Difference between revisions of "Power Mean Inequality"
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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 Inequalitythe 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, so 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. | 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 Inequalitythe 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, so 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: | + | [[Category:Algebra]] |
− | [[Category: | + | [[Category:Inequalities]] |
Revision as of 16:54, 29 December 2021
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 Inequalitythe inequality is flipped. Thus, the inequality sign in
is flipped, but as
,
is a decreasing function, so the inequality sign is flipped again after applying
, resulting in
as desired.