% EJC papers *must* begin with the following two lines. \documentclass[12pt]{article} \usepackage{e-jc} \specs{P3}{20(4)}{2013} % Please remove all other commands that change parameters such as % margins or pagesizes. % only use standard LaTeX packages % only include packages that you actually need % we recommend these ams packages \usepackage{amsthm,amsmath,amssymb} % we recommend the graphicx package for importing figures \usepackage{graphicx} % use this command to create hyperlinks (optional and recommended) \usepackage[colorlinks=true,citecolor=black,linkcolor=black,urlcolor=blue]{hyperref} % all overfull boxes must be fixed; % i.e. there must be no text protruding into the margins % declare theorem-like environments \theoremstyle{plain} \newtheorem{theorem}{Theorem} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{fact}[theorem]{Fact} \newtheorem{observation}[theorem]{Observation} \newtheorem{claim}[theorem]{Claim} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{open}[theorem]{Open Problem} \newtheorem{problem}[theorem]{Problem} \newtheorem{question}[theorem]{Question} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newtheorem{note}[theorem]{Note} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % if needed include a line break (\\) at an appropriate place in the title \title{\bf A criterion for the log-convexity of combinatorial sequences} % input author, affilliation, address and support information as follows; % the address should include the country, and does not have to include % the street address \author{Ernest X.~W. Xia \thanks{Supported by the National Natural Science Foundation of China (11201188).}\\ \small Department of Mathematics\\[-0.8ex] \small Jiangsu University\\[-0.8ex] \small Zhenjiang, Jiangsu, P. R. China\\ \small\tt ernestxwxia@163.com\\ \and Olivia X.~M. Yao\\ \small Department of Mathematics\\[-0.8ex] \small Jiangsu University\\[-0.8ex] \small Zhenjiang, Jiangsu, P. R. China\\ \small\tt yaoxiangmei@163.com } % \date{\dateline{submission date}{acceptance date}\\ % \small Mathematics Subject Classifications: comma separated list of % MSC codes available from http://www.ams.org/mathscinet/freeTools.html} \date{\dateline{May 28, 2013 }{Sep 30, 2013}{Oct 14, 2013}\\ \small Mathematics Subject Classifications: 05A20, 11B83} \begin{document} \maketitle % E-JC papers must include an abstract. The abstract should consist of a % succinct statement of background followed by a listing of the % principal new results that are to be found in the paper. The abstract % should be informative, clear, and as complete as possible. Phrases % like "we investigate..." or "we study..." should be kept to a minimum % in favor of "we prove that..." or "we show that...". Do not % include equation numbers, unexpanded citations (such as "[23]"), or % any other