Difference between revisions of "Power Mean Inequality"
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− | + | The '''Power Mean Inequality''' is a generalized form of the multi-variable [[Arithmetic Mean-Geometric Mean]] Inequality. | |
− | + | == Inequality == | |
+ | For <math>n</math> positive real numbers <math>a_i</math> and <math>n</math> positive real weights <math>w_i</math> with sum <math>\sum_{i=1}^n w_i=1</math>, the power mean with exponent <math>t</math>, where <math>t\in\mathbb{R}</math>, is defined by | ||
+ | <cmath> | ||
+ | M(t)= | ||
+ | \begin{cases} | ||
+ | \prod_{i=1}^n a_i^{w_i} &\text{if } t=0 \\ | ||
+ | \left(\sum_{i=1}^n w_ia_i^t \right)^{\frac{1}{t}} &\text{otherwise} | ||
+ | \end{cases}. | ||
+ | </cmath> | ||
− | + | (<math>M(0)</math> is the [[AM-GM_Inequality#Weighted_AM-GM_Inequality|weighted geometric mean]].) | |
− | + | 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 | |
− | M( | + | <cmath> |
− | / | + | \left( \frac{1}{n} \sum_{i=1}^n a_{i}^{k_1} \right)^{\frac{1}{k_1}} \ge \left( \frac{1}{n} \sum_{i=1}^n a_{i}^{k_2} \right)^{\frac{1}{k_2}}. |
+ | </cmath> | ||
− | ( | + | Considering the limiting behavior, we also have <math>\lim_{t\rightarrow +\infty} M(t)=\max\{a_i\}</math>, <math>\lim_{t\rightarrow -\infty} M(t)=\min\{a_i\}</math> and <math>\lim_{t\rightarrow 0} M(t)= M(0)</math>. |
− | + | The Power Mean Inequality follows from [[Jensen's Inequality]]. | |
− | + | == Proof == | |
+ | We prove by cases: | ||
+ | |||
+ | 1. <math>M(t)\ge M(0)\ge M(-t)</math> for <math>t>0</math> | ||
+ | |||
+ | 2. <math>M(k_1)\ge M(k_2)</math> for <math>k_1 \ge k_2</math> with <math>k_1k_2>0</math> | ||
+ | |||
+ | Case 1: | ||
+ | |||
+ | Note that | ||
+ | <cmath> | ||
+ | \begin{align*} | ||
+ | && \left(\sum_{i=1}^n w_ia_i^{t} \right)^{\frac{1}{t}} &\ge \prod_{i=1}^n a_i^{w_i} \\ | ||
+ | \Longleftarrow && \frac{1}{t} \ln\left( \sum_{i=1}^n w_i a_i^{t} \right) &\ge \sum_{i=1}^n w_i \ln{a_i} && \text{as } e^x \text{ is increasing} \\ | ||
+ | \Longleftarrow && \ln\left( \sum_{i=1}^n w_i a_i^{t} \right) &\ge \sum_{i=1}^n w_i \ln{a_i^t} && \text{as } t>0 | ||
+ | \end{align*} | ||
+ | </cmath> | ||
+ | As <math>\ln(x)</math> is concave, by [[Jensen's Inequality]], the last inequality is true, proving <math>M(t)\ge M(0)</math>. By replacing <math>t</math> by <math>-t</math>, the last inequality implies <math>M(0)\ge M(-t)</math> as the inequality signs are flipped after multiplication by <math>-\frac{1}{t}</math>. | ||
+ | |||
+ | |||
+ | Case 2: | ||
+ | |||
+ | For <math>k_1\ge k_2>0</math>, | ||
+ | <cmath> | ||
+ | \begin{align} | ||
+ | && \left(\sum_{i=1}^n w_ia_i^{k_1} \right)^{\frac{1}{k_1}} &\ge \left(\sum_{i=1}^n w_ia_i^{k_2} \right)^{\frac{1}{k_2}} \nonumber \\ | ||
+ | \Longleftarrow && \left(\sum_{i=1}^n w_ia_i^{k_1} \right)^{\frac{k_2}{k_1}} &\ge \sum_{i=1}^n w_ia_i^{k_2} \label{eq} | ||
+ | \end{align} | ||
+ | </cmath> | ||
+ | As the function <math>f(x)=x^{\frac{k_2}{k_1}}</math> is concave for all <math>x > 0</math>, by [[Jensen's Inequality]], | ||
+ | <cmath> | ||
+ | \left(\sum_{i=1}^n w_i a_i^{k_1} \right)^{\frac{k_2}{k_1}} | ||
+ | = f\left(\sum_{i=1}^n w_i a_i^{k_1} \right) | ||
+ | \geq \sum_{i=1}^n w_i f\left(a_i^{k_1}\right) | ||
+ | =\sum_{i=1}^n w_i a_{i}^{k_2} | ||
+ | </cmath> | ||
+ | For <math>0>k_1\ge k_2</math>, <math>f(x)</math> becomes convex as <math>|k_1|\le |k_2|</math>, so the inequality sign when applying Jensen's Inequality is flipped. Thus, the inequality sign in <math>(1)</math> is flipped, but as <math>k_2<0</math>, <math>x^\frac{1}{k_2}</math> is a decreasing function, the inequality sign is flipped again after applying <math>x^{\frac{1}{k_2}}</math>, resulting in <math>M(k_1)\ge M(k_2)</math> as desired. | ||
+ | |||
+ | [[Category:Algebra]] | ||
+ | [[Category:Inequalities]] |
Latest revision as of 12:57, 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.)
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.