Let ${x,y,z}$ be positive real numbers.
Prove that
$\sum\limits_{cyc}\frac{x^2}{yz+\frac{x^4}{y^2}+\frac{x^4}{z^2}} \leq 1$
Here’s my try:
By A.M.-G.M. we know that
$\frac{yz}{2}+\frac{yz}{2}+\frac{x^4}{y^2}+\frac{x^4}{z^2}\geq$ $4\sqrt[4]{\frac{1}{4}}x^2.$
So we get $L.H.S \leq \frac{3}{4\sqrt[4]{\frac{1}{4}}}.$
But this inequality is never hold
How can I prove that it is less than $1$?