references to things in the paper that are not defined in % the abstract. The abstract will be distributed without the rest of the % paper so it must be entirely self-contained. \begin{abstract} Recently, Do\v{s}li\'{c}, and Liu and Wang developed techniques for dealing with the log-convexity of sequences. In this paper, we present a criterion for the log-convexity of some combinatorial sequences. In order to prove the log-convexity of a sequence satisfying a three-term recurrence, by our method, it suffices to compute a constant number of terms at the beginning of the sequence. For example, in order to prove the log-convexity of the Ap\'{e}ry numbers $A_n$, by our method, we just need to evaluate the values of $A_n$ for $0\leq n \leq 6$. As applications, we prove the log-convexity of some famous sequences including the Catalan-Larcombe-French numbers. This confirms a conjecture given by Sun. % keywords are optional \bigskip\noindent \textbf{Keywords:} log-convexity; three-term recurrence; combinatorial sequences \end{abstract} \section{Introduction} A positive sequence $\{S_n\}_{n=0}^\infty $ is said to be log-convex (respectively log-concave) if for $n\geq 1$, \begin{align} \frac{S_n}{S_{n-1}}\leq \frac{S_{n+1}}{S_n} \qquad ({\rm respectively }\ \ \frac{S_n}{S_{n-1}}\geq \frac{S_{n+1}}{S_n}). \label{1-1} \end{align} Meanwhile, the sequence $\{S_n\}_{n=0}^\infty $ is called strictly log-convex (log-concave) if the inequality in (1.1) is strict for all $n\geq 1$. In 1994, Engel \cite{Engel} proved the log-convexity of the Bell numbers. Recently, Do\v{s}li\'{c} \cite{D,D3,D4}, Do\v{s}li\'{c} and Veljan \cite{D2}, and Liu and Wang \cite{Liu} developed techniques for proving the log-convexity of sequences. Do\v{s}li\'{c} \cite{D,D3,D4} presented several methods for dealing with the log-convexity of combinatorial sequences. He proved that the Motzkin numbers, the Fine numbers, the Franel numbers of order 3 and 4, the Ap\'{e}ry numbers, the large Schr\"{o}der numbers, the derangements numbers and the central Delannoy numbers are log-convex. In their wonderful paper \cite{Liu}, Liu and Wang proved that the log-convexity is preserved under componentwise sum, under binomial convolution, and by the linear transformations given by the matrices of binomial coefficients and Stirling numbers of two kinds. Many combinatorial sequences satisfy a three-term recurrence. Liu and Wang \cite{Liu} presented some criterions for the log-convexity of the sequences $\{z_n\}_{n=0}^\infty$ satisfying the following recurrence \begin{align} a(n)z_{n+1}=b(n)z_n+c(n)z_{n-1}, \label{1-2} \end{align} where $a(n)$, $b(n)$ and $c(n)$ are positive for $n\geq 1$, Liu and Wang \cite{Liu} proved the following theorem. \begin{theorem} \label{Th-1} Let $\{z_n\}_{n=0}^\infty$ be defined by \eqref{1-2} and \begin{align} \lambda_n=\frac{b(n)+\sqrt{b^2(n)+4a(n)c(n)}}{2a(n)}.