Mahler’s inequality

I ran across a reference to Mahler the other day, not the composer Gustav Mahler but the mathematician Kurt Mahler, and looked into his work a little. A number of things have been named after Kurt Mahler, including Mahler’s inequality.

Mahler’s inequality says the geometric mean of a sum bounds the sum of the geometric means. In detail, the geometric mean of a list of n non-negative real numbers is the nth root of their product. If x and y are two vectors of length n containing non-negative components, Mahler’s inequality says

G(x + y) ≥ G(x) + G(y)

where G is the geometric mean. The left side is strictly larger than the right unless x and y are proportional, or x and y both have a zero component in the same position.

I’m curious why this inequality is named after Mahler. The classic book Inequalities by Hardy, Littlewood, and Polya list the inequality but call it Hölder’s inequality. In a footnote they note that the inequality above appears in a paper by Minkowski in 1896 (seven years before Kurt Mahler was born). Presumably the authors file the inequality under Hölder’s name because it follows easily from Hölder’s inequality.

I imagine Mahler made good use of his eponymous inequality, i.e. that the inequality became associated with him because he applied it well rather than because he discovered it.

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