Difference between revisions of "Homogenization"
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If <math>a,b,c>0</math> and <math>a+b+c=1</math>, prove that <math>a^2+b^2+c^2+1\ge 4(ab+bc+ca)</math>. | If <math>a,b,c>0</math> and <math>a+b+c=1</math>, prove that <math>a^2+b^2+c^2+1\ge 4(ab+bc+ca)</math>. | ||
==Solution== | ==Solution== | ||
| − | So all the terms except for the <math>1</math> are of the second degree. We substituting <math>a+b+c</math> for | + | So all the terms except for the <math>1</math> are of the second degree. We substituting <math>a+b+c</math> for <math>1</math> |
. The inequality still gives a non-homogeneous inequality. So instead we square the condition to make | . The inequality still gives a non-homogeneous inequality. So instead we square the condition to make | ||
Revision as of 20:54, 22 January 2014
Homogenizing is a useful technique to solve certain multivariate inequalities. Given an inequality of the form 
, where 
is a homogenous polynomial (that is, the degree of any term in the polynomial is the same), then we can arbitrarily impose a restraint of one order.
Example
If 
and 
, prove that 
.
Solution
So all the terms except for theare of the second degree. We substituting
for
for . The inequality still gives a non-homogeneous inequality. So instead we square the condition to make
it second degree and get 
. Now plugging this for 
in the
inequality and simplifying gives 
, which is well-known by
Cauchy-Schwarz Inequality.
We can use homogenization to help us solve these types of problems, especially inequalities however
it's use is not limited. After making something homogenous we can often apply well known inequalities
to solve problems.
This article is a stub. Help us out by expanding it.