\label{1-3} \end{align} Suppose that $z_0,\ z_1,\ z_2,\ z_3$ is log-convex and that the inequality \begin{align} a(n)\lambda_{n-1}\lambda_{n+1}-b(n)\lambda(n-1)-c(n)\geq 0\label{1-4} \end{align} is true for $n\geq 2$. Then the sequence $\{z_n\}_{n=0}^\infty$ is log-convex. \end{theorem} Liu and Wang \cite{Liu} also considered the log-convexity of the sequence $\{z_n\}_{n=0}^\infty $ defined by \begin{align} (\alpha_n+\alpha_0)z_{n+1}=(\beta_1n+\beta_0)z_n-(\gamma_1n+\gamma_0)z_{n-1}. \label{1-5} \end{align} for $n\geq 1$. They gave criterions for the log-convexity of the sequences $\{z_n\}_{n=0}^\infty$. Employing their criterions, they proved the log-convexity of some combinatorial sequences. Liu \cite{Liu-0} gave sufficient conditions for the positivity of the sequences defined by \eqref{1-5}. Motivated by these results established by Liu and Wang \cite{Liu}, in this paper, we investigate the log-convexity problem of the sequence $\{S_n\}_{n=0}^\infty$ having the following three-term recurrence \begin{align}\label{1-6} S_n=\frac{\sum_{i=0}^k a_i n^i}{\sum_{i=0}^kb_in^i}S_{n-1}-\frac{\sum_{i=0}^l c_i n^i}{\sum_{i=0}^l d_i n^i}S_{n-2} \qquad (n\geq 2), \end{align} where $\gcd(\sum_{i=0}^k a_i n^i, \sum_{i=0}^kb_in^i)=\gcd(\sum_{i=0}^l c_i n^i,\sum_{i=0}^l d_i n^i)=1$ and $k$, $l$, $a_k $, $b_k$, $c_l$ and $d_l$ are positive numbers. The authors \cite{Xia} gave a criterion for the positivity of the sequence $\{S_n\}_{n=0}^\infty$ defined by \eqref{1-6}. The aim of this paper is to present a criterion for the log-convexity of some famous combinatorial sequences. By our method, in order to determine the log-convexity of the sequence $\{S_n\}_{n=0}^\infty$ defined by \eqref{1-6}, it suffices to compute a constant number of terms at the beginning of the sequence $\{S_n\}_{n=0}^\infty$. As applications, we prove some famous combinatorial sequences are strictly log-convex. Specially, we show that the Catalan-Larcombe-French numbers $\{P_n\}_{n=0}^\infty$ is strictly log-convex which confirms a conjecture given by Sun \cite{Sun-2}. In order to state our main result, we first introduce some notations. Given a polynomial $f(n)$ defined by \begin{align} f(n)=\sum_{i=0}^k f_i n^i,\label{1-7} \end{align} where $f_i$ $(0\leq i \leq k)$ are real numbers and $f_k>0$. Define an operator $L$ on $f(n)$ by \begin{align} L(f(n))=\frac{1}{f_k}\sum_{0 \leq i\leq k-1,\ f_i<0} |f_i|. \label{1-8} \end{align} For example, \begin{align} L(5n^4-2n^3+4n^2-6n-3)= \frac{11}{5}. \label{1-9} \end{align} It is easy to see that $f(n)>0$ for $n\geq \left[L(f(n))\right]+1$. Throughout this paper, we always let \begin{align} \frac{\sum_{i=0}^k a_i (n+2)^i}{\sum_{i=0}^kb_i(n+2)^i}&-\frac{\sum_{i=0}^k a_i (n+1)^i}{\sum_{i=0}^kb_i(n+1)^i}=\frac{\sum_{j=0}^r e_j n^j}{\sum_{t=0}^s