Mixed arithmetic and geometric means

Billie Chandler Carlson, Robert K. Meany and Stuart Alan Nelson

Vol. 38 (1971), No. 2, 343–349

DOI: 10.2140/pjm.1971.38.343

Abstract

Consider all ordinary arithmetic and geometric means of n real nonnegative numbers taken k at a time. Let α_k be the geometric mean of all the arithmetic means and γ_k the arithmetic mean of all the geometric means. It is proved that α_k increases with k,γ_k decreases, and γ_h ≦ α_k if h + k > n. These results are generalized to mixed means of any real orders. Comparison of α_k and γ_k with elementary symmetric functions suggests a conjecture.

Mathematical Subject Classification

Primary: 26A87

Milestones

Received: 22 July 1970

Published: 1 August 1971