\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2012 (2012), No. 195, pp. 1--15.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2012 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2012/195\hfil Meromorphic coefficients of (p,q)-order] {Solutions for linear differential equations \\ with meromorphic coefficients of (p,q)-order\\ in the plane} \author[L.-M. Li, T.-B. Cao \hfil EJDE-2012/195\hfilneg] {Lei-Min Li, Ting-Bin Cao} % in alphabetical order \address{Lei-Min Li \newline Department of Mathematics, Nanchang University, Nanchang, Jiangxi 330031, China} \email{leiminli@hotmail.com} \address{Ting-Bin Cao \newline Department of Mathematics, Nanchang University, Nanchang, Jiangxi 330031, China} \email{tbcao@ncu.edu.cn, tingbincao@hotmail.com (corresponding author)} \thanks{Submitted January 6, 2012. Published November 8, 2012.} \subjclass[2000]{34M10, 30D35, 34M05} \keywords{Linear differential equation; meromorphic function; (p,q)-order;\hfill\break\indent Nevanlinna theory} \begin{abstract} In this article we study the growth of meromorphic solutions of high order linear differential equations with meromorphic coefficients of $(p,q)$-order. We extend some previous results due to Bela\"{\i}di, Cao-Xu-Chen, Kinnunen, Liu- Tu -Shi, and others. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{definition}[theorem]{Definition} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction and main results} For $k\geq 2$, consider the linear differential equations \begin{gather}\label{E-1} f^{(k)}+A_{k-1}(z)f^{(k-1)}+\dots+A_1(z)f'+A_0(z)f=0,\\ \label{E-2} f^{(k)}+A_{k-1}(z)f^{(k-1)}+\dots+A_1(z)f'+A_0(z)f=F(z), \end{gather} where $A_0\not\equiv 0$ and $F\not\equiv 0$. When the coefficients $A_0, A_1, \dots, A_{k-1}$ and $F$ are entire functions, it is well known that all solutions of \eqref{E-1} and \eqref{E-2} are entire functions, and that if some coefficients of \eqref{E-1} are transcendental then \eqref{E-1} has at least one solution with infinite order. We refer to \cite{12} for the literature on the growth of entire solutions of \eqref{E-1} and \eqref{E-2}. As far as we known, Bernal \cite{1} firstly introduced the idea of iterated order to express the fast growth of solutions of complex linear differential equations. Since then, many authors obtained further results on iterated order of solutions of \eqref{E-1} and \eqref{E-2}, see e.g. \cite{belaidi-1, belaidi-2, 1, 3, 5, 11, 15}. Recently, Liu, Tu and Shi \cite{13} firstly introduced the concept of (p, q)-order for the case $p\geq q\geq 1$ to investigate the entire solutions of \eqref{E-1} and \eqref{E-2}, and obtained some results which improve and generalize some previous results. \begin{theorem}[{\cite[Theorems 2.2-2.3]{13}}] \label{T-A} Let $p\geq q\geq 1$, and let $A_0, A_1, \dots, A_{k-1}$ be entire functions such that either $$\max\{\sigma_{(p,q)}(A_j): j \neq 0\} < \sigma_{(p,q)}(A_0) < +\infty,$$ or \begin{gather*} \max\{\sigma_{(p,q)}(A_j): j \neq 0\} \leq \sigma_{(p,q)}(A_0) < +\infty,\\ \max\{\tau_{(p,q)}(A_j): \sigma_{(p,q)}(A_j) = \sigma_{(p,q)}(A_0) > 0\} < \tau_{(p,q)}(A_0), \end{gather*} then every nontrivial solution $f$ of \eqref{E-1} satisfies $\sigma_{(p+1,q)}(f) = \sigma_{(p,q)}(A_0)$. \end{theorem} Recently, Cao, Xu and Chen \cite{3} considered the growth of meromorphic solutions of equations \eqref{E-1} and \eqref{E-2} with meromorphic coefficients of finite iterated order, and obtained some results which improve and generalize some previous results. \begin{theorem}[{\cite[Theorem 2.1]{3}}] \label{T-B} Let $A_0, A_1, \dots, A_{k-1}$ be meromorphic functions in the plane, and let $i(A_0)=p$ $(00\} <\tau_{(p,q)}(A_0). \end{gather*} Then any nonzero meromorphic solution$f$whose poles are of uniformly bounded multiplicities of \eqref{E-1} satisfies $\sigma_{(p+1,q)}(f)=\sigma_{(p,q)}(A_0).