{\prime}=r_{0}^{\prime}(\psi _{0})>1$ such that for all $z$ satisfying $\arg z=\psi _{0}$ and $| z|=r\geq r_{0}^{\prime}$, \begin{equation*} \big| \frac{w^{(k)}(z)}{w^{(q)}(z)}\big| \leq c\big[ \exp _{p-1}\{ (\alpha r)^{\sigma +\varepsilon ^{\prime}}\} \exp _{p-2}\{ ( \alpha r)^{\sigma +\varepsilon ^{\prime}}\} \big] ^{k-q}. \end{equation*} Then, there exists a constant $\varepsilon >\varepsilon ^{\prime}>0$ and $R_{0}$ large enough, such that for $| z| =r\geq R_{0}$ \begin{equation*} \big| \frac{w^{(k)}(z)}{w^{(q)}(z)}\big| \leq [ \exp _{p-1}\{ r^{\sigma +\varepsilon }\}] ^{k-q}. \end{equation*} \end{proof} \begin{lemma}[\cite{gund1,gund3,lain.yan}] \label{lem2.3} Let $w(z)$ be an entire function and suppose that $| w^{(k)}(z)|$ is unbounded on some ray $\arg z=\theta$. Then there exists an infinite sequence of points $z_{j}=r_{j}e^{i\theta }$ $(j=1,2,\dots )$, where $r_{j}\to +\infty$, such that $w^{(k)}(z_{j})\to \infty$ and \begin{equation*} \big| \frac{w^{(q)}(z_{j})}{w^{(k)}(z_{j})}\big| \leq \frac{1}{(k-q)!} (1+o(1))| z_{j}| ^{k-q}\quad q=0,\dots ,k-1. \end{equation*} \end{lemma} \section{Proof of Theorems} \begin{proof}[Proof of Theorem \protect\ref{thm1.4}] From \eqref{homg} we can write \begin{aligned} |A_{s_{i}}| & \leq | \frac{w^{(n)}}{ w^{(s_{i})}}| +| A_{n-1}|| \frac{w^{(n-1)}}{w^{(s_{i})}}| +\dots +| A_{s_{i}+1}| | \frac{w^{(s_{i}+1)}}{w^{(s_{i})}}| \\ &\quad +| A_{s_{i}-1}| | \frac{w^{(s_{i}-1)}}{w^{(s_{i})}}| +\dots +| A_{1}| | \frac{w'}{w^{(s_{i})}}| +| A_{0}| | \frac{w}{w^{(s_{i})}}| . \end{aligned} \label{12b} By Lemma \ref{lem2.1} and taking into account $\frac{1}{r}\log ^{2}(r)<1$ and \begin{equation*} \log T(2r,w^{(s_{i})})0$and a set$E\subset \lbrack 0,2\pi )$that has linear measure zero, such that if$\theta \in \lbrack 0,2\pi )-E$, then there is a constant$r_{1}=r_{1}(\theta )>1$such that for all$z$satisfying$\arg z=\theta $and$|z|=r\geq r_{1}$, we have $$\big|\frac{w^{(k)}(z)}{w^{(s_{i})}(z)}\big|\leq c[T(2r,w^{(s_{i})})]^{2n},\quad k=s_{i}+1,\dots ,n. \label{13}$$ Since$w$is transcendental, then there exist a real constants$\varphi _{1},\varphi _{2}$, with$0\leq \varphi _{1}<\varphi _{2}\leq 2\pi $, such that for all$\theta \in (\varphi _{1},\varphi _{2})$,$|w^{(s_{i})}(z)|$is unbounded as$z\to \infty $along$\arg z=\theta $. The sector$(\varphi _{1},\varphi _{2})$will certainly be intersected with, at least, one of the sectors$(\theta _{i},\theta _{i+1})(i=1,\dots ,m-1)$in a sector$(\varphi _{1}^{\prime },\varphi _{2}^{\prime })$,$(\varphi _{1}\leq \varphi _{1}^{\prime }<\varphi _{2}^{\prime }\leq \varphi _{2})$. By Lemma \ref{lem2.3}, if$\theta \in (\varphi _{1}^{\prime },\varphi _{2}^{\prime })$, there exists an infinite sequence of points$z_{j}=r_{j}e^{i\theta }$where$r_{j}\to \infty $, such that$w^{(s_{i})}(z_{j})\to \infty $and $$|\frac{w^{(k)}(z_{j})}{w^{(s_{i})}(z_{j})}|\leq (1+o(1))|z_{j}|^{s_{i}-k} \label{14}$$ as$z_{j}\to \infty $, for all$k$satisfying$0\leq k\leq s_{i}-1$. Combining \eqref{4}, \eqref{5}, \eqref{13} and \eqref{14} with \eqref{12b}, we obtain, for$r_{j}$large enough, \begin{equation*} \exp _{p}\{\alpha _{i}r_{j}^{\mu _{i}}\}\leq A[T(2r_{j},w^{(s_{i})})]^{2n}\exp _{p}\{\beta _{i}r_{j}^{\mu _{i}}\}, \end{equation*} where$A>0;$so that \begin{equation*} \exp _{p}\{\alpha _{i}r_{j}^{\mu _{i}}\}\exp \{-\exp _{p-1}\{\beta _{i}r_{j}^{\mu _{i}}\}\}\leq A[T(2r_{j},w^{(s_{i})})]^{2n}, \end{equation*} which implies \begin{equation*} \limsup_{r\to +\infty }\frac{\log _{p+1}T(r,w^{(s_{i})})}{\log r} \geq \mu _{i}. \end{equation*} Since \begin{equation*} T(r,w^{(s_{i})})\leq (s_{i}+1)T(r,w)+S(r,w), \end{equation*} where$S(r,w)=o\{T(r,w)\}$as$r\to \infty $, it follows that$\sigma _{p+1}(w)\geq \mu _{i}$for, at least, one$i\in \{1,\dots ,m-1\}$. Thus, in general, we have \begin{equation*} \sigma _{p+1}(w)\geq \min_{1\leq i\leq m-1}\{\mu _{i}\}. \end{equation*} \end{proof} \begin{proof}[Proof of Theorem \protect\ref{thm1.5}] Suppose that$w$is a transcendental solution with$\sigma _{p}(w)=\sigma <\infty of \eqref{9s}. From \eqref{9s}, we can write \begin{aligned} &w^{(s)}\Big[ \frac{w^{(n)}}{w^{(s)}} \frac{1}{A_{s}}+\frac{w^{(n-1)}}{w^{(s)}}\frac{ A_{n-1}}{A_{s}}+\dots +\frac{w^{(s+1)}}{w^{(s)}}\frac{ A_{s+1}}{A_{s}}\\ &+1 \frac{w^{(s-1)}}{w^{(s)}}\frac{A_{s-1}}{ A_{s}}+\dots +\frac{w}{w^{(s)}}\frac{A_{0}}{A_{s}}\Big] =\frac{B}{A_{s}} \end{aligned} \label{15} From Lemma \ref{lem2.2}, there exists a setE\subset \lbrack 0,2\pi )$of linear measure zero such that for all$k=s+1,\dots ,n$and all$r\geq r_{2}$large enough, we have $$\big|\frac{w^{(k)}(z)}{w^{(s)}(z)}\big|\leq \lbrack \exp _{p-1}\{r^{\sigma +\varepsilon }\}]^{k-s}, \label{16}$$ along any ray$\arg z=\psi \in \lbrack 0,2\pi )-E$, provided$\varepsilon >0$. If$|w^{(s)}(z)|$is unbounded on some ray$\arg z=\phi \in S(0)-E$, then by Lemma \ref{lem2.3}, there exists a sequence of points$z_{j}=r_{j}e^{i\phi }$,$r_{j}\to \infty $such that$w^{(s)}(z_{j})\to \infty $and $$\big|\frac{w^{(q)}(z_{j})}{w^{(s)}(z_{j})}|\leq \frac{1}{(s-q)!}% (1+o(1))|z_{j}\big|^{s-q}\leq 2|z_{j}|^{n} \label{17}$$ for all$q=0,\dots ,s-1$and all$j$large enough. Combining now \eqref{6s}, \eqref{7s}, \eqref{8s}, \eqref{16} and \eqref{17} with \eqref{15} yelds$w^{(s)}(z_{j})\to 0$as$z_{j}\to \infty $, a contradiction. Hence$| w^{(s)}(z_{j})| $must be bounded on any$\arg z=\phi \in S(0)-E$. By Phragm\'{e}n-Lindel\"{o}f theorem, we conclude that$| w^{(s)}(z)| $is bounded, say$|w^{(s)}(z)| s$, \begin{equation*} | w^{(l)}(z)| 0$ small enough), we obtain |w^{(l)}(z)|