Boletín de la Sociedad Matemática Mexicana

, Volume 23, Issue 2, pp 713–736 | Cite as

The harmonic and geometric means are Bernstein functions

Original Article


In the paper, the authors present by several approaches that both the harmonic mean and the geometric mean of two positive numbers are Bernstein functions and establish their integral representations.


Bernstein function Harmonic mean Geometric mean  Integral representation Stieltjes function Induction  Cauchy integral formula Stieltjes-Perron inversion formula 

Mathematics Subject Classification

Primary 26E60 Secondary 26A48 30E20 44A10 65R10 



This first author was partially supported by the National Natural Science Foundation of China under Grant No. 11361038 and by the Foundation of the Research Program of Science and Technology at Universities of Inner Mongolia Autonomous Region under Grant No. NJZY14192, China. The authors thank anonymous referees for their careful corrections to and valuable comments on the original version of this paper.


  1. 1.
    Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards, Applied Mathematics Series 55, 10th printing, Washington (1972)Google Scholar
  2. 2.
    Atanassov, R.D., Tsoukrovski, U.V.: Some properties of a class of logarithmically completely monotonic functions. C. R. Acad. Bulgare Sci. 41(2), 21–23 (1988)MathSciNetMATHGoogle Scholar
  3. 3.
    Berg, C.: Integral representation of some functions related to the gamma function. Mediterr. J. Math. 1(4), 433–439 (2004).
  4. 4.
    Berg, C.: Stieltjes-Pick-Bernstein-Schoenberg and their connection to complete monotonicity, In: Mateu, S., Porcu E. (eds) : Positive Definite Functions: from Schoenberg to Space-Time Challenges. Dept. Math., Univ. Jaume I, p. 24. Castellón de la Plana, Spain (2008).
  5. 5.
    Bochner, S.: Harmonic Analysis and the Theory of Probability. University of California Press, Berkeley-Los Angeles (1960)MATHGoogle Scholar
  6. 6.
    Bullen, P.S.: Handbook of means and their inequalities, mathematics and its applications, Volume 560, Kluwer Academic Publishers, Dordrecht-Boston-London (2003).
  7. 7.
    Chen, C.-P., Qi, F., Srivastava, H.M.: Some properties of functions related to the gamma and psi functions, Integral Transforms Spec. Funct. 21(2), 153–164 (2010).
  8. 8.
    Gamelin, T.W.: Complex Analysis, Undergraduate Texts in Mathematics, Springer, New York-Berlin-Heidelberg (2001).
  9. 9.
    Guo, B.-N., Qi, F.: A completely monotonic function involving the tri-gamma function and with degree one. Appl. Math. Comput. 218(19), 9890–9897 (2012).
  10. 10.
    Guo, B.-N., Qi, F.: A property of logarithmically absolutely monotonic functions and the logarithmically complete monotonicity of a power-exponential function. Politeh. Univ. Buchar. Sci. Bull. Ser. A Appl. Math. Phys. 72(2), 21–30 (2010)Google Scholar
  11. 11.
    Guo, B.-N., Qi, F.: A simple proof of logarithmic convexity of extended mean values. Numer. Algorithms 52(1), 89–92 (2009).
  12. 12.
    Guo, B.-N., Qi, F.: On the degree of the weighted geometric mean as a complete Bernstein function. Afr. Mat. 26(7), 1253–1262 (2015).
  13. 13.
    Guo, B.-N., Qi, F.: The function\((b^x-a^x)/x\): Logarithmic convexity and applications to extended mean values. Filomat 25(4), 63–73 (2011).
  14. 14.
    Henrici, P.: Applied and Computational Complex Analysis, Vol. 2, Wiley, NY (1977)Google Scholar
  15. 15.
    Kalugin, G.A., Jeffrey, D.J., Corless, R.M.: Bernstein, Pick, Poisson and related integral expressions for Lambert W, Integral Transforms Spec. Funct. 23(11), 817–829 (2012).
  16. 16.
    Kalugin, G.A., Jeffrey, D.J., Corless, R.M., Borwein, P.B.: Stieltjes and other integral representations for functions of Lambert W. Integral Transf. Spec. Funct. 23(8), 581–593 (2012).
  17. 17.
    Mitrinović, D.S., Pečarić, J.E., Fink, A.M.: Classical and New Inequalities in Analysis, Kluwer Academic Publishers (1993).