$ \end{theorem} Obviously, Theorems \ref{T-2} and \ref{T-3} are a generalization of Theorems \ref{T-A} and \ref{T-B}. Considering nonhomogeneous linear differential equation \eqref{E-2}, we obtain the following three results. \begin{theorem}\label{T-4} Assume that$A_0,A_1,\dots,A_{k-1},F\not\equiv 0$be meromorphic functions in the plane satisfying $\max\{\sigma_{(p,q)}(A_j), \lambda_{(p,q)}(\frac{1}{A_0}), \sigma_{(p+1,q)}(F):j=1,2,\dots,k-1\}<\sigma_{(p,q)}(A_0),$ then all meromorphic solutions$f$whose poles are of uniformly bounded multiplicities of \eqref{E-2} satisfy $\overline{\lambda}_{(p+1,q)}(f)=\lambda_{(p+1,q)}(f) =\sigma_{(p+1,q)}(f)=\sigma_{(p,q)}(A_0)$ with at most one exceptional solution$f_0$satisfying$\sigma_{(p+1,q)}(f_0)<\sigma_{(p,q)}(A_0)$. \end{theorem} \begin{theorem}\label{T-5} Let$A_0,A_1,\dots,A_{k-1},F\not\equiv 0$be meromorphic functions in the plane satisfying $\max\{\sigma_{(p,q)}(A_j):j=0,1,\dots,k-1\}<\sigma_{(p+1,q)}(F).$ Suppose that all solutions of \eqref{E-2} are meromorphic functions whose poles are of uniformly bounded multiplicities, then$\sigma_{(p+1,q)}(f)=\sigma_{(p+1,q)}(F)$holds for all solutions of \eqref{E-2}. \end{theorem} \begin{theorem}\label{T-6} Let$H\subset (1,\infty)$be a set satisfying$\overline{\log dens}\{|z|: |z|\in H\}>0$and let$A_0,A_1,\dots$,$A_{k-1}$,$F\not\equiv 0$be meromorphic functions in the plane satisfying $\max\{\sigma_{(p,q)}(A_j):j=1,2,\dots,k-1\}<\alpha_1,$ where$\alpha_1$is a constant, and there exists another constant$\alpha_2$($\alpha_2<\alpha_1$) such than for any given$\epsilon$($0<\epsilon<\alpha_1-\alpha_2$), we have $|A_0{(z)}|\geq\exp_{p+1}\{(\alpha_1-\epsilon)\log_qr\}, |A_j{(z)}|\leq\exp_{p+1}\{\alpha_2\log_qr\}$ for$|z|\in H$,$j=1,2,\dots,k-1$. Then we have: (i) If$\sigma_{(p+1,q)}(F)\geq \alpha_1$, then all meromorphic solutions whose poles are of uniformly bounded multiplicities of \eqref{E-2} satisfy $\sigma_{(p+1,q)}(f)=\sigma_{(p+1,q)}(F).$ (ii) If$\sigma_{(p+1,q)}(F)< \alpha_1$,then all meromorphic solutions whose poles are of uniformly bounded multiplicities of \eqref{E-2} satisfy $\overline{\lambda}_{(p+1,q)}(f)=\lambda_{(p+1,q)}(f)=\sigma_{(p+1,q)}(f)=\alpha_1$ with at most one exceptional solution$f_2$satisfying $\sigma_{(p+1,q)}(f_2)<\alpha_1.$ \end{theorem} Recently, B. Bela\"{\i}di \cite{belaidi-3} investigated the growth of solutions of differential equations \eqref{E-1} and \eqref{E-2} with analytic coefficients of$(p,q)$-order in the unit disc. So, it is also interesting to consider the growth of meromorphic solutions of differential equations with coefficients of$(p,q)$-order in the unit disc? \section{Preliminaries and some lemmas} We shall introduce some notation. Let us define inductively, for$r\in [0,+\infty)$,$\exp_1r=e^{r}$and$\exp_{n+1}r=\exp(\exp_nr)$,$n\in\mathbb{N}$. For all$r$sufficiently large, we define$\log_1r=\log^{+}r=\max\{\log r,0\}$and$\log_{n+1}r=\log(\log_nr)$,$n\in\mathbb{N}$. We also denote$\exp_0r=r=\log_0r$,$\log_{-1}r=\exp_1r$and$\exp_{-1}r=\log_1r$. Moreover, we denote the linear measure and the logarithmic measure of a set$E\subset (1,\infty)$by$mE=\int_{E}dt$and$m_{l}E=\int_{E}\frac{dt}{t}$. The upper logarithmic density of$E\subset (1,\infty)$is defined by $\overline{\log {\rm dens}}{E}=\limsup_{r\to \infty} \frac{m_{l}(E\cap\mathcal[1,r])}{\log r }.$ We assume that the reader is familiar with the fundamental results and the standard notations of the Nevanlinna's value distribution theory of meromorphic functions (e.g. see \cite{8,17}), such as$T(r,f)$,$m(r,f)$, and$N(r,f)$. In this section, a meromorphic function$f$means meromorphic in the complex plane$\mathbb{C}$. To express the rate of fast growth of meromorphic functions, we recall the following definitions (e.g. see \cite{1,3,11,12,14}). \begin{definition}\label{D-2.1} \rm The iterated$p$-order$\sigma_p(f)$of a meromorphic function$f$is defined by \begin{equation*} \sigma_p(f)=\limsup_{r\to \infty}\frac{\log_pT(r,f)}{\log r } \quad(p\in\mathbb{N}). \end{equation*} If$f$is an entire function, then \begin{equation*} \sigma _{p,M}(f)=\limsup_{r\to \infty}\frac{\log_{p+1}M(r,f)}{\log r }\quad(p\in\mathbb{N}). \end{equation*} \end{definition} \begin{definition} \rm The growth index of the iterated order of a meromorphic function$f$is defined by $i(f)=\begin{cases} 0 & \text{if f is rational}, \\ \min\{n\in\mathbb{N}:\sigma_n(f)<\infty\} & \text{if f is transendental and \sigma_n(f)<\infty}\\ &\text{for some n\in\mathbb{N}}, \\ \infty & \text{if \sigma_n(f)=\infty for all n\in\mathbb{N}.