Proceedings - Mathematical Sciences

, Volume 127, Issue 4, pp 551–564 | Cite as

Some inequalities for the Bell numbers

Article

Abstract

In this paper, we present derivatives of the generating functions for the Bell numbers by induction and by the Faà di Bruno formula, recover an explicit formula in terms of the Stirling numbers of the second kind, find the (logarithmically) absolute and complete monotonicity of the generating functions, and construct some inequalities for the Bell numbers. From these inequalities, we derive the logarithmic convexity of the sequence of the Bell numbers.

Keywords

Bell number determinant product inequality generating function derivative absolutely monotonic function completely monotonic function logarithmically absolutely monotonic function logarithmically completely monotonic function Stirling number of the second kind induction Faà di Bruno formula logarithmic convexity 

2010 Mathematics Subject Classification

Primary: 11B73 Secondary: 26A48 26A51 33B10 

Notes

Acknowledgements

The author would like to thank the anonymous referees for their careful corrections and valuable comments on the original version of this paper.

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Copyright information

© Indian Academy of Sciences 2017

Authors and Affiliations

  1. 1.Institute of MathematicsHenan Polytechnic UniversityJiaozuo CityChina
  2. 2.College of MathematicsInner Mongolia University for NationalitiesTongliao CityChina
  3. 3.Department of Mathematics, College of ScienceTianjin Polytechnic UniversityTianjin CityChina

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