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If $2$ finite sets of positive integers have different cardinality but the same arithmetic mean, does the set with the greater number of elements always have a lower geometric mean?

5xum
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Simon
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1 Answers1

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The simplest counter example for the case of positive integers is probably $$ A=\{1,3\} \qquad B=\{1,2,3\}, $$ both with arithmetic mean $2$. The geometric mean of $A$ is $\sqrt{3}\approx 1.73$, while that of $B$ is $6^{1/3}\approx 1.81$. Hence the geometric mean of the larger set ($B$) is larger than that of the smaller set ($A$).

Hrodelbert
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