Difference between revisions of "User:Temperal/Inequalities"
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'''Problem (me):''' Prove for <math>x,y,z\in\mathbb{R}^ + ,x + y + z = \frac {1}{2}</math> that | '''Problem (me):''' Prove for <math>x,y,z\in\mathbb{R}^ + ,x + y + z = \frac {1}{2}</math> that | ||
| − | + | <cmath> | |
\left(\sum_{sym}\frac {x}{y}\right)\left(\sqrt {2x^2 + 2y^2 + 2z^2}\right)\ge 1 | \left(\sum_{sym}\frac {x}{y}\right)\left(\sqrt {2x^2 + 2y^2 + 2z^2}\right)\ge 1 | ||
| − | + | </cmath> | |
Revision as of 13:48, 25 December 2007
Problem (me): Prove for 
that
![\[\left(\sum_{sym}\frac {x}{y}\right)\left(\sqrt {2x^2 + 2y^2 + 2z^2}\right)\ge 1\]](http://latex.artofproblemsolving.com/3/b/a/3ba61b4841f12f8037f32be749db2374e9e638c6.png)
Solution (Altheman): By AM-GM, 
, and my RMS-AM 
, thus the inequality is true.
Solution (me): By Jensen's Inequality, 
and by the Cauchy-Schwarz Inequality, 
