TITLE: How to show this inequality $\sum\sqrt{\frac{x}{x+2y+z}}\le 2$
QUESTION [6 upvotes]: Let $x,y,z,w>0$ show that
$$\sqrt{\dfrac{x}{x+2y+z}}+\sqrt{\dfrac{y}{y+2z+w}}+\sqrt{\dfrac{z}{z+2w+x}}+\sqrt{\dfrac{w}{w+2x+y}}\le 2$$
I tried C-S, but without success.

REPLY [2 votes]: By C-S:
$(LHS)^2\le \sum_{cyc}a(b+2c+d) \sum_{cyc}\frac{1}{(a+2b+c)(b+2c+d)}$
$\sum_{cyc}a(b+2c+d) \sum_{cyc}\frac{1}{(a+2b+c)(b+2c+d)}\le 4 \ \  \iff \ \ (a-c)^2(b-d)^2\ge 0$