Bell numbers, log-concavity, and log-convexity

Abstract

Let $\{b_k(n){\}}_{n=0}^{\infty }$ be the Bell numbers of order $k$. It is proved that the sequence $\{b_k(n)/n!{\}}_{n=0}^{\infty }$ is log-concave and the sequence $\{b_k(n){\}}_{n=0}^{\infty }$ is log-convex, or equivalently, the following inequalities hold for all $n\ge 0$, $$1\le \frac{b_k(n+2)b_k(n)}{b_k(n+1{)}^2}\le \frac{n+2}{n+1}.$$ Let $\{\a(n)\}_{n=0}^{\infty}$ be a sequence of positive numbers with $\a(0)=1$. We show that if $\{\a(n)\}_{n=0}^{\infty}$ is log-convex, then $$\a (n) \a (m) \leq \a(n+m), \quad \forall n, m\geq 0.$$ On the other hand, if $\{\a(n)/n!\}_{n=0}^{\infty}$ is log-concave, then $$\a (n+m) \leq {n+m \choose n} \a (n) \a (m), \quad \forall n, m\geq 0.$$ In particular, we have the following inequalities for the Bell numbers $$b_k(n)b_k(m)\le b_k(n+m)\le \left(\genfrac{}{}{0ex}{}{n+m}{n}\right)b_k(n)b_k(m),\forall n,m\ge 0.$$ Then we apply these results to white noise distribution theory.

Louisiana state university preprint (1999)