Convexity properties of generalizations of the arithmetic-geometric mean

Roger D. Nussbaum, Joel E. Cohen

Research output: Contribution to journalArticlepeer-review

1 Scopus citations


In the eighteenth century, Landen, Lagrange and Gauss studied a function of two positive real numbers that has become known as the arithmetic-geometric mean (AGM). In the nineteenth century, Borchardt generalized the AGM to a function of any 2n(n = 1,2,3,…) positive real numbers. In this paper, we generalize the AGM to a function of any even number of positive real numbers. If M(a, b) is the original AGM then M(a, b) is concave in the pair (a, b) of positive numbers and log M(eα, eβ) is convex in the pair (α, β) of real numbers; all our generalizations of the AGM behave similarly. We generalize this analysis extensively.

Original languageEnglish (US)
Pages (from-to)33-44
Number of pages12
JournalNumerical Functional Analysis and Optimization
Issue number1-2
StatePublished - Jan 1 1990

All Science Journal Classification (ASJC) codes

  • Analysis
  • Signal Processing
  • Computer Science Applications
  • Control and Optimization

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