Abstract
Let {b k (n)} n=0 ∞ be the Bell numbers of order k. It is proved that the sequence {b k (n)/n!} n=0 ∞ is log-concave and the sequence {b k (n)} n=0 ∞ is log-convex, or equivalently, the following inequalities hold for all n⩾0,
. 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
. In particular, we have the following inequalities for the Bell numbers
. Then we apply these results to characterization theorems for CKS-space in white noise distribution theory.
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Asai, N., Kubo, I. & Kuo, HH. Bell Numbers, Log-Concavity, and Log-Convexity. Acta Applicandae Mathematicae 63, 79–87 (2000). https://doi.org/10.1023/A:1010738827855
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DOI: https://doi.org/10.1023/A:1010738827855