  18. 18.
    Qi, F.: A new lower bound in the second Kershaw’s double inequality. J. Comput. Appl. Math. 214(2), 610–616 (2008).
  19. 19.
    Qi, F.: Properties of modified Bessel functions and completely monotonic degrees of differences between exponential and trigamma functions. Math. Inequal. Appl. 18(2), 493–518 (2015).
  20. 20.
    Qi, F.: The extended mean values: Definition, properties, monotonicities, comparison, convexities, generalizations, and applications. Cubo Mat. Educ. 5(3), 63–90 (2003)MathSciNetMATHGoogle Scholar
  21. 21.
    Qi, F., Chen, C.-P.: A complete monotonicity property of the gamma function. J. Math. Anal. Appl. 296(2), 603–607 (2004).
  22. 22.
    Qi, F., Chen, S.-X.: Complete monotonicity of the logarithmic mean. Math. Inequal. Appl. 10(4), 799–804 (2007).
  23. 23.
    Qi, F., Guo, B.-N.: Complete monotonicities of functions involving the gamma and digamma functions. RGMIA Res. Rep. Coll. 7(1) Art. 8, 63–72 (2004).
  24. 24.
    Qi, F., Guo, S., Chen, S.-X.: A new upper bound in the second Kershaw’s double inequality and its generalizations. J. Comput. Appl. Math. 220(1–2), 111–118 (2008).
  25. 25.
    Qi, F., Luo, Q.-M., Guo, B.-N.: The function\((b^x-a^x)/x\): Ratio’s properties. In: Milovanović, G.V., Rassias, M. Th. (eds) Analytic Number Theory, Approximation Theory, and Special Functions, pp. 485–494. Springer (2014).
  26. 26.
    Qi, F., Wang, S.-H.: Complete monotonicity, completely monotonic degree, integral representations, and an inequality related to the exponential, trigamma, and modified Bessel functions. Glob. J. Math. Anal. 2(3), 91–97 (2014).
  27. 27.
    Qi, F., Wei, C.-F., Guo, B.-N.: Complete monotonicity of a function involving the ratio of gamma functions and applications. Banach J. Math. Anal. 6(1), 35–44 (2012).
  28. 28.
    Qi, F., Zhang, X.-J., Li, W.-H.: An elementary proof of the weighted geometric mean being a Bernstein function. Politeh. Univ. Buchar. Sci. Bull. Ser. A Appl. Math. Phys. 77(1), 35–38 (2015)Google Scholar
  29. 29.
    Qi, F., Zhang, X.-J., Li, W.-H.: An integral representation for the weighted geometric mean and its applications. Acta Math. Sin. (Engl. Ser.) 30(1), 61–68 (2014).
  30. 30.
    Qi, F., Zhang, X.-J., Li, W.-H.: Lévy-Khintchine representation of the geometric mean of many positive numbers and applications. Math. Inequal. Appl. 17(2), 719–729 (2014).
  31. 31.
    Qi, F., Zhang, X.-J., Li, W.-H.: Lévy-Khintchine representations of the weighted geometric mean and the logarithmic mean. Mediterr. J. Math. 11(2), 315–327 (2014).
  32. 32.
    Qi, F., Zhang, X.-J., Li, W.-H.: Some Bernstein functions and integral representations concerning harmonic and geometric means. arXiv preprint (2013).
  33. 33.
    Schilling, R.L., Song, R., Vondraček, Z.: Bernstein Functions, de Gruyter Studies in Mathematics 37. De Gruyter, Berlin (2010)Google Scholar
  34. 34.
    Stolarsky, K.B.: Generalizations of the logarithmic mean. Math. Mag. 48, 87–92 (1975)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Widder, D.V.: The Laplace Transform. Princeton University Press, Princeton (1946)MATHGoogle Scholar
  36. 36.
    Zhang, X.-J.: Integral Representations, Properties, and Applications of Three Classes of Functions. Thesis supervised by Professor Feng Qi and submitted for the Master Degree of Science in Mathematics at Tianjin Polytechnic University (2013) (Chinese) Google Scholar

Copyright information

© Sociedad Matemática Mexicana 2016

Authors and Affiliations

  1. 1.Institute of Mathematics and InformaticsHenan Polytechnic UniversityJiaozuoChina
  2. 2.College of MathematicsInner Mongolia University for NationalitiesTongliaoChina
  3. 3.Department of Mathematics, School of ScienceTianjin Polytechnic UniversityTianjinChina
  4. 4.The 59th Middle SchoolLuoyangChina

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