GeneralizedMean 2004-08-06 15:41:25 2006-10-24 22:29:46 Definition % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here {\bf 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 \emph{generalized mean} is defined as \[ f^{-1}\Big( \sum_{i=1}^n p_i f(x_i) \Big). \] {\bf Special cases} \begin{enumerate} \item $f(x)=x$, $p_i=1/n$ for all $i$: arithmetic mean \item $f(x)=x$: weighted mean \item $f(x)=\log(x)$, $p_i=1/n$ for all $i$: geometric mean \item $f(x)=1/x$ and $p_i=1/n$ for all $i$: harmonic mean \item $f(x)=x^2$ and $p_i=1/n$ for all $i$: root-mean-square \item $f(x)=x^d$ and $p_i=1/n$ for all $i$: power mean \item $f(x)=x^d$: weighted power mean \item $f(x)=2^{(1-\alpha)x}$, $\alpha\neq 1$, $x_i=-\log_2 p_i$: R\'enyi's $\alpha$-entropy \end{enumerate}