} \end{cases}$ \end{definition} \begin{definition} \rm The iterated$p$-type of a meromorphic function$f$with iterated order$p$-order$0<\sigma_p(f)<\infty$is defined by \begin{equation*} \tau_p(f)=\limsup_{r\to \infty}\frac{\log_{p-1}T(r,f)}{r^{\sigma_p(f)}} \quad(p\in\mathbb{N}). \end{equation*} If$f$is an entire function, then \begin{equation*} \tau_{p,M}(f)=\limsup_{r\to \infty}\frac{\log_pM(r,f)}{\ r^{\sigma_p(f)}} \quad(p\in\mathbb{N}). \end{equation*} \end{definition} \begin{definition} \rm The iterated convergence exponent of the sequence of zeros of a meromorphic function$f$is defined by $$\lambda_p(f)=\limsup_{r\to \infty} \frac{\log_pN(r,\frac{1}{f})}{\log r}\quad(p\in\mathbb{N}).$$ \end{definition} \begin{definition} \rm The growth index of the iterated convergence exponent of the sequence of zeros of a meromorphic function$f$with iterated order is defined by $i_{\lambda}(f)=\begin{cases} 0 & \text{if } n(r,\frac{1}{f})=O(\log r), \\ \min\{n\in\mathbb{N}:\lambda _n(f)<\infty\} & \text{if \lambda_n(f)<\infty for some n\in\mathbb{N},} \\ \infty & \text{if \lambda _n(f)=\infty for all n\in\mathbb{N}.} \end{cases}$ \end{definition} Similarly, we can use the notation$\overline{\lambda}_p(f)$to denote the iterated convergence exponent of the sequence of distinct zeros, and use the notation$i_{\overline{\lambda}}(f)$to denote the growth index of$\overline{\lambda}_p(f)$. Now, we shall introduce the definition of meromorphic functions of$(p,q)$-order, where$p$,$q$are positive integers satisfying$p\geq q\geq 1$. In order to keep accordance with Definition \ref{D-2.1}, we will give a minor modification to the original definition of$(p,q)$-order (e.g. see \cite{9,10}). \begin{definition}\label{D-2.6} \rm The$(p,q)$-order of a transcendental meromorphic function$f$is defined by \begin{equation*} \sigma_{(p,q)}(f)=\limsup_{r\to \infty}\frac{\log_pT(r,f)}{\log_qr}. \end{equation*} If$f$is a transcendental entire function, then \begin{equation*} \sigma _{(p,q)}(f)=\limsup_{r\to \infty}\frac{\log_{p+1}M(r,f)}{\log_qr }. \end{equation*} \end{definition} It is easy to show that$0\leq\sigma _{(p,q)}\leq \infty$. By Definition \ref{D-2.6} we note that$\sigma_{(1,1)}(f)=\sigma_1(f)=\sigma(f)$,$\sigma_{(2,1)}(f)=\sigma_2(f)$and$\sigma_{(p,1)}(f)=\sigma_p(f)$. \begin{remark} \rm If$f$is a meromorphic function satisfying$0\leq\sigma _{(p,q)}\leq \infty$, then (i)$\sigma _{(p-n,q)}= \infty(np-q$and$\sigma _{(p_1,q_1)}=0$for$p_1-q_1>p-q$. \end{remark} \begin{remark}\rm Suppose that$f_1$is a meromorphic function of$(p,q)$-order$\sigma _1$and$f_2$is a meromorphic function of$(p_1,q_1)$-order$\sigma _2$, let$p\leq p_1$. We can easily deduce the result about their comparative growth: (i) If$p_1-q_1>p-q$, then the growth of$f_1$is slower than the growth of$f_2$. (ii) If$p_1-q_10$, then the growth of$f_1$is slower than the growth of$f_2$if$\sigma _2\geq 1$,and the growth of$f_1$is faster than the growth of$f_2$if$\sigma _2< 1$. (iv) Especially, when$p_1=p$and$q_1=q$then$f_1$and$f_2$are of the same index-pair$(p,q)$. If$\sigma _1>\sigma _2$, then$f_1$grows faster than$f_2$; and if$\sigma _1<\sigma_2$, then$f_1$grows slower than$f_2$. If$\sigma _1=\sigma_2$, Definition 1.6 does not show any precise estimate about the relative growth of$f_1$and$f_2$. \end{remark} \begin{definition}\label{D-2.7} \rm The$(p,q)$-type of a meromorphic function$f$with$(p,q)$-order$\sigma_{(p,q)}(f)\in(0, \infty)$is defined by \begin{equation*} \tau_{(p,q)}(f)=\limsup_{r\to \infty}\frac{\log_{p-1}T(r,f)}{(\log _{q-1}r)^{\sigma_{(p,q)}(f)}}. \end{equation*} \end{definition} \begin{definition} \rm The$(p,q)$convergence exponent of the sequence of zeros of a meromorphic function$f$is defined by $$\lambda_{(p,q)}(f)=\limsup_{r\to \infty} \frac{\log_pN(r,\frac{1}{f})}{\log_qr}.