I know the problem is repeated but my question is somehow different. I want to know whether my proof is correct because I have troubles with the last part.

Since $abc=1$ we can homogenize the variables being $$a=\frac{x}{y}$$ $$b=\frac{y}{z}$$ $$c=\frac{z}{x}$$

Hence the asked inequality is converted into $$(x+y-z)(y+x-z)(z+y-x)\leq xyz$$ But I searched on the internet and it seems that it has only a solution for $x,y,z$ when they are sides of a triangle. Can anyone explain it to me?