generalized mean

Definition

Let $x_1$, $x_2,\mathrm{\dots },x_n$ be real numbers, and $f$ a continuousand strictly increasing or decreasing function on the realnumbers. If each number $x_i$ is assigned a weight $p_i$, with${\sum }_{i=1}^n{p}_i=1$, for $i=1,\mathrm{\dots },n$, then the generalized meanis defined as

$$f^{-1}\left(\sum _{i=1}^n{p}_{i}f(x_i)\right).$$

Special cases

1. $f(x)=x$, $p_i=1/n$ for all $i$: arithmetic mean

2. $f(x)=x$: weighted mean

3. $f(x)=\log (x)$, $p_i=1/n$ for all $i$: geometric mean

4. $f(x)=1/x$ and $p_i=1/n$ for all $i$: harmonic mean

5. $f(x)=x^2$ and $p_i=1/n$ for all $i$: root-mean-square

6. $f(x)=x^d$ and $p_i=1/n$ for all $i$: power mean

7. $f(x)=x^d$: weighted power mean

8. $f(x)=2^{(1-\alpha )x}$, $\alpha \ne 1$, $x_i=-{\log }_2{p}_i$:Rényi’s $\alpha$-entropy