Vol. 38, No. 2, 1971

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Mixed arithmetic and geometric means

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

Vol. 38 (1971), No. 2, 343–349

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

Mathematical Subject Classification
Primary: 26A87
Received: 22 July 1970
Published: 1 August 1971
Billie Chandler Carlson
Robert K. Meany
Stuart Alan Nelson