Acta Applicandae Mathematica

, Volume 63, Issue 1–3, pp 79–87

# Bell Numbers, Log-Concavity, and Log-Convexity

• Nobuhiro Asai
• Izumi Kubo
• Hui-Hsiung Kuo
Article

## 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,
$$1 \leqslant \frac{{b_k (n + 2)b_k (n)}}{{b_k (n + 1)^2 }} \leqslant \frac{{n + 2}}{{n + 1}}$$
. 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
$$\alpha (n + m) \leqslant \left( {\begin{array}{*{20}c} {n + m} \\ n \\ \end{array} } \right)\alpha (n)\alpha (m),{\text{ }}\forall n,m \geqslant 0$$
. In particular, we have the following inequalities for the Bell numbers
$$b_k (n)b_k (m) \leqslant b_k (n + m) \leqslant \left( {\begin{array}{*{20}c} {n + m} \\ n \\ \end{array} } \right)b_k (n)b_k (m),{\text{ }}\forall n,m \geqslant 0$$
. Then we apply these results to characterization theorems for CKS-space in white noise distribution theory.
Bell numbers log-concavity log-convexity CKS-space characterization theorem white noise distribution theory

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## Authors and Affiliations

• Nobuhiro Asai
• 1
• Izumi Kubo
• 2
• Hui-Hsiung Kuo
• 3
1. 1.Graduate School of MathematicsNagoya UniversityNagoyaJapan
2. 2.Department of Mathematics, Graduate School of ScienceHiroshima UniversityHigashi-HiroshimaJapan
3. 3.Department of MathematicsLouisiana State UniversityBaton RougeU.S.A.