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

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?

• Take a look at en.wikipedia.org/wiki/Harmonic_mean and en.wikipedia.org/wiki/Root_mean_square and see if you can generalize the concepts to functions. Mar 30, 2022 at 23:58
• For example the harmonic average of a function is presumably the reciprocal of the mean of the reciprocal of the function Mar 30, 2022 at 23:59

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).
• "Quadratic average" is $\phi(x) = x^2$, also called "root mean square": $$\left(\int |f|^2 d\mu\right)^{1/2}$$ Stated with absolute value in it so that the definition can be used for cases where $f$ has complex values. Mar 31, 2022 at 1:17