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\vskip 1cm{\LARGE\bf
On Engel's Inequality for Bell Numbers}
\vskip 1cm
\large
Horst Alzer\\
Morsbacher Stra{\ss}e 10\\
51545 Waldbr\"ol\\
Germany\\
\href{mailto:h.alzer@gmx.de}{\tt  h.alzer@gmx.de}  \\
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\begin{abstract}
We prove that for all integers $n\geq 2$ the expression $B_{n-1} B_{n+1}-B_n^2$ can be represented as an infinite series with nonnegative terms. Here $B_k$ denotes the $k$-th Bell number. It follows that the sequence $(B_n)_{n\geq 0}$ is strictly log-convex. This result refines Engel's inequality 
$B_n^2 \leq B_{n-1} B_{n+1}$.
\end{abstract}

\section{Introduction}

A partition of a set $S$ with $n$ elements is a collection of nonempty, pairwise disjoint subsets whose union is equal to $S$. The Bell number  $B_n$,
  named after the British mathematician Eric T. Bell, is the number of partitions of $S$.
The first few numbers are
$$
B_0=1, \quad B_1=1, \quad B_2=2, \quad B_3=5, \quad B_4=15, \quad B_5=52.
$$
The Bell numbers are given by the exponential generating function
$$
\exp(e^x-1)=\sum_{n=0}^\infty \frac{B_n}{n!} x^n
$$
and they satisfy the recurrence relation
$$
B_{n+1}=\sum_{k=0}^n {n\choose k}  B_k \quad{(n\geq 0)}.
$$
Moreover, the Bell numbers can be expressed in terms of the Stirling numbers of the second kind,
$$
B_n=\sum_{k=1}^n S(n,k) \quad{(n\geq 1)}.
$$
The following remarkable series representation is due to Dobi\'nski \cite{D},
\begin{equation}\label{E1}
B_n=\frac{1}{e} \sum_{k=0}^\infty \frac{k^n}{k!} \quad{(n\geq 0)}.
\end{equation}
In 1994, Engel \cite{E} proved that the sequence $(B_n)_{n\geq 0}$ is log-convex, that is,
\begin{equation}\label{E2}
B_n^2 \leq B_{n-1} B_{n+1} \quad{(n\geq 1)}.
\end{equation}
 Canfield \cite{C}, Asai, Kubo, and Kuo \cite{AKK} and  Bouroubi \cite{Bo} published further proofs of \eqref{E2}.
For more information and references on Bell numbers we  refer to \cite{Br} and  \cite{S}.


Richard E. Bellman (1920--1984), who was one of the leading mathematicians in the field of inequalities,  pointed out that ``every inequality should come from an equality which makes the inequality obvious" \cite[p. 449]{Be}. In view of this statement it is natural to ask whether Engel's inequality is a consequence of an equality. It is the aim of this note to give an affirmative answer to this question. 
In particular, we prove that for all $n\geq 1$ strict inequality holds in \eqref{E2}.
We show that an application of Dobi\'nski's formula leads to the following result.

\section{The main result}

\begin{theorem}
{For all natural numbers $n\geq 2$ we have}
\begin{equation}\label{E3}
B_{n-1} B_{n+1} - B_n^2 =\frac{1}{2e^2} \sum_{k=2}^\infty \sum_{j=1}^{k-1} \frac{j^{n-1} (k-j)^{n-1}}{j! (k-j)!} (k-2j)^2.
\end{equation}
\end{theorem}



