Bell Numbers, Log-Concavity, and Log-Convexity

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, % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 % qacaaIXaGaeyizIm6aaSaaa8aabaWdbiaadkgapaWaaSbaaSqaa8qa % caWGRbaapaqabaGcpeGaaiikaiaad6gacqGHRaWkcaaIYaGaaiykai % aadkgapaWaaSbaaSqaa8qacaWGRbaapaqabaGcpeGaaiikaiaad6ga % caGGPaaapaqaa8qacaWGIbWdamaaBaaaleaapeGaam4AaaWdaeqaaO % WdbiaacIcacaWGUbGaey4kaSIaaGymaiaacMcapaWaaWbaaSqabeaa % peGaaGOmaaaaaaGccqGHKjYOdaWcaaWdaeaapeGaamOBaiabgUcaRi % aaikdaa8aabaWdbiaad6gacqGHRaWkcaaIXaaaaiaac6caaaa!52BA! $$1 \leqslant \frac{{{b_k}(n + 2){b_k}(n)}}{{{b_k}{{(n + 1)}^2}}} \leqslant \frac{{n + 2}}{{n + 1}}.$$ 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 % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeqabeGaaa % qaaiabeg7aHjaacIcacaWGUbGaaiykaiabeg7aHjaacIcacaWGTbGa % aiykaiabgsMiJkabeg7aHjaacIcacaWGUbGaey4kaSIaamyBaiaacM % cacaGGSaaabaGaeyiaIiIaamOBaiaacYcacaWGTbGaeyyzImRaaGim % aiaac6caaaaaaa!4C91! $$\begin{array}{*{20}{c}} {\alpha (n)\alpha (m) \leqslant \alpha (n + m),}&{\forall n,m \geqslant 0.} \end{array}$$ On the other hand, if {α(n)/n!} n=0 ∞ is log-concave, then % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeqabeGaaa % qaaiabeg7aHjaacIcacaWGUbGaey4kaSIaamyBaiaacMcacqGHKjYO % daqadaqaauaabeqaceaaaeaacaWGUbGaey4kaSIaamyBaaqaaiaad6 % gaaaaacaGLOaGaayzkaaGaeqySdeMaaiikaiaad6gacaGGPaGaeqyS % deMaaiikaiaad2gacaGGPaGaaiilaaqaaiabgcGiIiaad6gacaGGSa % GaamyBaiabgwMiZkaaicdacaGGUaaaaaaa!51E1! $$\begin{array}{*{20}{c}} {\alpha (n + m) \leqslant \left( {\begin{array}{*{20}{c}} {n + m} \\ n \end{array}} \right)\alpha (n)\alpha (m),}&{\forall n,m \geqslant 0.} \end{array}$$ In particular, we have the following inequalities for the Bell numbers % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeqabeGaaa % qaaiaadkgadaWgaaWcbaGaam4AaaqabaGccaGGOaGaamOBaiaacMca % caWGIbWaaSbaaSqaaiaadUgaaeqaaOGaaiikaiaad2gacaGGPaGaey % izImQaamOyamaaBaaaleaacaWGRbaabeaakiaacIcacaWGUbGaey4k % aSIaamyBaiaacMcacqGHKjYOdaqadaqaauaabeqaceaaaeaacaWGUb % Gaey4kaSIaamyBaaqaaiaad6gaaaaacaGLOaGaayzkaaGaamOyamaa % BaaaleaacaWGRbaabeaakiaacIcacaWGUbGaaiykaiaadkgadaWgaa % WcbaGaam4AaaqabaGccaGGOaGaamyBaiaacMcacaGGSaaabaGaeyia % IiIaamOBaiaacYcacaWGTbGaeyyzImRaaGimaiaac6caaaaaaa!5D91! $$\begin{array}{*{20}{c}} {{b_k}(n){b_k}(m) \leqslant {b_k}(n + m) \leqslant \left( {\begin{array}{*{20}{c}} {n + m} \\ n \end{array}} \right){b_k}(n){b_k}(m),}&{\forall n,m \geqslant 0.} \end{array}$$ Then we apply these results to characterization theorems for CKS-space in white noise distribution theory.