Difference between revisions of "Power Mean 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 \\ | + | \displaystyle\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} | + | \left(\displaystyle\sum_{i=1}^n w_ia_i^t \right)^{\frac{1}{t}} &\text{otherwise} |
\end{cases}. | \end{cases}. | ||
</cmath> | </cmath> | ||
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=\sum_{i=1}^n w_i a_{i}^{k_2} | =\sum_{i=1}^n w_i a_{i}^{k_2} | ||
</cmath> | </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 | + | 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:Algebra]] | ||
[[Category:Inequalities]] | [[Category:Inequalities]] |
Latest revision as of 23:35, 24 June 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
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.