$$ \end{definition} Similarly, we can use the notation$\overline{\lambda}_{(p,q)}(f)$to denote the$(p,q)$convergence exponent of the sequence of distinct zeros of$f$. To prove our results, we need the following lemmas. \begin{lemma}[\cite{6}]\label{L-1} Let$f_1,f_2,\dots,f_{k}$be linearly independent meromorphic solutions of the differential equation \eqref{E-1} with meromorphic functions$A_0,A_1,\dots,A_{k-1}$as the coefficients, then $m(r,A_j)=O\{\log(\max_{1\leq n\leq k}T(r,f_n))\}\quad (j=0,1,\dots,k-1).$ \end{lemma} \begin{lemma}[\cite{18}] \label{L-2} Let$f$be a meromorphic solution of equation \eqref{E-1}, assuming that not all coefficients$A_j$are constants. Given a real constant$\gamma>1$, and denoting$T(r)=\Sigma^{k-1}_{j=0}T(r,A_j)$, we have \begin{gather*} \log m(r,f)0 \end{gather*} outside of an exceptional set$E_s$with$\int_{E_s}t^{s-1}dt<\infty$. \end{lemma} By inequalities in \cite[Chapter 6]{23} and in \cite[Corollary 2.3.5]{12}, we obtain the following lemma. \begin{lemma}\label{L-3} If$f$is a meromorphic function, then $\sigma_{(p.q)}(f)=\sigma_{(p.q)}(f').$ \end{lemma} \begin{lemma}[\cite{7}] \label{L-4} Let$f$be a transcendental meromorphic function, and let$\alpha$be a given constant. Then there exist a set$E_1\subset (1,\infty)$that has finite logarithmic measure and a constant$B>0$depending only on$\alpha$and$(m,n)(m,n\in\{0,1,\dots ,k\})$,$m0$, then for any given$\tau_{(p.q)}(f)>\beta$, there exists a set$E_4\subset (1,\infty)$that has infinite logarithmic measure such that for all$r \in E_4$, we have $\log_{p-1}T(r,f)>\beta(\log_{q-1}{r})^{\sigma_{(p.q)}(f)}.$. \end{lemma} \begin{proof} (i) (see \cite{3}) when$q=1$, it holds absolutely. (ii) when$q \geq 2$, by Definition \ref{D-2.7}, there exists an increasing sequence$\{r_m\}(r_m\to \infty) satisfying (1+\frac{1}{m})r_mm_0 and for any given \epsilon  (0<\epsilon<\tau_{(p.q)}(f)-\beta) we have $$\label{E-3} \log_{p-1}T(r_m,f)>(\tau_{(p.q)}(f)-\epsilon)(\log _{q-1}r_m)^{\sigma_{(p.q)}(f)}.$$ For any r \in [r_m,(1+\frac{1}{m})r_m], we have $\lim_{r_m\to +\infty}\frac{\log _{q-1}r_m}{\log _{q-1}r}=1.$ Since \beta<\tau_{(p.q)}(f)-\epsilon, there exists a positive constant m_1 such that for all m>m_1, we have $(\frac{\log _{q-1}r_m}{\log _{q-1}r})^{\sigma_{(p.q)}(f)}>\frac{\beta}{\tau_{(p.q)}(f)-\epsilon};$ i.e., $$\label{E-4} (\tau_{(p.q)}(f)-\epsilon){(\log _{q-1}r_m)}^{\sigma_{(p.q)}(f)}>\beta{(\log _{q-1}r)}^{\sigma_{(p.q)}(f)}.$$ Now we take m_2=\max\{m_0,m_1\} and E_4=\cup_{m=m_2}^{\infty}[r_m,(1+\frac{1}{m})r_m], then by \eqref{E-3}-\eqref{E-4}, for any r \in E_4, we have \begin{align*} \log_{p-1}T(r, f) &\geq \log_{p-1}T(r_m,f)\\ &>(\tau_{(p.q)}(f)-\epsilon){(\log _{q-1}r_m)}^{\sigma_{(p.q)}(f)}\\ &>\beta{(\log _{q-1}r)}^{\sigma_{(p.q)}(f)}, \end{align*} where $m_{l}E_4=\Sigma_{m=m_2}^{\infty}\int_{r_m}^{(1+\frac{1}{m})r_m}\frac{dt}{t} =\Sigma_{m=m_2}^{\infty}\log (1+\frac{1}{m})=\infty. \qedhere$ \end{proof} \begin{lemma}[\cite{20}]\label{L-9} Let g(r) and h(r) be monotone nondecreasing functions on [0,\infty) such that g(r) \leq h(r) for all r \not\in[0,1]\cup E_{5}, where E_{5}\in (1,\infty) is a set of finite logarithmic measure. Then for any constant \alpha >1, there exists r_0=r_0(\alpha)>0 such that g(r) \leq h(\alpha r) for all r\geq r_0. \end{lemma} \begin{lemma}\label{L-10} Let A_0,A_1,\dots,A_{k-1}, F\not\equiv 0 be meromorphic functions and let f be a meromorphic solution of equation \eqref{E-2}. If $\max\{\sigma_{(p+1,q)}(A_j),\sigma_{(p+1,q)}(F):j=0,1,\dots,k-1\} <\sigma_{(p+1,q)}(f),$ then we have $\overline{\lambda}_{(p+1,q)}(f)=\lambda_{(p+1,q)}(f)=\sigma_{(p+1,q)}(f).