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    • Some inequalities for the Bell numbers

      FENG QI

      Research Article Volume 127 Issue 4 September 2017 pp 551-564

    • Fulltext

       

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      Permanent link:
      https://www.ias.ac.in/article/fulltext/pmsc/127/04/0551-0564

    • 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

    • 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.

    • Author Affiliations

       

      FENG QI1 2 3

      1. Institute of Mathematics, Henan Polytechnic University, Jiaozuo City 454010, Henan Province, China
      2. College of Mathematics, Inner Mongolia University for Nationalities, Tongliao City 028043, Inner Mongolia Autonomous Region, China
      3. Department of Mathematics, College of Science, Tianjin Polytechnic University, Tianjin City 300387, China

    • Dates

       
      Published
      1 September 2017
      Manuscript received
      28 February 2016
      Manuscript revised
      21 June 2016
      Accepted
      27 June 2016
      Early published
      Unedited version published online
      Final version published online
      19 August 2017

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