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What are some uses of the generalized f-mean outside of the geometric mean and the power means?

Also, is there a known way to compare two functions and find out which will yield a larger f-mean (ex: we know that the function $f(x)=x^2$ will yield a greater f-mean than $f(x)=x$)?

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Regarding #2: one way to state Jensen's inequality is that the $f$-mean is greater than or equal to the $g$-mean if $g$ is invertible and $f(g^{-1})$ is convex. – Qiaochu Yuan Jul 6 '11 at 17:35

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