Inequality for a rational function of three variables $x,y,z$ are positive real numbers such that $$x^2+y^2+z^2=1$$ Prove that $\dfrac{x^2}{1+2yz}+\dfrac{y^2}{1+2xz}+\dfrac{z^2}{1+2xy} \geqslant \dfrac{3}{5}$.Again, I try with Engel form of Cauchy inequality... 

Hint: By AM-GM:
$$\frac{x^2}{1+2yz} \ge \frac{x^2}{1+y^2+z^2} = \frac{x^2}{2-x^2}$$
Now apply Jensen to the convex function $t \mapsto \dfrac{t^2}{2-t^2}$ to conclude...
