\documentclass[12pt]{article} \usepackage{e-jc} \specs{P4.7}{24(4)}{2017} \usepackage{mathrsfs} \usepackage{amsmath} \usepackage{amssymb,amsfonts,amsmath,amsthm} \usepackage{latexsym,amsmath,amssymb,amsfonts,epsfig,graphicx,cite,psfrag} \usepackage{eepic,color,colordvi,amscd,amsthm} \usepackage{pstcol,pstricks,color} \usepackage{epsfig} %-------------------------------------------------------------------- %-------------------------------------------------------------------- \newtheorem{theorem}{Theorem}[section] \newtheorem{thm}[theorem]{Theorem} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{rem}[theorem]{Remark} \newtheorem{defi}[theorem]{Definition} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{question}[theorem]{Question} \newtheorem{ex}[theorem]{Example} \newtheorem{claim}[theorem]{Claim} \newtheorem{conj}[theorem]{Conjecture} \newtheorem{continuation}[theorem]{Continuation of Example} \newtheorem{alg}{Algorithm}[section] \def\theequation{\thesection.\arabic{equation}} \makeatletter \@addtoreset{equation}{section} \newcommand{\pf}{\proof} \makeatother %-------------------------------------------------------------------- %-------------------------------------------------------------------- \title{\bf A Short Proof of Moll's Minimal Conjecture} \author{ Lun Lv\\ \small School of Sciences\\[-0.8ex] \small Hebei University of Science and Technology\\[-0.8ex] \small Shijiazhuang 050018, P.R. China \\ \small \tt klunlv@163.com} \date{\dateline{Mar 1, 2017}{Sep 25, 2017}{Oct 6, 2017}\\ \small Mathematics Subject Classifications: 05A20, 11B83, 33F99} %-------------------------------------------------------------------- \begin{document} \maketitle \begin{abstract} We give a short proof of Moll's minimal conjecture, which has been confirmed by Chen and Xia. \bigskip\noindent \textbf{Keywords:} Boros-Moll polynomial; Moll's minimal conjecture; spiral property \end{abstract} \section{Introduction} The Boros-Moll polynomials, denoted by $P_m(a)$, arise in the evaluation of the following quartic integral, see \cite{Bor-Mol99a,Bor-Mol99b,Bor-Mol99c, Bor-Mol01,Bor-Mol04,Mol02}. For any $a>-1$ and any nonnegative integer $m$, \begin{equation*} \int_{0}^{\infty}\frac{1}{(x^4+2ax^2+1)^{m+1}}dx=\frac{\pi}{2^{m+3/2}(a+1)^{m+1/2}}P_m(a), \end{equation*} where \begin{equation}\label{ss} P_m(a)=2^{-2m}\sum\limits_{k}2^k{2m-2k\choose m-k}{m+k\choose k}(a+1)^k. \end{equation} Let $d_l(m)$ be the coefficient of $a^l$ in $P_m(a)$. Then (\ref{ss}) gives \begin{equation}\label{eq1.4} d_{l}(m)=2^{-2m}\sum\limits_{k=l}^m 2^k{2m-2k\choose m-k}{m+k\choose k}{k\choose l}. \end{equation} Much progress has been made since Boros and Moll \cite{Amd-Mol08} proved the positivity of the coefficients of $P_m(a)$. Boros and Moll \cite{Bor-Mol99c} have proved that the sequence $\{d_l(m)\}_{0\leq l \leq m}$ is unimodal. The log-concavity of the sequence $\{d_l(m)\}_{1\leq l \leq m-1}$ was conjectured by Moll \cite{Mol02}, and it was proved by Kauers and Paule\cite{Kau-Pau07} based on recurrence relations. Chen and Xia \cite{Chen-Xia08} showed that the sequence $d_l(m)$ satisfies the strongly ratio monotone property which implies the log-concavity and the spiral property. Chen and Gu \cite{Chen-Gu09} proved the reverse ultra log-concavity of the Boros-Moll polynomials. By introducing the structure of partially 2-colored permutations, Chen, Pang and Qu \cite{cpq} found a combinatorial proof of the log-concavity of the Boros-Moll polynomials. Moll also posed a conjecture that is stronger than the log-concavity of the polynomials $P_m(a)$. This conjecture was called Moll's minimum conjecture, and has been confirmed by Chen and Xia \cite{Chen-Xia09}. The main objective of this paper is to give a short proof of the following equivalent form of Moll's minimal conjecture, which was confirmed by Chen and Xia \cite{Chen-Xia09}. \begin{theorem}[Theorem 2.1 \cite{Chen-Xia09}]\label{theorem1} Given $m\geq2$, for $1\leq l\leq m$, $l(l+1)(d^{2}_{l}(m)-d_{l+1}(m)d_{l-1}(m))$ attains its minimum at $l=m$ with $m(m+1)d^2_{m}(m)$. \end{theorem} \section{The Proof of Theorem \ref{theorem1}} Chen and Gu \cite{Chen-Gu09} proved the following theorem, which gave a lower bound of $\frac{d_{l}^{2}(m)}{d_{l+1}(m)d_{l-1}(m)}.