Bell numbers, log-concavity, and log-convexity

Let {bk(n)}n=0∞ be the Bell numbers of order k. It is proved that the sequence {bk(n)/n!}n=0∞ is log-concave and the sequence {bk (n)}n=0∞ is log-convex, or equivalently, the following inequalities hold for all n ≥ 0, 1 ≤ bk(n+2)bk(n)/bk(n+1)2 ≤ n+2/n+1. Let {α(n)}n=0∞ be a sequence of positive numbers with α(0) = 1. We show that if {α(n)}n=0∞ is log-convex, then α(n)α(m) ≤ α(n+m), ∀n, m ≥ 0. On the other hand, if {α(n)/n!}n=0∞ is log-concave, then α(n + m) ≤ (nn+m) α(n)α(m), ∀n, m ≥ 0. In particular, we have the following inequalities for the Bell numbers bk(n)bk(m) ≤ bk(n+m) ≤ (nn+m)bk(n)bk(m), ∀n, m ≥ 0. Then we apply these results to characterization theorems for CKS-space in white noise distribution theory