$ \end{lemma} \begin{proof} By \eqref{E-1}, we have $$\label{E-5} \frac{1}{f}=\frac{1}{F}(\frac{f^{(k)}}{f}+A_{k-1}\frac{f^{(k-1)}}{f}+\dots +A_0).$$ It is easy to see that if f has a zero at z_0 of order \beta (\beta >k) and if A_0,A_1,\dots,A_{k-1} are all analytic at z_0, then F has a zero at z_0 of order at least \beta-k. Hence \begin{gather}\label{E-6} n(r ,\frac{1}{f}) \leq k \overline n(r ,\frac{1}{f})+n(r ,\frac{1}{F})+\Sigma^{k-1}_{j=0}n(r ,A_j), \\ \label{E-7} N(r ,\frac{1}{f}) \leq k \overline N(r ,\frac{1}{f})+N(r ,\frac{1}{F})+\Sigma^{k-1}_{j=0}N(r ,A_j). \end{gather} By the lemma of the logarithmic derivative and \eqref{E-5}, we have $$\label{E-8} m(r ,\frac{1}{f}) \leq m(r ,\frac{1}{F})+\Sigma^{k-1}_{j=0}m(r ,A_j)+O(\log T(r,f)+ \log r)$$ holds for all |z|=r\not\in E_{6}, where E_{6} is a set of finite linear measure. By \eqref{E-7},\eqref{E-8} and the first main theorem, we have $$\label{E-9} T(r,f)=T(r,\frac{1}{f})+O(1)\leq k \overline N(r ,\frac{1}{f})+T(r,F)+\Sigma^{k-1}_{j=0}T(r ,A_j)+O(\log(r T(r,f)))$$ holds for all sufficiently r \not\in E_{6}. Assume that \max\{\sigma_{(p+1,q)}(A_j),\sigma_{(p+1,q)}(F):j=0,1,\dots,k-1\} <\sigma_{(p+1,q)}(f). By Lemma \ref{L-6}, there exists a sequence \{r_n\}, r_n\not\in E_{6} such that $\lim_{r_n\to \infty}\frac{\log_{p+1}T(r_n,f)}{\log_qr_n }=\sigma_{(p+1,q)}(f)=:\sigma_1.$ Hence , if r_n \not\in E_{6} is sufficiently large, since \sigma_1>0, then we have $$\label{E-10} T(r_n,f) \geq \exp_{p+1}\{(\sigma_1-\epsilon) \log_qr_n\}$$ holds for any given \epsilon (0 <2\epsilon <\sigma_1-\sigma_2), where \sigma_2=\max\{\sigma_{(p+1,q)}(A_j),\sigma_{(p+1,q)}(F):j=0,1,\dots,k-1\}. We have $$\label{E-11} \max\{T(r_n,F),T(r_n,A_j):j=0,1,\dots,k-1\} \leq \exp_{p+1}\{(\sigma_2+\epsilon) \log_qr_n\}.$$ Since \epsilon (0 <2\epsilon <\sigma_1-\sigma_2), then from \eqref{E-10} and \eqref{E-11} we obtain $$\label{E-12} \max\{\frac{T(r_n,F)}{T(r_n,f)},\frac{T(r_n,A_j)}{T(r_n,f)}: j=0,1,\dots,k-1\}\to0 \quad (r_n \to \infty).$$ For sufficiently large r_n, we have O(\log (r_nT(r_n, f))) = O (T (r_n, f)). Hence, by \eqref{E-9} and \eqref{E-12}, we obtain that for sufficiently large r_n\not\in E_{6}, there holds $(1-o(1))T(r_n,f) \leq k \overline {N}(r_n,\frac{1}{f}).$ Then we have \overline{\lambda}_{(p+1,q)}(f) \geq \sigma_{(p+1,q)}(f), and by definitions we have \overline{\lambda}_{(p+1,q)}(f)\leq \lambda_{(p+1,q)}(f)\leq \sigma_{(p+1,q)}(f). Therefore $\overline{\lambda}_{(p+1,q)}(f)=\lambda_{(p+1,q)}(f)=\sigma_{(p+1,q)}(f). \qedhere$ \end{proof} \section{Proofs of Theorems \ref{T-1}-\ref{T-6}} \subsection*{Proof of Theorem \ref{T-1}} We shall divide the proof into two parts. \bullet Firstly, we prove that \sigma_{(p+1,q)}(f)\leq \sigma_{(p,q)}(A_s)\leq \sigma_{(p,q)}(f) holds for every transcendental meromorphic function f of \eqref{E-1}. By \eqref{E-1}, we know that the poles of f can only occur at the poles of A_0, A_1, \dots, A_{k-1}, note that the multiplicities of poles of f are uniformly bounded, then we have \begin{align*} N(r,f) &\leq C_1\overline{N}(r,f) \\ &\leq C_1\Sigma_{j=0}^{k-1}\overline{N}(r,A_j)\\ &\leq C_2\max\{N(r,A_j): j=0,1,\dots,k-1\} \leq O(T(r,A_s)), \end{align*} where C_1 and C_2 are suitable positive constants. Then we have $$\label{E-13} \log T(r,f)\leq\log m(r,f)+\log N(r,f)+\log 2\leq \log m(r,f)+ O\{\log T(r,A_s)\}.