Difference between revisions of "Power Mean Inequality"
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Algebraically, <math>k_1\ge k_2</math> implies that | Algebraically, <math>k_1\ge k_2</math> implies that | ||
<cmath> | <cmath> | ||
| − | \ | + | \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} \right)} |
| + | </cmath> | ||
| + | |||
| + | which can be written more concisely as | ||
| + | <cmath> | ||
| + | \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} \right)} | ||
</cmath> | </cmath> | ||
Revision as of 21:12, 11 January 2014
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
\[\sqrt[k_1]{\frac{a_{1}^{k_1}+a_{2}^{k_1}+\cdots +a_{n}^{k_1}}{n}}\ge \sqrt[k_2]{\frac{a_{1}^{k_2}+a_{2}^{k_2}+\cdots +a_{n}^{k_2}}{n} \right)}\] (Error compiling LaTeX. Unknown error_msg)
which can be written more concisely as
\[\sqrt[k_1]{\frac{\sum\limits_{i=1}^n a_{i}^{k_1}}{n}}\ge \sqrt[k_2]{\frac{\sum\limits_{i=1}^n a_{i}^{k_2}}{n} \right)}\] (Error compiling LaTeX. Unknown error_msg)
The Power Mean Inequality follows from the fact that 
(where 
is the 
th power mean) together with Jensen's Inequality.
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