h_t n^t },\label{1-10}\\[6pt] \frac{\sum_{i=0}^l c_i (n+2)^i}{\sum_{i=0}^l d_i (n+2)^i}&- \frac{\sum_{i=0}^l c_i (n+1)^i}{\sum_{i=0}^l d_i (n+1)^i} =\frac{\sum_{j=0}^u p_j n^j}{\sum_{t=0}^v q_t n^t }, \label{1-11} \end{align} and \begin{align} &\frac{\sum_{i=0}^k a_i (n+2)^i}{\sum_{i=0}^kb_i(n+2)^i}-\frac{\sum_{i=0}^l c_i (n+2)^i}{\sum_{i=0}^l d_i (n+2)^i}\frac{\left(\sum_{t=0}^v q_t n^t\right)\left(\sum_{j=0}^r e_j n^j\right) }{\left(\sum_{j=0}^u p_j n^j\right)\left(\sum_{t=0}^s h_t n^t \right)}\nonumber\\[6pt] &\qquad \qquad\qquad -\frac{\left(\sum_{j=0}^u p_j (n+1)^j\right)\left(\sum_{t=0}^s h_t (n+1)^t \right)}{\left(\sum_{t=0}^v q_t (n+1)^t\right)\left(\sum_{j=0}^r e_j (n+1)^j\right) }=\frac{\sum_{i=0}^\alpha x_i n^i }{\sum_{i=0}^\beta y_i n^i}, \label{1-12} \end{align} where $h_s>0$, $q_v>0$, $y_\beta>0$ and \begin{align*}\gcd\left( \sum_{j=0}^r e_j n^j,\sum_{t=0}^s h_t n^t \right)=\gcd\left( \sum_{j=0}^u p_j n^j, \sum_{t=0}^v q_t n^t \right)=\gcd\left(\sum_{i=0}^\alpha x_i n^i, \sum_{i=0}^\beta y_i n^i \right)=1. \end{align*} Our main result can be stated as follows. \begin{theorem} \label{Th-2} Let $\{S_n\}_{n=0}^\infty$ be a positive sequence and satisfy \eqref{1-6}. If $p_u>0$, $e_r>0$, $x_\alpha >0$ and there exists an integer $N_0$ such that \begin{align} N_0\geq r_1&= \max \bigg\{\left[L(\sum_{i=0}^l c_i n^i)\right], \left[L(\sum_{i=0}^l d_i n^i)\right], \left[L(\sum_{j=0}^r e_j n^j)\right], \left[L(\sum_{t=0}^s h_t n^t)\right],\nonumber\\[6pt] &\qquad \quad \left[L(\sum_{j=0}^u p_j n^j)\right], \left[L(\sum_{t=0}^v q_t n^t)\right], \left[L(\sum_{i=0}^\alpha x_i n^i)\right], \left[L(\sum_{i=0}^\beta y_i n^i)\right] \bigg\}+1\label{1-13} \end{align} and \begin{align} \frac{S_{N_0}}{S_{N_0-1}} &< \frac{S_{N_0+1}}{S_{N_0}},\label{1-16}\\[6pt] \frac{S_{N_0+1}}{S_{N_0}}&> \frac{\left(\sum_{j=0}^u p_j N_0^j\right)\left(\sum_{t=0}^s h_t N_0^t \right)}{\left(\sum_{t=0}^v q_t N_0^t\right)\left(\sum_{j=0}^r e_j N_0^j\right) },\label{1-17} \end{align} then the sequence $\{S_n\}_{n=N_0}^\infty $ is strictly log-convex, namely, \begin{align} \frac{S_n}{S_{n-1}} < \frac{S_{n+1}}{S_n}, \qquad (n\geq N_0).\label{1-18} \end{align} \end{theorem} This paper is organized as follows. We give the proof of Theorem \ref{Th-2} in Sections 2. As applications of Theorem \ref{Th-2}, in Section 3, we prove the log-convexity of some famous sequences including the Catalan-Larcombe-French numbers. This confirms a conjecture given by Sun \cite{Sun-2}. