Power Mean Inequality
The Power Mean Inequality is a generalized form of the multi-variable Arithmetic Mean-Geometric Mean Inequality.
Inequality
For real numbers 
and positive real numbers 
, 
implies the 
th power mean is greater than or equal to the 
th.
Algebraically, implies that
![\[\sqrt[k_1]{\frac{a_{1}^{k_1}+a_{2}^{k_1}+\cdots +a_{n}^{k_1}}{n}}\ge \sqrt[k_2]{\frac{a_{1}^{k_2}+a_{2}^{k_2}+\cdots +a_{n}^{k_2}}{n}}\]](http://latex.artofproblemsolving.com/5/1/c/51ca907a73ebeaf6f520c77f7281c0c0b5f662ec.png)
which can be written more concisely as
![\[\sqrt[k_1]{\frac{\sum\limits_{i=1}^n a_{i}^{k_1}}{n}}\ge \sqrt[k_2]{\frac{\sum\limits_{i=1}^n a_{i}^{k_2}}{n}}\]](http://latex.artofproblemsolving.com/6/f/b/6fb4cbf03b94f3e494719f49c0cf2504c93add08.png)
The Power Mean Inequality follows from the fact that 
(where 
is the 
th power mean) together with Jensen's Inequality.
Proof
![\[\left(\sum_{i=1}^n \frac{a_{i}^{k_1}}{n}\right)^{\frac{1}{k_1}}\ge \left(\sum_{i=1}^n \frac{a_{i}^{k_2}}{n}\right)^{\frac{1}{k_2}}\]](http://latex.artofproblemsolving.com/0/5/9/059dcc989c9c5b8bdd5c2388a1aec7ba518b8948.png)
Raising both sides to the th power, we get
![\[\sum_{i=1}^n \frac{a_{i}^{k_1}}{n}\ge \left(\sum_{i=1}^n \frac{a_{i}^{k_2}}{n}\right)^{\frac{k_1}{k_2}}\]](http://latex.artofproblemsolving.com/1/2/2/122c41965c1491041edd9c9426eba00acf1922bd.png)
We can see that the function 
is convex for all 
, and so we can apply Jensen's Inequality. Therefore,
![\[\left(\sum_{i=1}^n \frac{a_{i}^{k_2}}{n}\right)^{\frac{k_1}{k_2}}= f\left(\sum_{i=1}^n \frac{a_{i}^{k_2}}{n}\right)\le \sum_{i=1}^n f\left(\frac{a_i}{n}\right)= \sum_{i=1}^n \frac{a_{i}^{k_1}}{n}\]](http://latex.artofproblemsolving.com/8/6/a/86a1eb018c8dabb433cc9229b690d9dd960a609e.png)
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