\begin{proof}
Applying \eqref{E1}  and the Cauchy product for infinite series we obtain for $n\geq 2$,
\begin{align} 
\nonumber
e^2 \bigl(  B_{n-1} B_{n+1} - B_n^2 \bigr)
 & =     \sum_{k=0}^\infty \sum_{j=0}^{k} \frac{j^{n-1} (k-j)^{n+1}}{j! (k-j)!} 
-\sum_{k=0}^\infty \sum_{j=0}^{k} \frac{j^{n} (k-j)^{n}}{j! (k-j)!} \\
\label{E4}
& =  \sum_{k=2}^\infty \bigl( L_n(k) - R_n(k) \bigr)  
\end{align}
with
$$
L_n(k)= \sum_{j=1}^{k-1} \frac{j^{n-1} (k-j)^{n+1}}{j! (k-j)!} 
\quad\mbox{and}
\quad
R_n(k)=
 \sum_{j=1}^{k-1} \frac{j^{n} (k-j)^{n}}{j! (k-j)!}.
$$
Since
$$
 \sum_{j=1}^{k-1} \frac{j^{n-1} (k-j)^{n+1}}{j! (k-j)!} = \sum_{j=1}^{k-1} \frac{j^{n+1} (k-j)^{n-1}}{j! (k-j)!},
$$
we get
\begin{align}
 \nonumber
L_n(k)-R_n(k) & =  \frac{1}{2} \sum_{j=1}^{k-1} \frac{1}{j! (k-j)!} \bigl(  j^{n-1} (k-j)^{n+1}+j^{n+1}(k-j)^{n-1}-2j^n (k-j)^n \bigr) \\ 
\label{E5}
& =  \frac{1}{2}  \sum_{j=1}^{k-1} \frac{j^{n-1} (k-j)^{n-1}}{j! (k-j)!} (k-2j)^2.   
\end{align}
From \eqref{E4} and \eqref{E5}  we conclude that \eqref{E3} holds.
\end{proof}







\begin{remark}
Using $B_0 B_2-B_1^2=1$ and \eqref{E3} reveals that the sequence $(B_n)_{n\geq 0}$ is not only log-convex, but even strictly log-convex,
\begin{equation}\label{E6}
B_n^2 < B_{n-1} B_{n+1} \quad{(n\geq 1)}.
\end{equation}
\end{remark}




\begin{remark}
 Asai, Kubo, and Kuo \cite{AKK} used Engel's inequality to prove that
$$
B_m B_n \leq B_{m+n} \quad{(m,n\geq 0)}.
$$
An application of \eqref{E6} gives for $m,n\geq 1$,
$$
B_m=\prod_{\nu=1}^m \frac{B_{\nu}}{B_{\nu -1}} < \prod_{\nu=1}^m \frac{B_{n+\nu}}{B_{n+\nu-1}}=\frac{B_{m+n}}{B_n}.
$$
Thus,
$$
B_m B_n < B_{m+n} \quad{(m,n\geq 1)}.
$$
\end{remark}




\begin{thebibliography}{9}

\bibitem{AKK}
N. Asai, I. Kubo, and H.-H. Kuo, Bell numbers, log-concavity, and log-convexity, {\it Acta Appl. Math.} {\textbf 63} (2000), 79--87.

\bibitem{Be}
R. Bellman, Why study  inequalities?, in E. F. Beckenbach, ed.,
{\it General Inequalities 2}, Int. Ser. Numer. Math. {\bf 47}, Birkh\"auser,
1980, p.~449.

\bibitem{Bo}
S. Bouroubi, Bell numbers and Engel's conjecture, {\it
Rostock. Math. Kolloq.} {\textbf 62} (2007), 61--70.

\bibitem{Br}
D.  Branson, Stirling numbers and Bell numbers: their role in combinatorics and probability, {\it Math. Scientist} {\textbf 25} (2000), 1--31.

\bibitem{C}
E. R. Canfield, Engel's inequality for Bell numbers,
{\it J. Combin. Th. Ser. A} {\textbf 72} (1995), 184--187.

\bibitem{D}
G. Dobi\'nski, Summierung der Reihe $\sum n^m/n!$ f\"ur $m=1,2,3,4,5,\ldots$, {\it Archiv Math. Phys.} {\textbf 61} (1877), 333--336.

\bibitem{E}
K. Engel, On the average rank of an element  in a filter of the partition lattice, {\it J. Combin. Th. Ser. A} {\textbf 65}  (1994), 67--78.

\bibitem{S}
N. J. A. Sloane et al.,
{\it The On-Line Encyclopedia of Integer Sequences},
OEIS Foundation, Inc., 2019.  Available at 
\url{https://oeis.org}.

\end{thebibliography}



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\noindent {\it 2010 Mathematics Subject Classification:}  Primary 05A19;
Secondary 05A20,  11B73, 26A51.  

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\noindent  {\emph Keywords:} Bell number, Engel's inequality,
strictly log-convex, identity.

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\noindent (Concerned with sequence   \seqnum{A000110}.)

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\vspace*{+.1in}
\noindent
Received June 26 2019;
revised version received August 25 2019.
Published in {\it Journal of Integer Sequences}, September 25 2019.

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\noindent
Return to
\htmladdnormallink{Journal of Integer Sequences home page}{http://www.cs.uwaterloo.ca/journals/JIS/}.
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