$ \begin{theorem}[Theorem 1.2 \cite{Chen-Gu09}]\label{eq1} For $m\geq2$ and $1\leq l\leq m-1$, we have \begin{align} \frac{d^{2}_{l}(m)}{d_{l+1}(m)d_{l-1}(m)} >\frac{(m-l+1)(m+l)(l+1)}{l(m-l)(m+l+1)}. \end{align} \end{theorem} Multiplying both sides of \eqref{eq1} by $l$ and then plusing $ld^{2}_{l}(m)$ to the two sides gives the following result. \begin{theorem} For $m\geq2$ and $1\leq l\leq m-1$, we have \begin{align}\label{eq2} l(l+1)\left(d^{2}_{l}(m)-d_{l+1}(m)d_{l-1}(m)\right) >\left(l+\frac{2l^3}{(m+l)(m-l+1)}\right) d^{2}_{l}(m). \end{align} \end{theorem} On the other hand, Chen and Xia \cite{Chen-Xia08} have shown the spiral property of sequence $\{d_{l}(m)\}_{1\leq l\leq m-1}$, that is \begin{align}\label{eq11} d_{m-1}(m)0.$$ Restricting $l\in N^{+}$, we see that the sequence $\{l+\frac{2l^3}{(m+l)(m-l+1)}\}_{1\leq l\leq m-1}$ is strictly monotone increasing. Combining \eqref{eq2} and \eqref{eq11}, we get \begin{align}\label{eq3} &l(l+1)(d^{2}_{l}(m)-d_{l+1}(m)d_{l-1}(m))>(l+\frac{2l^3}{(m+l)(m-l+1)})d^{2}_{l}(m)\nonumber\\ &\qquad\qquad\geq\min\{(1+\frac{2}{(m+1)m})d^{2}_{1}(m),(m-1+\frac{(m-1)^3}{2m-1})d^{2}_{m-1}(m)\}. \end{align} By direct computation we may deduce from \eqref{eq1.4} that \begin{align*} &(1+\frac{2}{(m+1)m})d^{2}_{1}(m)\geq m(m+1)d^2_{m}(m),\\ &(m-1+\frac{(m-1)^3}{2m-1})d^{2}_{m-1}(m)\geq m(m+1)d^2_{m}(m). \end{align*} It follows by \eqref{eq3} that \begin{align*} l(l+1)(d^{2}_{l}(m)-d_{l+1}(m)d_{l-1}(m))>m(m+1)d^2_{m}(m),\quad 1\leq l\leq m-1. \end{align*} This completes the proof. \qed %\vskip 3mm \noindent {\bf Acknowledgments.} \subsection*{Acknowledgements} This work was supported by the National Natural Science Foundation of China, the Natural Science Foundation of Hebei Province (A2014208152) and the Top Young-aged Talents Program of Hebei Province. \begin{thebibliography}{99} \bibitem{Amd-Mol08} T. Amdeberhan and V. Moll. \newblock A formula for a quartic integral: A survey of old proofs and some new ones. \newblock {\em Ramanujan J.}, 18: 91--102, 2009. \bibitem{Bor-Mol99a} G. Boros and V. Moll. \newblock An integral hidden in Gradshteyn and Ryzhik. \newblock {\em J. Comput. Appl. Math.}, 106: 361--368, 1999. \bibitem{Bor-Mol99b} G. Boros and V. Moll. \newblock A sequence of unimodal polynomials. \newblock {\em J. Math. Anal. Appl.}, 237: 272--285, 1999. \bibitem{Bor-Mol99c} G. Boros and V. Moll. \newblock A criterion for unimodality. \newblock {\em Electron. J. Combin.}, 6: \#R10, 1999. \bibitem{Bor-Mol01} G. Boros and V. Moll. \newblock The double square root, Jacobi polynomials and Ramanujan's Master Theorem. \newblock {\em J. Comput. Appl. Math.}, 130: 337--344, 2001. \bibitem{Bor-Mol04} G. Boros and V. Moll. \newblock {\em Irresistible Integrals}, Cambridge University Press, Cambridge, 2004. \bibitem{Chen-Gu09} W.Y.C. Chen and C.C.Y Gu. \newblock The reverse ultra log-concavity of the Boros-Moll polynomials. \newblock {\em Proc. Amer. Math. Soc.}, 137: 3991--3998, 2009. \bibitem{cpq} W.Y.C. Chen, S.X.M. Pang and E.X.Y. Qu. \newblock Partially 2-colored permutations and the Boros-Moll polynomials. \newblock {\em Ramanujan J.}, 27: 297--304, 2012. \bibitem{Chen-Xia08} W.Y.C. Chen and E.X.W. Xia. \newblock The ratio monotonicity of the Boros-Moll polynomials. \newblock {\em Math. Comput.}, 78: 2269--2282, 2009. \bibitem{Chen-Xia09} W.Y.C. Chen and E.X.W. Xia. \newblock Proof of Moll's minimum conjecture. \newblock {\em European J. Combin.}, 34: 787--791, 2013. \bibitem{Kau-Pau07} M. Kauers and P. Paule. \newblock A computer proof of Moll's log-concavity conjecture. \newblock {\em Proc. Amer. Math. Soc.}, 135: 3847--3856, 2007. \bibitem{Mol02} V. Moll. \newblock The evaluation of integrals: A personal story. \newblock {\em Notices Amer. Math. Soc.}, 49: 311--317, 2002. \end{thebibliography} \end{document}