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Proof of Theorem \ref{Th-2}} In this section, we present the proof of Theorem \ref{Th-2}. \begin{proof} By the definition of $r_1$, we see that for all $n\geq N_0\geq r_1$, \begin{align} \frac{\sum_{i=0}^l c_i n^i}{\sum_{i=0}^l d_i n^i}&>0, \label{2-1}\\[6pt] \frac{\sum_{i=0}^k a_i (n+2)^i}{\sum_{i=0}^kb_i(n+2)^i}-&\frac{\sum_{i=0}^k a_i (n+1)^i}{\sum_{i=0}^kb_i(n+1)^i}=\frac{\sum_{j=0}^r e_j n^j}{\sum_{t=0}^s h_t n^t }>0, \label{2-2}\\[6pt] \frac{\sum_{i=0}^k c_i (n+2)^i}{\sum_{i=0}^k d_i (n+2)^i}-& \frac{\sum_{i=0}^k c_i (n+1)^i}{\sum_{i=0}^k d_i (n+1)^i}=\frac{\sum_{j=0}^u p_j n^j}{\sum_{t=0}^v q_t n^t }>0, \label{2-3} \end{align} and \begin{align} \frac{\sum_{i=0}^\alpha x_i n^i }{\sum_{i=0}^\beta y_i n^i}>0. \label{2-4} \end{align} We first give a lower bound for $\frac{S_{n+1}}{S_{n}}$. Moreover, we prove that for $n\geq N_0$, \begin{align} \frac{S_{n+1}}{S_{n}}> \frac{\left(\sum_{j=0}^u p_j n^j\right)\left(\sum_{t=0}^s h_t n^t \right)}{\left(\sum_{t=0}^v q_t n^t\right)\left(\sum_{j=0}^r e_j n^j\right) }. \label{2-5} \end{align} We prove \eqref{2-5} by induction on $n$. By \eqref{1-17}, we see that \eqref{2-5} holds for $n=N_0$. Suppose that \eqref{2-5} holds for $n=m\geq N_0$, that is, \begin{align} \frac{S_{m+1}}{S_{m}}>\frac{\left(\sum_{j=0}^u p_j m^j\right)\left(\sum_{t=0}^s h_t m^t \right)}{\left(\sum_{t=0}^v q_t m^t\right)\left(\sum_{j=0}^r e_j m^j\right) }.\label{2-6} \end{align} It follows from \eqref{2-1}, \eqref{2-2} and \eqref{2-6} that for $m\geq N_0$, \begin{align} -\frac{\sum_{i=0}^l c_i (m+2)^i}{\sum_{i=0}^l d_i (m+2)^i}\frac{S_{m}}{S_{m+1}}>- \frac{\sum_{i=0}^l c_i (m+2)^i}{\sum_{i=0}^l d_i (m+2)^i}\frac{\left(\sum_{t=0}^v q_t m^t\right)\left(\sum_{j=0}^r e_j m^j\right) }{\left(\sum_{j=0}^u p_j m^j\right)\left(\sum_{t=0}^s h_t m^t \right)}.\label{2-7} \end{align} Now, we are ready to show that \eqref{2-5} also holds for $n=m+1$. Employing \eqref{1-6} and \eqref{2-7}, we deduce that \begin{align} \frac{S_{m+2}}{S_{m+1}}&=\frac{\sum_{i=0}^k a_i (m+2)^i}{\sum_{i=0}^kb_i(m+2)^i} -\frac{\sum_{i=0}^l c_i (m+2)^i}{\sum_{i=0}^l d_i (m+2)^i}\frac{S_{m}}{S_{m+1}}\nonumber\\[6pt] &> \frac{\sum_{i=0}^k a_i (m+2)^i}{\sum_{i=0}^kb_i(m+2)^i}-\frac{\sum_{i=0}^l c_i (m+2)^i}{\sum_{i=0}^l d_i (m+2)^i}\frac{\left(\sum_{t=0}^v q_t m^t\right)\left(\sum_{j=0}^r e_j m^j\right) }{\left(\sum_{j=0}^u p_j m^j\right)\left(\sum_{t=0}^s h_t m^t \right)}.\label{2-8} \end{align} In view of \eqref{1-12}, \eqref{2-4} and \eqref{2-8}, we find that for $m\geq N_0$ \begin{align} &\frac{S_{m+2}}{S_{m+1}}-\frac{\left(\sum_{j=0}^u p_j (m+1)^j\right)\left(\sum_{t=0}^s h_t (m+1)^t \right)}{\left(\sum_{t=0}^v q_t (m+1)^t\right)\left(\sum_{j=0}^r e_j (m+1)^j\right) }\nonumber\\[6pt] > & \frac{\sum_{i=0}^k a_i (m+2)^i}{\sum_{i=0}^kb_i(m+2)^i}-\frac{\sum_{i=0}^l c_i (m+2)^i}{\sum_{i=0}^l d_i (m+2)^i}\frac{\left(\sum_{t=0}^v q_t m^t\right)\left(\sum_{j=0}^r e_j m^j\right) }{\left(\sum_{j=0}^u p_j m^j\right)\left(\sum_{t=0}^s h_t m^t \right)}\nonumber\\[6pt] &\qquad -\frac{\left(\sum_{j=0}^u p_j (m+1)^j\right)\left(\sum_{t=0}^s h_t (m+1)^t \right)}{\left(\sum_{t=0}^v q_t (m+1)^t\right)\left(\sum_{j=0}^r