$$ By \eqref{E-13} and Lemma \ref{L-2}, we obtain \begin{align*} \log T(r,f) &\leq \log m(r,f)+ O\{\log T(r,A_s)\}\\ &= O\left(T(r, A_s)\{(\log r)\log T(r, A_s)\}^{\lambda}\right) \end{align*} outside of an exceptional set E_0 with \int_{E_0}\frac{dt}{t}<\infty, this implies \sigma_{(p+1,q)}(f)\leq \sigma_{(p,q)}(A_s). On the other hand, by \eqref{E-1}, we obtain \begin{align*} -A_s &=\frac{f^{(k)}}{f^{(s)}}+A_{k-1}\frac{f^{(k-1)}}{f^{(s)}}+\dots +A_{s+1}\frac{f^{(s+1)}}{f^{(s)}}+ A_{s-1}\frac{f^{(s-1)}}{f^{(s)}}+\dots+A_0\frac{f}{f^{(s)}} \\ &=\frac{f}{f^{(s)}}\{\frac{f^{(k)}}{f}+A_{k-1}\frac{f^{(k-1)}}{f}+\dots +A_{s+1}\frac{f^{(s+1)}}{f}+ A_{s-1}\frac{f^{(s-1)}}{f}+\dots+A_0\}. \end{align*} Since $m(r,\frac{f}{f^{(s)}})\leq T(r,f)+T(r,\frac{1}{f^{(s)}})=T(r,f)+T(r,f^{(s)})+O(1)=O(T(r,f)),$ then by the lemma of logarithmic derivative we have $$\label{E-14} T(r ,A_s)\leq N(r ,A_s)+\Sigma_{j\neq s}m(r ,A_j)+O(\log r T(r,f))+O(T(r,f))$$ hold for all |z|=r\not\in E_{7}, where E_{7} is a set of finite linear measure. By Lemma \ref{L-6} and similar discussion as in the proof of Lemma \ref{L-10}, we see that there exists a sequence \{r_n\} (r_n\to \infty) such that $\sigma_1:=\sigma_{(p,q)}(A_s)=\lim_{r_n\to \infty}\frac{\log_pT(r_n, A_s)}{\log_qr_n }$ and \begin{gather}\label{E-15} T(r_n, A_s) \geq \exp_p\{(\sigma_1-\epsilon) \log_qr_n\},\\ \label{E-16} N(r_n,A_s) \leq \exp_p\{(\sigma_2+\epsilon) \log_qr_n\}, \\ \label{E-17} m(r_n, A_j)\leq \exp_p\{(\sigma_2+\epsilon) \log_qr_n\}\quad (j\neq s), \end{gather} where \sigma_2:=\max\{\sigma_{(p,q)}(A_j), \lambda_{(p,q)}(\frac{1}{A_s}): j\neq s\} and 0<2\epsilon<\sigma_1-\sigma_2. By \eqref{E-14}-\eqref{E-17}, we obtain $(1-o(1))\exp_p\{(\sigma_1-\epsilon) \log_qr_n\}\leq O\{\log r_n T(r_n,f)\}+O(T(r_n,f)).$ Hence we have \sigma_{(p,q)}(A_s)=\sigma_1\leq \sigma_{(p,q)}(f). \bullet Secondly, we prove that there exists at least one meromorphic solution that satisfies \sigma_{(p+1,q)}(f)=\sigma_{(p,q)}(A_s). $$Now we can assume that \{f_1,f_2,\dots,f_{k} \} is a meromorphic solution base of \eqref{E-1}. By Lemma \ref{L-1}, $m(r,A_s) \leq O\left(\log(\max_{1\leq n\leq k}T(r,f_n))\right).$ Now we assert that m(r,A_s)>N(r,A_s) holds for sufficiently large r. Indeed, if m(r,A_s) \leq N(r,A_s), then $T(r,A_s)=m(r,A_s)+N(r,A_s) \leq 2 N(r,A_s),$ so $\limsup_{r\to \infty}\frac{\log_p T(r,A_s)}{\log_q r } \leq \limsup_{r\to \infty}\frac{\log_p 2 N(r,A_s)}{\log_q r },$ then we have \sigma_{(p,q)}(A_s) \leq \lambda_{(p,q)}(\frac{1}{A_s}), which contradicts the condition \lambda_{(p,q)}(\frac{1}{A_s})<\sigma_{(p,q)}(A_s). Hence,$$ T(r,A_s)=O(m(r,A_s))\leq O\Big(\log(\max_{1\leq n\leq k}T(r,f_n))\Big). $$By Lemma \ref{L-6}, there exists a set E_9\subset (0,\infty) has finite linear measure , and a sequence \{r_n\}, r_n\not\in E_9, such that $\lim_{r_n\to \infty}\frac{\log_pT(r_n,A_s)}{\log_qr_n }=\sigma_{(p,q)}(A_s).$ Set $T_n=\{r:r \in (0,\infty)\setminus E_9,\quad T(r,A_s) \leq O(\log(T(r,f_n)))\quad (n=1,2,\dots,k)$ By Lemma \ref{L-1}, we have \cup_{n=1}^{k}T_n=(0,\infty)\setminus E_9. It is easy to see that there exists at least one T_n, say T_1\subset (0,\infty)\setminus E_9, that has infinite linear measure and satisfies $$\label{E-18} T(r,A_s) \leq O(\log T(r,f_1)).$$ From \eqref{E-18}, we have \sigma_{(p+1,q)}(f_1) \geq \sigma_{(p,q)}(A_s). In the first part we have proved that \sigma_{(p+1,q)}(f_1) \leq \sigma_{(p,q)}(A_s). Therefore, we have that there is at least one meromorphic solution f_1 satisfies $\sigma_{(p+1,q)}(f_1)=\sigma_{(p,q)}(A_s).$ \subsection*{Proof of Theorem \ref{T-2}} Suppose that f is a nonzero meromorphic solution whose poles are of uniformly bounded multiplicities of \eqref{E-1}, then \eqref{E-1} can be written $$\label{E-19} -A_0=\frac{f^{(k)}}{f}+ A_{k-1}\frac{f^{(k-1)}}{f}+\dots +A_1\frac{f'}{f}.$$ By the lemma of the logarithmic derivative and \eqref{E-19}, we have $$\label{E-20} \begin{split} m(r,A_0) &\leq \sum_{j=1}^{k-1}m(r,A_j)+\sum_{j=1}^{k}m(r,\frac{f^{(j)}}{f})+O(1)\\ &=\sum_{j=1}^{k-1}m(r,A_j)+O\{\log (r T(r,f))\} \end{split}$$ holds for all sufficiently large r \not\in E_{10}, where E_{10}\subset (0,\infty) has finite linear measure. Hence $$\label{E-21} T(r,A_0)=m(r,A_0)+N(r,A_0) \leq N(r,A_0)+\sum_{j=1}^{k-1}m(r,A_j)+O\{\log (r T(r,f))\}$$ holds for all sufficiently large |r|=r \not\in E_{10}. Since \max\{\sigma_{(p,q)}(A_j):j\neq 0\}<\sigma_{(p,q)}(A_0)<\infty, by Lemma \ref{L-5}, there exist a set E_{11}\subset (1,\infty) having infinite logarithmic measure such that for all z satisfying |z|=r \in E_{11}, we have $$\label{E-22} \lim_{r\to \infty}\frac{\log_pT(r, A_0)}{\log_qr }=\sigma_{(p,q)}(A_0),\, \frac{m(r,A_j)}{m(r,A_0)}=o(1)\quad(r \in E_{10}, j=1,2,\dots,k-1).