# A symmetric inequality [duplicate]

Is it true for all $x, y, z > 0$ that

$$x + y + z \leq 2 \left\{ \frac{x^2}{y+z} + \frac{y^2}{x+z} + \frac{z^2}{x+y} \right\}$$

This is an exercise (1.4) in "The Cauchy-Schwarz Master Class: An Introduction to the Art of mathematical Inequalities"

The solution suggests applying C-S to $$x + y + z = \frac{x}{\sqrt{y+z}}\sqrt{y+z} + \frac{y}{\sqrt{x+z}}\sqrt{x+z} + \frac{z}{\sqrt{x+y}}\sqrt{x+y}$$

-

## marked as duplicate by Martin Sleziak, Simon S, Rudy the Reindeer, Asaf Karagila, Henning MakholmFeb 3 '12 at 14:05

It is correct $x + y + z = \frac{x}{\sqrt{y+z}}\sqrt{y+z} + \frac{y}{\sqrt{x+z}}\sqrt{x+z} + \frac{z}{\sqrt{x+y}}\sqrt{x+y}\le \left(\frac{x^2}{y+z}+\frac{y^2}{x+z}+\frac{z^2}{y+x}\right)^{1/2}(2x+2y+2z)^{1/2}.$ Square both sides, and cancel out $x+y+z$.