\[
x^{2}+y^{2}\geq 2xy\Longrightarrow x^{2}z+y^{2}z\geq 2xyz
\]
\[
y^{2}+z^{2}\geq 2yz\Longrightarrow y^{2}x+z^{2}x\geq 2xyz
\]
\[
z^{2}+x^{2}\geq 2zx\Longrightarrow z^{2}y+x^{2}y\geq 2xyz
\]
O halde
\[
x^{2}y+x^{2}z+xy^{2}+xz^{2}+y^{2}z+yz^{2}\geq 6xyz
\]
Her iki tarafa $2xyz$ eklenirse%
\[
\left( x+y\right) \left( y+z\right) \left( z+x\right) \geq 8xyz
\]