$$ By \eqref{E-20} and \eqref{E-22}, for all sufficiently large r \in E_{11}\setminus E_{10}, we have $$\label{E-23} \frac{1}{2} m(r,A_0)\leq O\{\log (r T(r,f))\}.$$ Using a similar discussion as in second part of proof of Theorem \ref{T-1}, we can get that $$\label{E-24} m(r,A_0)>N(r,A_0),$$ hence,$$ T(r, A_0)=m(r,A_0)+N(r, A_0)=O(m(r, A_0))=O(\log rT(r, f)) $$for all sufficiently large r \in E_{11}\setminus E_{10}, this means $\sigma_{(p+1,q)}(f) \geq \sigma_{(p,q)}(A_0).$ On the other hand, by Theorem \ref{T-1}, we have $\sigma_{(p+1,q)}(f) \leq \sigma_{(p,q)}(A_0).$ Therefore, every meromorphic solution f whose poles are of uniformly bounded multiplicities of \eqref{E-1} satisfies $\sigma_{(p+1,q)}(f)=\sigma_{(p,q)}(A_0).$ \subsection*{Proof of Theorem \ref{T-3}} When A_0,A_1,\dots,A_{k-1} satisfy $\max\{\sigma_{(p,q)}(A_j):j\neq 0\}<\sigma_{(p,q)}(A_0),$ then by Theorem \ref{T-2}, it is easy to see that Theorem \ref{T-3} holds. Now we assume that there exists at least one of A_j (j=1,2,\dots,k-1) satisfies \sigma_{(p,q)}(A_j)=\sigma_{(p,q)}(A_0). Suppose that f is a nonzero meromorphic solution of \eqref{E-1}, we have $$\label{E-25} |A_0| \leq |\frac{f^{(k)}(z)}{f(z)}|+ |A_{k-1}||\frac{f^{(k-1)}(z)}{f(z)}|+ \dots +|A_1||\frac{f'(z)}{f(z)}|.$$ Using a similar discussion as in the proof of Theorem \ref{T-2}, we can get that \eqref{E-20} and \eqref{E-21} hold for all sufficiently large r \not\in E_{12}, where E_{12}\subset (0,\infty) has finite linear measure. Since $\max\{\sigma_{(p,q)}(A_j):j=1,2,\dots,k-1\}=\sigma_{(p,q)}(A_0)$ and $\max\{\tau_{(p,q)}(A_j):\sigma_{(p,q)}(A_j) =\sigma_{(p,q)}(A_0)>0\}<\tau_{(p,q)}(A_0),$ then there exists a set J \subset\{1,2,\dots,k-1\} such that for j \in J, we have \sigma_{(p,q)}(A_j)=\sigma_{(p,q)}(A_0) and \tau_{(p,q)}(A_j)<\tau_{(p,q)}(A_0). Hence, there exist two constants \beta_1 and \beta_2 satisfying \max\{\tau_{(p,q)}:j \in J\}<\beta_1<\beta_2\leq\tau_{(p,q)}(A_0). By Definitions \ref{D-2.6} and \ref{D-2.7}, we obtain that $$\label{E-26} m(r,A_j) \leq T(r,A_j)<\exp_{p-1}\{\beta_1 (\log_{q-1}r)^{\sigma_{(p,q)}(A_0)}\}.$$ Since \lambda_{(p,q)}(\frac{1}{A_0})<\sigma_{(p,q)}(A_0), we have $$\label{E-27} N(r,A_0) \leq \exp_p\{(\lambda_(\frac{1}{A_0})+\epsilon)\log_qr \} \leq \exp_{p-1}\{\beta_1 (\log_{q-1}r)^{\sigma_{(p,q)}(A_0)}\}.$$ By Lemma \ref{L-8}, there exists a set of E_{13} having infinite logarithmic measure such that for all r\in E_{13}, we have $$\label{E-28} T(r,A_0) \geq \exp_{p-1}\{\beta_2 (\log_{q-1}r)^{\sigma_{(p,q)}(A_0)}\}.$$ Now, substituting \eqref{E-26}-\eqref{E-28}into \eqref{E-21}, we have$$ (1-o(1))\exp_{p-1}\{\beta_2(\log_{q-1}r)^{\sigma_{(p,q)(A_0)}}\} \leq O(\log(r T(r, f)))  for all $r\in E_{13}\setminus E_{12}$, this implies $\sigma_{(p+1,q)}(f) \geq \sigma_{(p,q)}(A_0).$ On the other hand ,by Theorem \ref{T-1}, we have $\sigma_{(p+1,q)}(f) \leq \sigma_{(p,q)}(A_0).$ Then we have that $\sigma_{(p+1,q)}(f)=\sigma_{(p,q)}(A_0)$ holds for any nonzero meromorphic solution $f$ whose poles are of uniformly bounded multiplicities of \eqref{E-1}. \subsection*{Proof of Theorem \ref{T-4}} Since all solutions of equation \eqref{E-2} are meromorphic functions, all solutions of the homogeneous differential equation \eqref{E-1} corresponding to equation \eqref{E-2} are still meromorphic functions. Now we assume that $\{f_1,f_2,\dots,f_{k}\}$ is a meromorphic solution base of \eqref{E-1}, then by the elementary theory of differential equations (see, e.g. \cite{12}), any solution of \eqref{E-2} has the form $$\label{E-29} f=c_1(z)f_1+c_2(z)f_2+ \dots +c_{k}(z)f_{k},$$ where $c_1,c_2,\dots,c_{k}$ are suitable meromorphic functions satisfying $$\label{E-30} c'_j=FG_j(f_1,f_1,\dots,f_{k}) W(f_1,f_1,\dots,f_{k})^{-1}\quad (j=1,2,\dots,k),$$ where $G_j(f_1,f_1,\dots,f_{k})$ are differential polynomials in $\{f_1,f_2,\dots,f_{k}\}$ and their derivatives, and $W(f_1,f_1,\dots,f_{k})^{-1}$ is the Wronskian of $\{f_1,f_2,\dots,f_{k}\}$. By Theorem \ref{T-2}, we have $\sigma_{(p+1,q)}(f_j)=\sigma_{(p,q)}(A_0)\quad (j=1,2,\dots,k).