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 $n\geq 0$, $$1\leq {b_{k}(n+2) b_{k}(n) \over b_{k}(n+1)^{2}} \leq {n+2 \over n+1}.$$ 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 $$b_{k}(n) b_{k}(m) \leq b_{k}(n+m) \leq {n+m \choose n} b_{k}(n) b_{k}(m), \quad \forall n, m\geq 0.$$ Then we apply these results to white noise distribution theory.