What are the geometric, harmonic, and quadratic averages of a function?

Question

In Mean of a function, they describe the arithmetic mean of a function and at the bottom of the article they said:

There is also a harmonic average of functions and a quadratic average (or root mean square) of functions.

My question is what is the form of these averages?

Answer 1

Typically, if $(X,\mathscr{B},\mu)$ is a probability space, and $\phi$ is nice bijective function from $(a, b)$ to another interval $(c, d)$, then the $\phi$-mean, defined for measurable functions $f: X\rightarrow (c,d)$ for which $\phi\circ f$ is integrable is defined as $$\phi^{-1}\Big(\int\phi\circ f\,d\mu\Big)$$ The most common examples are

1. $\phi(x)=x$ (arirthmetic mean)
2. $\phi(x)=\log(x)$, $x>0$, (geometric mean)
3. $\phi(x)=\frac{1}{x}$, $x>0$ (harmonic mean)
4. and $\phi(x)=x^p$, $x\geq 0$ (the $p$-mean).

In many applications of the means described in 1-4, one considers $|f|$, which is nonnegative, and defined $$|f|_\phi:=\phi^{-1}\Big(\phi(|f|)\,d\mu\Big)$$ When $\phi(x)=x^p$, the space $L_p(\mu):=\{f:|f|_\phi<\infty\}$ is a (complex) linear space and $|\;|_p$ defines a (pseudo)-norm that is complete. As Prof. GEdgar mentioned in his comment below, $\phi(x)=x^2$ corresponds to the quadratic mean (or quadratic norm).