Bell Numbers, Log-Concavity, and Log-Convexity
Bell Numbers, Log-Concavity, and Log-Convexity
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贝尔数、对数凹性和对数凸性
DOI：
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发表时间：
2001
期刊：
影响因子：
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通讯作者：
H. Kuo
中科院分区：
文献类型：
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作者：
Nobuhiro Asai;I. Kubo;H. Kuo
AbstractLet {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,
$$1 leqslant frac{{b_k (n + 2)b_k (n)}}{{b_k (n + 1)^2 }} leqslant frac{{n + 2}}{{n + 1}}$$
. 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
$$alpha (n + m) leqslant left( {egin{array}{*{20}c} {n + m} \ n \ end{array} }
ight)alpha (n)alpha (m),{ ext{ }}forall n,m geqslant 0$$
. In particular, we have the following inequalities for the Bell numbers
$$b_k (n)b_k (m) leqslant b_k (n + m) leqslant left( {egin{array}{*{20}c} {n + m} \ n \ end{array} }
ight)b_k (n)b_k (m),{ ext{ }}forall n,m geqslant 0$$
. Then we apply these results to characterization theorems for CKS-space in white noise distribution theory.
AbstractLet {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,
$$1 leqslant frac{{b_k (n + 2)b_k (n)}}{{b_k (n + 1)^2 }} leqslant frac{{n + 2}}{{n + 1}}$$
. 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
$$alpha (n + m) leqslant left( {egin{array}{*{20}c} {n + m} \ n \ end{array} }
ight)alpha (n)alpha (m),{ ext{ }}forall n,m geqslant 0$$
. In particular, we have the following inequalities for the Bell numbers
$$b_k (n)b_k (m) leqslant b_k (n + m) leqslant left( {egin{array}{*{20}c} {n + m} \ n \ end{array} }
ight)b_k (n)b_k (m),{ ext{ }}forall n,m geqslant 0$$
. Then we apply these results to characterization theorems for CKS-space in white noise distribution theory.