Find the maximum of the value a such $\sum_{cyc}\sqrt{\frac{x}{ax+y+z}}\le 3\sqrt{\frac{1}{2+a}}$ Find the maximum of the value $a$ such foy any real postive numbers $x,y,z$ have
$$\sqrt{\dfrac{x}{ax+y+z}}+\sqrt{\dfrac{y}{ay+z+x}}+\sqrt{\dfrac{z}{az+x+y}}\le3\sqrt{\dfrac{1}{2+a}}$$
I conjecture  $a>0?$

It's wrong for $a\rightarrow0^+$. Try $x\rightarrow+\infty.$
By the way, just by Jensen your inequality is true for all $a\geq\frac{3}{4}.$ 
Also, by Vasc's LCF Theorem it's enough to prove your inequality for $z=y$ 
and since our inequality is homogeneous, we can assume $y=z=1$, 
which gives a minimal value of $a$, for which our inequality is true: $a=\frac{1}{3}$ 
For $a=\frac{1}{3}$ we get the following interesting inequality.
Let $a$, $b$ and $c$ be positive numbers. Prove that:
$$\sqrt{\frac{a}{a+3b+3c}}+\sqrt{\frac{b}{b+3c+3a}}+\sqrt{\frac{c}{c+3a+3b}}\leq\frac{3}{\sqrt7}$$
The equality occurs also for $a=b=1$ and $c=8$.
