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Afrika Matematika

, Volume 26, Issue 7–8, pp 1253–1262 | Cite as

On the degree of the weighted geometric mean as a complete Bernstein function

  • Bai-Ni GuoEmail author
  • Feng Qi
Article

Abstract

In the paper, the authors verify that the geometric mean of many positive numbers is a complete Bernstein function of degree \(0\) and, consequently, a positive operator monotone function.

Keywords

Geometric mean Completely monotonic function Completely monotonic degree Complete Bernstein function Operator monotone function 

Mathematics Subject Classification

Primary 26E60 Secondary 26A48 30E20 44A10 44A20 

Notes

Acknowledgments

The authors heartily express their thanks to the anonymous referee for his/her very helpful corrections and very valuable suggestions to the original version of this paper.

References

  1. 1.
    Abramowitz, M., Stegun, I.A. (eds.): Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards, Applied Mathematics Series 55, 10th printing. Dover Publications, New York and Washington (1972)Google Scholar
  2. 2.
    Besenyei, Á.: On complete monotonicity of some functions related to means. Math. Inequal. Appl. 16(1), 233–239 (2013). doi: 10.7153/mia-16-17 zbMATHMathSciNetGoogle Scholar
  3. 3.
    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). doi: 10.1016/j.amc.2012.03.075 zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Qi, F.: Completely monotonic degree of a function involving the tri- and tetra-gamma functions. http://arxiv.org/abs/1301.0154v3. Accessed 7 June 2014
  5. 5.
    Qi, F.: Integral representations and properties of Stirling numbers of the first kind. J. Number Theory 133(7), 2307–2319 (2013). doi: 10.1016/j.jnt.2012.12.015 zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Qi, F.: Properties of modified Bessel functions and completely monotonic degrees of differences between exponential and trigamma functions. Math. Inequal. Appl. 18 (2014). http://arxiv.org/abs/1302.6731v2
  7. 7.
    Qi, F., Chen, S.-X.: Complete monotonicity of the logarithmic mean. Math. Inequal. Appl. 10(4), 799–804 (2007). doi: 10.7153/mia-10-73 zbMATHMathSciNetGoogle Scholar
  8. 8.
    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). doi: 10.1007/s10114-013-2547-8 zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    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). doi: 10.7153/mia-17-53 zbMATHMathSciNetGoogle Scholar
  10. 10.
    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). doi: 10.1007/s00009-013-0311-z zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Qi, F., Zhang, X.-J., Li, W.-H.: Some Bernstein functions and integral representations concerning harmonic and geometric means. http://arxiv.org/abs/1301.6430. Accessed 28 Jan 2014
  12. 12.
    Schilling, R.L., Song, R., Vondraček, Z.: Bernstein Functions-Theory and Applications, 2nd ed., de Gruyter Studies in Mathematics 37. Walter de Gruyter, Berlin, Germany (2012). doi: 10.1515/9783110269338
  13. 13.
    Widder, D.V.: The Laplace Transform, Princeton Mathematical Series 6. Princeton University Press, Princeton (1941)Google Scholar

Copyright information

© African Mathematical Union and Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.School of Mathematics and InformaticsHenan Polytechnic UniversityJiaozuoChina
  2. 2.Department of Mathematics, College of ScienceTianjin Polytechnic UniversityTianjinChina

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