Proceedings - Mathematical Sciences

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

Some inequalities for the Bell numbers



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.


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 



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


  1. 1.
    Abramowitz M and Stegun I A (eds), Handbook of mathematical functions with formulas, graphs and mathematical tables, National Bureau of Standards, Applied Mathematics Series 55, 10th printing (1972) (New York and Washington: Dover Publications)Google Scholar
  2. 2.
    Asai N, Kubo I and Kuo H-H, Bell numbers, log-concavity and log-convexity, Acta Appl. Math. 63 (1–3) (2000) 79–87; available online at doi: 10.1023/A:1010738827855
  3. 3.
    Atanassov R D and Tsoukrovski U V, Some properties of a class of logarithmically completely monotonic functions, C. R. Acad. Bulgare Sci. 41(2) (1988) 21–23MathSciNetMATHGoogle Scholar
  4. 4.
    Berg C, Integral representation of some functions related to the gamma function, Mediterr. J. Math. 1(4) (2004) 433–439; available online at doi: 10.1007/s00009-004-0022-6
  5. 5.
    Canfield E R, Engel’s inequality for Bell numbers, J. Combin. Theory Ser. A 72(1) (1995) 184–187; available online at doi: 10.1016/0097-3165(95)90033-0
  6. 6.
    Chen C-P and Qi F, Completely monotonic function associated with the gamma function and proof of Wallis’ inequality, Tamkang J. Math. 36(4) (2005) 303–307; available online at doi: 10.5556/j.tkjm.36.2005.101
  7. 7.
    Comtet L, Advanced combinatorics: the art of finite and infinite expansions, revised and enlarged edition (1974) (Dordrecht and Boston: D. Reidel Publishing Co.)Google Scholar
  8. 8.
    Guo B-N and Qi F, A property of logarithmically absolutely monotonic functions and the logarithmically complete monotonicity of a power-exponential function, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys. 72(2) (2010) 21–30Google Scholar
  9. 9.
    Guo B-N and Qi F, An explicit formula for Bell numbers in terms of Stirling numbers and hypergeometric functions, Glob. J. Math. Anal. 2(4) (2014) 243–248; available online at doi: 10.14419/gjma.v2i4.3310
  10. 10.
    Feng Qi and Bai-Ni Guo, Explicit formulas for special values of the Bell polynomials of the second kind and for the Euler numbers and polynomials, Mediterranean J. Math. 14(3) (2017) Article 140, 14 pages; available online at doi: 10.1007/s00009-017-0939-1
  11. 11.
    Howard F T, A special class of Bell polynomials, Math. Comp. 35(151) (1980) 977–989; available online at doi: 10.2307/2006208
  12. 12.
    Mitrinović D S, Analytic inequalities (1970) (Berlin: Springer)Google Scholar
  13. 13.
    Mitrinović D S and Pečarić J E, On two-place completely monotonic functions, Anzeiger Öster. Akad. Wiss. Math.-Natturwiss. Kl. 126 (1989) 85–88Google Scholar
  14. 14.
    Mitrinović D S, Pečarić J E and Fink A M, Classical and new inequalities in analysis (1993) (Kluwer Academic Publishers); available online at doi: 10.1007/978-94-017-1043-5
  15. 15.
    Pečarić J E, Remarks on some inequalities of A M Fink, J. Math. Anal. Appl. 104(2) (1984) 428–431; available online at doi: 10.1016/0022-247X(84)90006-4
  16. 16.
    Qi F, An explicit formula for the Bell numbers in terms of the Lah and Stirling numbers, Mediterr. J. Math. 13(5) (2016) 2795–2800; available online at doi: 10.1007/s00009-015-0655-7
  17. 17.
    Qi F, Generalized weighted mean values with two parameters, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 454(1978) (1998) 2723–2732; available online at doi: 10.1098/rspa.1998.0277
  18. 18.
    Qi F, Some inequalities for the Bell numbers, ResearchGate Technical Report (2015), available online at doi: 10.13140/RG.2.1.2544.2721
  19. 19.
    Qi F and Chen C-P, A complete monotonicity property of the gamma function, J. Math. Anal. Appl. 296(2) (2004) 603–607; available online at doi: 10.1016/j.jmaa.2004.04.026
  20. 20.
    Qi F and Guo B-N, Complete monotonicities of functions involving the gamma and digamma functions, RGMIA Res. Rep. Coll. 7(1) (2004), Art. 8 63–72; available online at
  21. 21.
    Qi F, Guo S and Guo B-N, Complete monotonicity of some functions involving polygamma functions, J. Comput. Appl. Math. 233(9) (2010) 2149–2160; available online at doi: 10.1016/
  22. 22.
    Qi F, Luo Q-M and Guo B-N, Complete monotonicity of a function involving the divided difference of digamma functions, Sci. China Math. 56(11) (2013) 2315–2325; available online at doi: 10.1007/s11425-012-4562-0
  23. 23.
    Qi F, Wei C-F and Guo B-N, Complete monotonicity of a function involving the ratio of gamma functions and applications, Banach J. Math. Anal. 6(1) (2012) 35–44; available online at doi: 10.15352/bjma/1337014663
  24. 24.
    Qi F and Xu S-L, The function \((b^{x}-a^{x})/x\): inequalities and properties, Proc. Amer. Math. Soc. 126(11) (1998) 3355–3359; available online at doi: 10.1090/S0002-9939-98-04442-6
  25. 25.
    Schilling R L, Song R and Vondraček Z, Bernstein functions—theory and applications, 2nd ed., de Gruyter Studies in Mathematics 37 (2012) (Berlin, Germany: Walter de Gruyter); available online at doi: 10.1515/9783110269338
  26. 26.
    van Haeringen H, Completely monotonic and related functions, J. Math. Anal. Appl. 204(2) (1996) 389–408; available online at doi: 10.1006/jmaa.1996.0443
  27. 27.
    van Haeringen H, Inequalities for real powers of completely monotonic functions, J. Math. Anal. Appl. 210(1) (1997) 102–113; available online at doi: 10.1006/jmaa.1997.5376
  28. 28.
    Widder D V, The Laplace Transform (1946) (Princeton: Princeton University Press)Google Scholar

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

Personalised recommendations