TY - JOUR

T1 - Convexity properties of generalizations of the arithmetic-geometric mean

AU - Nussbaum, Roger D.

AU - Cohen, Joel E.

N1 - Funding Information:
ACKNOWLEDGMENTS R.D.N. was partially supported by N.S.F. grant DMS 85-03316. J.E.C. was partially supported by N.S.F. grant BSR 87-05047. J.E.C. is grateful for the hospitality of Mr. and Mrs. William T. Golden during this work.

PY - 1990/1/1

Y1 - 1990/1/1

N2 - 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.

AB - 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.

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U2 - 10.1080/01630569008816359

DO - 10.1080/01630569008816359

M3 - Article

AN - SCOPUS:84973001514

VL - 11

SP - 33

EP - 44

JO - Numerical Functional Analysis and Optimization

JF - Numerical Functional Analysis and Optimization

SN - 0163-0563

IS - 1-2

ER -