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Difference between revisions of "Homogenization"
(→‎Solution)
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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 the <math>1</math> in the inequality gives a non-homogeneous inequality. So instead we square the condition to make it second degree and get <math>a^2+b^2+c^2+2(ab+bc+ca)=1</math>. Now plugging this for <math>1</math> in the inequality and simplifying gives <math>a^2+b^2+c^2\ge ab+bc+ca</math>, which is well-known by Cauchy-Schwarz Inequality.
+
  So all the terms except for the <math>1</math> are of the second degree. We substituting <math>a+b+c</math> for the <math>1</math> in the  
  
−
  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.
+
inequality gives a non-homogeneous inequality. So instead we square the condition to make it second degree and
 +
 
 +
get <math>a^2+b^2+c^2+2(ab+bc+ca)=1</math>. Now plugging this for <math>1</math> in the inequality and simplifying gives
 +
 
 +
<math>a^2+b^2+c^2\ge ab+bc+ca</math>, 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.
 
{{stub}}
 
{{stub}}
  
 
[[Category:Inequality]]
 
[[Category:Inequality]]

Revision as of 20:52, 22 January 2014

Homogenizing is a useful technique to solve certain multivariate inequalities. Given an inequality of the form $P(a_1,a_2, \ldots, a_n) \ge 0$, where $P$ 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 $a,b,c>0$ and $a+b+c=1$, prove that $a^2+b^2+c^2+1\ge 4(ab+bc+ca)$.

Solution

So all the terms except for the $1$ are of the second degree. We substituting $a+b+c$ for the $1$ in the

inequality gives a non-homogeneous inequality. So instead we square the condition to make it second degree and 

get $a^2+b^2+c^2+2(ab+bc+ca)=1$. Now plugging this for $1$ in the inequality and simplifying gives

$a^2+b^2+c^2\ge ab+bc+ca$, 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.

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