Advertisement

The Shape of the Borwein–Affleck–Girgensohn Function Generated by Completely Monotone and Bernstein Functions

  • Hristo S. Sendov
  • Ričardas ZitikisEmail author
Article

Abstract

Fifteen years ago, J. Borwein, I. Affleck, and R. Girgensohn posed a problem concerning the shape (convexity, log-convexity, reciprocal concavity) of a certain function of several arguments that had manifested in a number of contexts concerned with optimization problems. In this paper we further explore the shape of the Borwein–Affleck–Girgensohn function as well as of its extensions generated by completely monotone and Bernstein functions.

Keywords

Borwein–Affleck–Girgensohn function Completely monotone function Bernstein function Harmonically convex function Laplace transform Quasi-arithmetic mean Kolmogorov–Nagumo mean 

Notes

Acknowledgements

We are grateful to Associate Editor Lionel Thibault and an anonymous referee for constructive criticism and suggestions. We are also indebted to Shen Shan of the University of Western Ontario for pointing out several omissions in an earlier version of the manuscript and for suggesting further improvements. This research was partially supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada.

References

  1. 1.
    Borwein, J., Affleck, I., Girgensohn, R.: Convex? Problem 99-002. Society for Industrial and Applied Mathematics, Philadelphia, Pennsylvania (2000). Online http://www.siam.org/journals/problems/downloadfiles/99-002.pdf
  2. 2.
    Affleck, I.: Convex! Solution of part (b) of Problem 99-002. Society for Industrial and Applied Mathematics, Philadelphia, Pennsylvania (2000). Online http://www.siam.org/journals/problems/downloadfiles/99-002s.pdf
  3. 3.
    Borwein, J., Hijab, O.: Convex! II Solution of Problem 99-002. Society for Industrial and Applied Mathematics, Philadelphia, Pennsylvania (2000). Online http://www.siam.org/journals/problems/downloadfiles/99-5sii.pdf
  4. 4.
    Affleck, I.A.: Minimizing expected broadcast time in unreliable networks. Ph.D. Thesis, Simon Fraser University, Canada (2000) Google Scholar
  5. 5.
    Nath, H.B.: Waiting time in the coupon-collector’s problem. Aust. J. Stat. 15, 132–135 (1973) CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Kuczma, M.: An Introduction to the Theory of Functional Equations and Inequalities. Cauchy’s Equation and Jensen’s Inequality, 2nd edn. Birkhäuser, Basel (2009) CrossRefzbMATHGoogle Scholar
  7. 7.
    Schilling, R.L., Song, R., Vondraček, Z.: Bernstein Functions. Theory and Applications. De Gruyter, Berlin (2010) zbMATHGoogle Scholar
  8. 8.
    Bernstein, S.: Sur la définition et les propriétés des fonctions analytiques d’une variable réelle. Math. Ann. 75, 449–468 (1914) CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Faraut, J.: Puissances fractionnaires d’un noyau de Hunt. Sémin. Brelot-Choquet-Deny. Théor. Potentiel 7, 1–12 (1965/1966) Google Scholar
  10. 10.
    Merkle, M.: Reciprocally convex functions. J. Math. Anal. Appl. 293, 210–218 (2004) CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Aczél, J.: Lectures on Functional Equations and Their Applications. Dover, Mineola (2006) Google Scholar
  12. 12.
    Kolmogorov, A.N.: Sur la notion de la moyenne. Atti Accad. Naz. Lincei, Rend. 12, 388–391 (1930) Google Scholar
  13. 13.
    Nagumo, M.: Über eine klasse von mittelwerten. Jpn. J. Math. 7, 71–79 (1930) zbMATHGoogle Scholar
  14. 14.
    de Finetti, B.: Sul concetto di media. G. Ist. Ital. Attuari 2, 369–396 (1931) Google Scholar
  15. 15.
    Jessen, B.: Über die verallgemeinerung des arthmetischen mittels. Acta Sci. Math. 5, 108–116 (1931) Google Scholar
  16. 16.
    Kitagawa, T.: On some class of weighted means. Proc. Phys. Math. Soc. Jpn. 16, 117–126 (1934) Google Scholar
  17. 17.
    Porcu, E., Mateu, J., Christakos, G.: Quasi-arithmetic means of covariance functions with potential applications to space–time data. J. Multivar. Anal. 100, 1830–1844 (2009) CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Denuit, M., Dhaene, J., Goovaerts, M., Kaas, R.: Actuarial Theory for Dependent Risks: Measures, Orders and Models. Wiley, Chichester (2005) CrossRefGoogle Scholar
  19. 19.
    Schaefer, M.: Note on the k-dimensional Jensen inequality. Ann. Probab. 4, 502–504 (1976) CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Doumas, A.V., Papanicolaou, V.G.: The coupon collector’s problem revisited: asymptotics of the variance. Adv. Appl. Probab. 44, 166–195 (2012) CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Borwein, J.M., Zhu, Q.J.: Variational methods in the presence of symmetry. Technical report, Physical Sciences, University of Newcastle, Callaghan NSW, Australia (2012). Online http://carma.newcastle.edu.au/jon/symmetry.pdf
  22. 22.
    Sendov, H.S., Shan, S.: Personal communication (2013) Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of Statistical and Actuarial SciencesUniversity of Western OntarioLondonCanada

Personalised recommendations