AoPSWiki

Want to learn how to tackle those tough AMC/AIME/Olympiad counting and probability problems? Check out Art of Problem Solving's Intermediate Counting & Probability by David Patrick.

Personal tools

Search

Toolbox

Power Mean Inequality

From AoPSWiki

Revision as of 02:07, 21 December 2007 by Minsoens (Talk | contribs)

(diff) ← Older revision | Current revision (diff) | Newer revision → (diff)

The Power Mean Inequality is a generalized form of the multi-variable Arithmetic Mean-Geometric Mean Inequality.

Inequality

For real numbers $k_1,k_2$ and positive real numbers $a_1, a_2, \ldots, a_n$ , $k_1\ge k_2$ implies the $k_1$ th power mean is greater than or equal to the $k_2$ th.

Algebraically, $k_1\ge k_2$ implies that $\left( \frac{\sum\limits_{i=1}^n a_{i}^{k_1}}{n} \right) ^ {\frac{1}{k_1}}\ge \left( \frac{\sum\limits_{i=1}^n a_{i}^{k_2}}{n...$

The Power Mean Inequality follows from the fact that $\frac{\partial M(t)}{\partial t}\geq 0$ (where $M(x)$ is the $t$ th power mean) together with Jensen's Inequality.

This article is a stub. Help us out by expanding it.

Retrieved from "http://www.artofproblemsolving.com/Wiki/index.php/Power_Mean_Inequality"

Categories: Stubs | Inequality | Theorems

Our Precalculus course starts on Dec. 4. Master trig, complex numbers, and vectors and matrices in 2 and 3 dimensions. Click here to enroll today!

© Copyright 2008 AoPS Incorporated. All Rights Reserved. • Foundation • Privacy • Contact Us