Find the maximum of the value a such $\sum_{cyc}\sqrt{\frac{x}{ax+y+z}}\le 3\sqrt{\frac{1}{2+a}}$

Question

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 $a>0?$

Answer 1

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 $c=8$.