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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

I've been pondering over the f-mean, and the best interpretation I can think of is that $f(x)$ is representative of a transform between two representations of the same set of data. In this sense, the generalized f-mean will be applicable whenever:

  1. The set of data is $x$ is meaningful, as well as the related set of data $y=f(x)$.
  2. The mean of $y$ is important, but is more easily used or interpreted when converted back to an $x$ form through $f^{-1}(\bar y)$

This is a narrow set of constraints, and makes examples outside of the more common geometric mean and power means difficult to come up with. If the data is much more meaningful or useful in either its $x$ or $y$ form, then operations might be done on the data in that form without ever converting it to the other form. In a similar vein, the idea that $f^{-1}(\bar y)$ is easier to interpret than $\bar y$ can be a reflection of what people find is easier to interpret, rather than it being more mathematically meaningful.

An example of how the power mean fits these requirements is in the context of circuits. Electrical signals are measured in both voltage amplitude $V$, as well as its square $V^2$, as functions of time. Amplitude $V$ is meaningful because circuits are often linear with respect to $V$, while $V^2$ is meaningful through its relation to energy. A value called the root mean square (RMS) is the same as the power mean of $V$ for an exponent of 2 (aka quadratic mean). This value $V_{RMS}$ has the properties that $V_{RMS}^2$ is a measure of the average energy transferred, and $V_{RMS}$ follows the same linear circuit laws as $V$.

However, this set of conditions where this f-mean is useful is specific enough that closely related fields won't ever use RMS values. Optical signals for example are more easily measured as $E^2$, and optical propagation is often as easily modeled in $E^2$ as $E$, so there's never a reason to calculate $E_{RMS}$. Similarly, when dealing with nonlinear circuits, $V_{RMS}$ no longer follows the same simple relations as $V$. In this case, energy transfer is calculated directly from $V$ without ever considering $V_{RMS}$.

Based on how narrowly useful the f-mean is, I can't think of any meaningful examples that aren't power means or geometric means.

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