Mixed arithmetic and geometric means

Abstract

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