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. | The '''Power Mean Inequality''' is a generalized form of the multi-variable [[Arithmetic Mean-Geometric Mean]] Inequality. | ||
− | == | + | == Description == |
− | 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>, | + | 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 function <math>M:\mathbb{R}\rightarrow\mathbb{R}</math> is defined by |
<cmath> | <cmath> | ||
M(t)= | M(t)= | ||
Line 11: | Line 11: | ||
</cmath> | </cmath> | ||
− | 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= | + | 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 |
<cmath> | <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}} | + | \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> | </cmath> | ||
+ | Considering the limiting behavior, we also have <math>\lim_{t\rightarrow +\infty} M(t)=\max\{x_i\}</math>, <math>\lim_{t\rightarrow -\infty} M(t)=\min\{x_i\}</math> and <math>\lim_{t\rightarrow 0} M(t)= M(0)</math>. | ||
The Power Mean Inequality follows from the fact that <math>\frac{\partial M(t)}{\partial t}\geq 0</math> together with [[Jensen's Inequality]]. | The Power Mean Inequality follows from the fact that <math>\frac{\partial M(t)}{\partial t}\geq 0</math> together with [[Jensen's Inequality]]. |
Revision as of 11:51, 30 July 2020
The Power Mean Inequality is a generalized form of the multi-variable Arithmetic Mean-Geometric Mean Inequality.
Description
For positive real numbers
and
positive real weights
with sum
, the power mean function
is defined by
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 the fact that together with 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 flip 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 Inequalitythe inequality sign is flipped. Thus, the inequality sign in
is also flipped, but as
,
is a decreasing function, so the inequality sign is flipped again, resulting in
as desired.