User:Temperal/Inequalities

Problem (me): Prove for $x,y,z\in\mathbb{R}^ + ,x + y + z = \frac {1}{2}$ that $\left(\sum_{sym}\frac {x}{y}\right)\left(\sqrt {2x^2 + 2y^2 + 2z^2}\right)\ge 1$

Solution (Altheman): By AM-GM, $\sum_{sym}\ge 6$ , and my RMS-AM $\sqrt {2x^2 + 2y^2 + 2z^2}\ge 1$ , thus the inequality is true.

Solution (me): By Jensen's Inequality, $\sum_{sym}\ge \frac{1}{\sum\limits_{sym}x\sqrt{y}}$ and by the Cauchy-Schwarz Inequality, $\frac{1}{\sum\limits_{sym}x\sqrt{y}}\ge\frac{1}{\sqrt{(2x^2 + 2y^2 + 2z^2)(2x+2y+2z)}}=\frac{1}{\sqrt{2x^2 + 2y^2 + 2z^2}}$

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User:Temperal/Inequalities