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.
Categories: math.CO