\documentclass[12pt,reqno]{article} \usepackage[usenames]{color} \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsthm} \usepackage{amsfonts} \usepackage{amscd} \usepackage{graphicx} \usepackage[colorlinks=true, linkcolor=webgreen, filecolor=webbrown, citecolor=webgreen]{hyperref} \definecolor{webgreen}{rgb}{0,.5,0} \definecolor{webbrown}{rgb}{.6,0,0} \usepackage{color} \usepackage{fullpage} \usepackage{float} \usepackage{psfig} \usepackage{graphics} \usepackage{latexsym} \usepackage{epsf} \usepackage{breakurl} \setlength{\textwidth}{6.5in} \setlength{\oddsidemargin}{.1in} \setlength{\evensidemargin}{.1in} \setlength{\topmargin}{-.1in} \setlength{\textheight}{8.4in} \newcommand{\seqnum}[1]{\href{https://oeis.org/#1}{\underline{#1}}} \begin{document} \begin{center} \epsfxsize=4in \leavevmode\epsffile{logo129.eps} \end{center} \theoremstyle{plain} \newtheorem{theorem}{Theorem} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{conjecture}[theorem]{Conjecture} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \begin{center} \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} \\ \end{center} \vskip .2 in \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} \bigskip \hrule \bigskip \noindent {\it 2010 Mathematics Subject Classification:} Primary 05A19; Secondary 05A20, 11B73, 26A51. \bigskip \noindent {\emph Keywords:} Bell number, Engel's inequality, strictly log-convex, identity. \bigskip \hrule \bigskip \noindent (Concerned with sequence \seqnum{A000110}.) \bigskip \hrule \bigskip \vspace*{+.1in} \noindent Received June 26 2019; revised version received August 25 2019. Published in {\it Journal of Integer Sequences}, September 25 2019. \bigskip \hrule \bigskip \noindent Return to \htmladdnormallink{Journal of Integer Sequences home page}{http://www.cs.uwaterloo.ca/journals/JIS/}. \vskip .1in \end{document}