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\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}
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\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)<d_1(m)<d_{m-2}(m)<d_2(m)<\cdots<d_{[\frac{m}{2}]}(m).
\end{align}

Now we are ready to prove Theorem \ref{theorem1}.
\pf[Proof of Theorem \ref{theorem1}]
Let $f(l)=l+\frac{2l^3}{(m+l)(m-l+1)}$.
Then for $1\leq l\leq m-1$,
$$f'(l)=1+\frac{6l^2}{(m+l)(m-l+1)}+\frac{2l^3(2l-1)}{(m+l)^2(m-l+1)^2}>0.$$
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}



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\bibitem{Chen-Gu09} W.Y.C. Chen and C.C.Y Gu. \newblock
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\bibitem{cpq}
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\bibitem{Chen-Xia08} W.Y.C. Chen and E.X.W. Xia. \newblock
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\bibitem{Chen-Xia09} W.Y.C. Chen and E.X.W. Xia. \newblock Proof of Moll's minimum conjecture. \newblock {\em
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\bibitem{Kau-Pau07} M. Kauers and P. Paule. \newblock
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\end{thebibliography}

\end{document}
