Difference between revisions of "Power Mean Inequality"
m (→Inequality) |
m (→Inequality) |
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M(t)= | M(t)= | ||
\begin{cases} | \begin{cases} | ||
− | + | \prod_{i=1}^n a_i^{w_i} &\text{if } t=0 \ | |
− | \left( | + | \left(\sum_{i=1}^n w_ia_i^t \right)^{\frac{1}{t}} &\text{otherwise} |
\end{cases}. | \end{cases}. | ||
</cmath> | </cmath> | ||
+ | |||
+ | (<math>M(0)</math> is the [[weighted geometric mean][AM-GM_Inequality#Weighted_AM-GM_Inequality]].) | ||
The Power Mean Inequality states that for all real numbers <math>k_1</math> and <math>k_2</math>, <math>M(k_1)\ge M(k_2)</math> if <math>k_1>k_2</math>. In particular, for nonzero <math>k_1</math> and <math>k_2</math>, and equal weights (i.e. <math>w_i=1/n</math>), if <math>k_1>k_2</math>, then | The Power Mean Inequality states that for all real numbers <math>k_1</math> and <math>k_2</math>, <math>M(k_1)\ge M(k_2)</math> if <math>k_1>k_2</math>. In particular, for nonzero <math>k_1</math> and <math>k_2</math>, and equal weights (i.e. <math>w_i=1/n</math>), if <math>k_1>k_2</math>, then |
Revision as of 12:53, 23 August 2024
The Power Mean Inequality is a generalized form of the multi-variable Arithmetic Mean-Geometric Mean Inequality.
Inequality
For positive real numbers and positive real weights with sum , the power mean with exponent , where , is defined by
( is the [[weighted geometric mean][AM-GM_Inequality#Weighted_AM-GM_Inequality]].)
The Power Mean Inequality states that for all real numbers and , if . In particular, for nonzero and , and equal weights (i.e. ), if , then
Considering the limiting behavior, we also have , and .
The Power Mean Inequality follows from Jensen's Inequality.
Proof
We prove by cases:
1. for
2. for with
Case 1:
Note that As is concave, by Jensen's Inequality, the last inequality is true, proving . By replacing by , the last inequality implies as the inequality signs are flipped after multiplication by .
Case 2:
For , As the function is concave for all , by Jensen's Inequality, For , becomes convex as , so the inequality sign when applying Jensen's Inequality is flipped. Thus, the inequality sign in is flipped, but as , is a decreasing function, the inequality sign is flipped again after applying , resulting in as desired.