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

  • Hristo S. Sendov
  • Ričardas ZitikisEmail author


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.


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



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.


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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

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

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