e_j (m+1)^j\right) }\nonumber\\[6pt] =& \frac{\sum_{i=0}^\alpha x_i m^i }{\sum_{i=0}^\beta y_i m^i}>0, \label{2-9} \end{align} which implies that \eqref{2-5} is true for $n=m+1$. By induction, we have proved \eqref{2-5} holds for $n\geq N_0$. Now, we turn to prove \eqref{1-18}. We also prove \eqref{1-18} by induction on $n$. It follows from \eqref{1-16} that \eqref{1-18} holds for $n=N_0$. Assume that \eqref{1-18} is true for $n=m\geq N_0$, namely, \begin{align} \frac{S_{m}}{S_{m-1}}< \frac{S_{m+1}}{S_m}.\label{2-10} \end{align} By \eqref{2-1} and \eqref{2-10}, we find that for $m\geq N_0$ \begin{align} \frac{\sum_{i=0}^l c_i (m+1)^i}{\sum_{i=0}^l d_i (m+1)^i} \frac{S_{m-1}}{S_{m}} > \frac{\sum_{i=0}^l c_i (m+1)^i}{\sum_{i=0}^l d_i (m+1)^i} \frac{S_{m}}{S_{m+1}}.\label{2-11} \end{align} Employing \eqref{1-6}, \eqref{1-10}, \eqref{1-11}, \eqref{2-3}, \eqref{2-5} and \eqref{2-11}, we deduce that for $m\geq N_0$, \begin{align} \frac{S_{m+2}}{S_{m+1}}-\frac{S_{m+1}}{S_{m}}&=\frac{\sum_{i=0}^k a_i(m+2)^i }{\sum_{i=0}^k b_i(m+2)^i} - \frac{\sum_{i=0}^l c_i(m+2)^i }{\sum_{i=0}^l d_i(m+2)^i} \frac{S_{m}}{S_{m+1}} \nonumber\\[6pt] &\qquad \qquad -\frac{\sum_{i=0}^k a_i(m+1)^i }{\sum_{i=0}^k b_i(m+1)^i} + \frac{\sum_{i=0}^l c_i(m+1)^i }{\sum_{i=0}^l d_i(m+1)^i} \frac{S_{m-1}}{S_{m}} \nonumber \\[6pt] &> \frac{\sum_{i=0}^k a_i(m+2)^i }{\sum_{i=0}^k b_i(m+2)^i} -\frac{\sum_{i=0}^k a_i(m+1)^i }{\sum_{i=0}^k b_i(m+1)^i} \nonumber\\[6pt] &\qquad \qquad + \left( \frac{\sum_{i=0}^l c_i(m+1)^i }{\sum_{i=0}^l d_i(m+1)^i} -\frac{\sum_{i=0}^l c_i(m+2)^i }{\sum_{i=0}^l d_i(m+2)^i} \right)\frac{S_m}{S_{m+1}} \nonumber\\[6pt] &= \frac{\sum_{j=0}^r e_j m^j}{\sum_{t=0}^s h_t m^t }- \frac{\sum_{j=0}^u p_j m^j}{\sum_{t=0}^v q_t m^t } \frac{S_m}{S_{m+1}}\nonumber\\[6pt] &> \frac{\sum_{j=0}^r e_j m^j}{\sum_{t=0}^s h_t m^t }- \frac{\sum_{j=0}^u p_j m^j}{\sum_{t=0}^v q_t m^t } \left( \frac{\left(\sum_{t=0}^v q_t m^t\right)\left(\sum_{j=0}^r e_j m^j\right) }{\left(\sum_{j=0}^u p_j m^j\right)\left(\sum_{t=0}^s h_t m^t \right)} \right)=0, \end{align} which implies that \eqref{1-18} holds for $n=m+1$. Theorem \ref{Th-2} is proved by induction. This completes the proof. \end{proof} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Applications of Theorem \ref{Th-2}} In this section, employing the criterion given in this paper, we prove some results on the log-convexity of some combinatorial sequences. The Catalan-Larcombe-French numbers $P_n$ for $n\geq 0$ were first defined by Catalan in \cite{Catalan}, in terms of the ``Segner numbers". Catalan stated that the $P_n$ could be defined by the following recurrence relation: \begin{align}\label{3-2} P_n=\frac{8(3n^2-3n+1)}{n^2}P_{n-1}-\frac{128(n-1)^2}{n^2}P_{n-2}, \end{align} for $n\geq 2$, with the initial values given by $P_0=1$ and $P_1=8$. Larcombe and French \cite{Larcombe} gave a detailed account of properties of $P_n$, and obtained the following formulas for these numbers: \begin{align}\label{v-1} P_n=\sum_{k=0}^n\frac{{2k\choose k}^2{2(n-k)\choose n-k}^2}{{n\choose k}}=2^n\sum_{k=0}^{\lfloor n/2\rfloor}{n\choose 2k} {2k\choose k}^2 4^{n-2k} \end{align} and \begin{align}\label{v-2} P_n=\frac{1}{n!