$ By Lemma \ref{L-3}, \eqref{E-29} and \eqref{E-30}, we obtain $\sigma_{(p+1,q)}(f)\leq \max\{\sigma_{(p+1,q)}(f_j),\sigma_{(p+1,q)}(F):j=1,2,\dots,k\} = \sigma_{(p,q)}(A_0).$ Now we assert that all solutions $f$ of \eqref{E-2} satisfy $\sigma_{(p+1,q)}(f)=\sigma_{(p,q)}(A_0)$ with at most one exceptional solution, say $f_0$, satisfying $\sigma_{(p+1,q)}(f_0)<\sigma_{(p,q)}(A_0)$. In fact, if there exists two distinct meromorphic functions $f_0$ and $f_1$ of \eqref{E-2} satisfying $\sigma_{(p+1,q)}(f_j)<\sigma_{(p,q)}(A_0)\quad (j=0,1),$ then $f=f_0-f_1$ is a nonzero meromorphic solution of \eqref{E-1}, and satisfying $\sigma_{(p+1,q)}(f)<\sigma_{(p,q)}(A_0)$, this contradicts Theorem \ref{T-2}. For all the solutions $f$ of \eqref{E-2} satisfying $\sigma_{(p+1,q)}(f)=\sigma_{(p, q)}(A_0)$, we have $\max\{\sigma_{(p+1,q)}(A_j),\sigma_{(p+1,q)}(F):j=0,1,\dots,k-1\} <\sigma_{(p+1,q)}(f).$ Thus by Lemma \ref{L-10}, we obtain $\overline{\lambda}_{(p+1,q)}(f)=\lambda_{(p+1,q)}(f)=\sigma_{(p+1,q)}(f).$ Therefore, Theorem \ref{T-4} is proved. \subsection{Proof of Theorem \ref{T-5}} Suppose that $\{g_1,g_2,\dots,g_{k}\}$ is a meromorphic solution base of \eqref{E-1} corresponding to \eqref{E-2}. By a similar discussion as in the proof of Theorem \ref{T-4}, we obtain $\sigma_{(p+1,q)}(f)\leq \max\{\sigma_{(p+1,q)}(g_j),\sigma_{(p+1,q)}(F):j=1,2,\dots,k\}$ By the first part of the proof of Theorem \ref{T-1}, we can get that $\sigma_{(p+1,q)}(g_j) \leq \max\{\sigma_{(p,q)}(A_j):j=0,1,\dots,k-1\} \leq \sigma_{(p+1,q)}(F),$ then we can get $$\label{E-31} \sigma_{(p+1,q)}(f)\leq \sigma_{(p+1,q)}(F).$$ On the other hand, by the simple order comparison from \eqref{E-2}, we have $\sigma_{(p+1,q)}(F)\leq \max\{\sigma_{(p+1,q)}(A_j),\sigma_{(p+1,q)}(f):j=0,1,\dots,k-1\}.$ Since $\sigma_{(p+1,q)}(A_j)< \sigma_{(p+1,q)}(F)$, we have $$\label{E-32} \sigma_{(p+1,q)}(F)\leq \sigma_{(p+1,q)}(f).$$ By \eqref{E-31}-\eqref{E-32}, we obtain $\sigma_{(p+1,q)}(F)=\sigma_{(p+1,q)}(f).$ Therefore, the proof of Theorem \ref{T-5} is complete. \subsection*{Proof of Theorem \ref{T-6}} (i) By the simple order comparison from \eqref{E-2} it is easy to see that all meromorphic solutions of \eqref{E-2} satisfy $\sigma_{(p+1,q)}(f)\geq \sigma_{(p+1,q)}(F).$ On the other hand, by the similar proof in \eqref{E-29}-\eqref{E-30}, we obtain that all meromorphic solutions of \eqref{E-2} satisfy $\sigma_{(p+1,q)}(f)\leq \sigma_{(p+1,q)}(F)$ if $\sigma_{(p+1,q)}(F)\geq \alpha_1$. Therefore, all meromorphic solutions whose poles are of uniformly bounded multiplicities of \eqref{E-2} satisfy $\sigma_{(p+1,q)}(f)= \sigma_{(p+1,q)}(F).$ (ii) By the hypotheses that $|A_0{(z)}|\geq\exp_{p+1}\{(\alpha_1-\epsilon)\log_qr\},$ and $|A_j{(z)}|\leq\exp_{p+1}\{\alpha_2\log_qr\}$, we can easily obtain that $\sigma_{(p+1,q)}(A_0)= \alpha_1$. Since $\sigma_{(p+1,q)}(F)< \alpha_1=\sigma_{(p+1,q)}(A_0)$, by the similar proof in Theorem \ref{T-4}, we obtain that all meromorphic solutions whose poles are of uniformly bounded multiplicities of \eqref{E-2} satisfy $\overline{\lambda}_{(p+1,q)}(f)=\lambda_{(p+1,q)}(f)=\sigma_{(p+1,q)}(f)=\alpha_1$ with at most one exceptional solution $f_2$ satisfying $\sigma_{(p+1,q)}(f_2)<\alpha_1$. Therefore, we completely prove Theorem \ref{T-6}. \subsection*{Acknowledgements} This research was supported by grants 11101201 from the NSFC, and 2010GQS0139 from the NSF of Jiangxi of China. 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