We need to prove the following inequality:

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> $$\frac{y^3xz}{x^3(xy+z^2)}+\frac{z^3xy}{y^3(x^2+yz)}+\frac{x^3yz}{z^3(xz+y^2)}\geq \frac{3}{2}$$
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This equation is convex in each of the variables $x,y,z$. Moreover, its minimum seems to be at the origin (the equation is homogeneous, and there are not displacements of the vertex). 

Can I then say that the minima is achieved when $x=y=z$? Clearly, when $x=y=z$, the value of the inequality is $\frac{3}{2}$