}\sum_{r+s=n}{2r\choose r}{2s\choose s} \frac{(2r)!(2s)!}{r!s!}=\sum_{r+s=n}\frac{{2r\choose r}^2{2s\choose s}^2 }{{n\choose r}} \end{align} for $n\geq 0$. The first few $P_n$ are $1, \ 8,\ 80,\ 896,\ 10816,\ 137728$. This is the sequence A053175 in Sloane's database \cite{Sloane}. The sequence $\{P_n\}_{n=0}^\infty$ is also related to the theory of modular forms; see \cite{Zagier}. Recently, Sun \cite{Sun-2} conjectured that \begin{conjecture}\label{c-2} The sequences $\{P_{n+1}/P_n\}_{n=0}^\infty $ and $\{\sqrt[n]{P_n}\}_{n=1}^\infty$ are strictly increasing. \end{conjecture} Employing Theorem \ref{Th-2}, we prove that \begin{corollary}\label{c-3} Conjecture \ref{c-2} is true. \end{corollary} \noindent{\it Proof.} By \eqref{1-13}, we find $r_1=3$. Set $N_0=3$. It is easy to check that \eqref{1-16} and \eqref{1-17} hold for $N_0=3$. By Theorem \ref{Th-2}, we see that the sequence $\{P_n\}_{n=3}^\infty$ is strictly log-convex. It is a routine to verify that $\frac{P_{i+1}}{P_i}> \frac{P_i}{P_{i-1}}$ for $1\leq i \leq 3$. Thus, the sequence $\{P_n\}_{n=0}^\infty$ is strictly log-convex and the sequence $\{P_{n+1}/P_n\}_{n=0}^\infty $ is strictly increasing, namely, \begin{align} \frac{P_{n+1}}{P_n}> \frac{P_n}{P_{n-1}}, \qquad n\geq 1.\label{3-3} \end{align} By \eqref{3-3} and the fact $P_0=1$, we deduce that \begin{align}\label{3-4} P_n=P_0\prod_{i=1}^n\frac{P_i}{P_{i-1}}<\left(\frac{P_{n+1}}{P_n}\right)^n, \end{align} which implies that \begin{align} P_n^{n+1} \frac{A_i}{A_{i-1}}$ for $1\leq i \leq 5$. Thus, the following corollary is true. \begin{corollary}\label{c-1} The sequence $\{A_{n}\}_{n=0}^\infty $ is strictly log-convex. \end{corollary} The central Delannoy number $D_n$ is defined by \begin{align} D_n=\frac{3(2n-1)}{n}D_{n-1}-\frac{n-1}{n}D_{n-2}, \qquad n\geq 2,\label{3-6} \end{align} with $D_0=1$ and $D_1=3$; see \cite{P}. Do\v{s}li\'{c} \cite{D} , and Liu and Wang \cite{Liu} proved the log-convexity of the sequence $\{D_n\}_{n=0}^\infty$. By \eqref{1-13}, we find $r_1=2$. Let $N_0=2$. It is easy to check that \eqref{1-16} and \eqref{1-17} hold for $N_0=2$. The following corollary follows from Theorem \ref{Th-2} and the fact $\frac{D_2}{D_1}>\frac{D_1}{D_0}$. \begin{corollary}\label{c-4} The sequence $\{D_{n}\}_{n=0}^\infty $ is strictly log-convex. \end{corollary} The little Schr\"{o}er number $s_n$ is defined by \begin{align} s_n=\frac{3(2n-1)}{n+1}s_{n-1}-\frac{n-2}{n+1}s_{n-2}, \qquad n\geq 2, \label{3-7} \end{align} with $s_0=1$ and $s_1=1$; see \cite{Foata,Sul}. Do\v{s}li\'{c} \cite{D} , and Liu and Wang \cite{Liu} proved the log-convexity of the sequence $\{s_n\}_{n=0}^\infty$. It is easy to see that $r_1=3$. Let $N_0=3$. We can check that \eqref{1-16} and \eqref{1-17} hold for $N_0=3$. The following corollary follows from Theorem \ref{Th-2} and the fact $\frac{s_3}{s_2}>\frac{s_2}{s_1}>\frac{s_1}{s_0}$. \begin{corollary}\label{c-5} The sequence $\{s_{n}\}_{n=0}^\infty $ is strictly log-convex. \end{corollary} Let $R_n$ be the number of the set of all tree-like polyhexes with $n+1$ hexagons. The sequence $\{R_n\}_{n=0}^\infty $ satisfies the recurrence \begin{align} R_n=\frac{3(2n-1)}{n+1}R_{n-1}-\frac{5(n-2)}{n+1}R_{n-2},\qquad n\geq 2, \label{3-8} \end{align} with $R_0=1$ and $R_1=1$; see \cite{Harary}. The sequence $\{R_n\}_{n=0}^\infty$ is the sequence A002212 in Sloane's database \cite{Sloane}. Liu and Wang \cite{Liu} proved the log-convexity of the sequence $\{R_n\}_{n=0}^\infty$. Let $N_0=3$. Employing Theorem \ref{Th-2} and evaluating the values of $R_2,\ R_3$ and $R_4$, we can prove the following corollary. \begin{corollary}\label{c-6} The sequence $\{R_{n}\}_{n=0}^\infty $ is strictly log-convex. \end{corollary} Let $w_n$ be the number of walks on cubic lattice with $n$ steps, starting and finishing on the $x-y$ plane and never going below it. The sequence $\{w_n\}_{n=0}^\infty $ has three-term recurrence relation \begin{align} w_n=\frac{4(2n+1)}{n+2}w_{n-1}-\frac{12(n-1)}{n+2}w_{n-2}, \qquad n\geq 2, \label{3-9} \end{align} with $w_0=1$ and $w_1=4$; see \cite{Guy}. The sequence $\{w_n\}_{n=0}^\infty$ is the sequence A005572 in Sloane's database \cite{Sloane}. Liu and Wang \cite{Liu} proved the log-convexity of the sequence $\{w_n\}_{n=0}^\infty$. Set $N_0=2$. The following corollary follows from Theorem \ref{Th-2} and the fact $\frac{w_{i+1}}{w_i}>\frac{w_i}{w_{i-1}}$ for $i=1,\ 2 $. \begin{corollary}\label{c-7} The sequence $\{w_{n}\}_{n=0}^\infty $ is strictly log-convex. \end{corollary} Let $F_n$ be defined by \begin{align} F_n=\frac{4n^4-n^3-n^2+3n+2 }{n^4+2n^2-1 } F_{n-1} -\frac{ 2n^3-5n^2-n+1 }{ 2n^3-3n^2+2n}F_{n-2}, \qquad n\geq 2, \label{3-10} \end{align} with $F_0=1$ and $F_1=1$. By \eqref{1-13}, we find $r_1=42$. Set $N_0=42$. It is easy to check that \eqref{1-16} and \eqref{1-17} hold for $N_0=42$. We can also verify that $\frac{F_{i+1}}{F_i}>\frac{F_i}{F_{i-1}}$ for $3\leq i \leq 42$. Hence, we can prove the following corollary. \begin{corollary}\label{c-8} The sequence $\{F_{n}\}_{n=2}^\infty $ is strictly log-convex. \end{corollary} To conclude this paper, we remark that the method presented in this paper can be used to prove the log-convexity of some combinatorial sequences satisfied longer recurrence relations. 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