# Inequality $\frac{x^{x^2}}{x^2+y^2}+\frac{y^{y^2}}{y^2+z^2}+\frac{z^{z^2}}{z^2+x^2}\geq \frac{3}{2}$ with $xyz=1$

I'm proud to present the work of my last week end and I think it's hard to prove :

Let $$x,y,z>0$$ such that $$xyz=1$$ then we have : $$\frac{x^{x^2}}{x^2+y^2}+\frac{y^{y^2}}{y^2+z^2}+\frac{z^{z^2}}{z^2+x^2}\geq \frac{3}{2}$$

I try the following inequality to prove my statement : $$\frac{x^{x^2}}{x^2+y^2}+\frac{y^{y^2}}{y^2+z^2}+\frac{z^{z^2}}{z^2+x^2}\geq \frac{x}{x+y}+\frac{y}{y+z}+\frac{z}{z+x}\geq \frac{3}{2}$$

But it doesn't work .

I try also to apply the well-know inequality :$$e^x\geq x+1$$ but it gives nothing consequent. So I'm a bit lost with this inequality ...

Any help is appeciated . Thanks

Prove that: $$x^{x^2}\geq 2x^3-4x^2+3x.$$ The rest is smooth.