Definition

Let $x_1$ , $x_2,\ldots, x_n$ be real numbers, and $f$ a continuous and strictly increasing or decreasing function on the real numbers. If each number $x_i$ is assigned a weight $p_i$ , with $\sum_{i=1}^n p_i= 1$ , for $i=1,\ldots,n$ , then the generalized mean is defined as $$ f^{-1}\Big( \sum_{i=1}^n p_i f(x_i) \Big). $$

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. $f(x)=2^{(1-\alpha)x}$ , $\alpha\neq 1$ , $x_i=-\log_2 p_i$ : Rényi's $\alpha$ -entropy