MHB Prove Inequality: (x^2+y^2+z^2)(x+y+z) + x^3+y^3+z^3 > 4(xy+yz+zx)

[anemone]

Here is this week's POTW:

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Prove that $(x^2+ y^2 + z^2)(x + y + z) + x^3+ y^3+ z^3> 4(xy + yz + zx)$ for all $x,\,y,\,z > 1$.

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[anemone]

No one answered this week's problem. You can read my solution below.

Spoiler

From the given constraint that says $x,\,y,\,z > 1$, it implies $ x^3+ y^3+ z^3> x^2+ y^2+ z^2$. Therefore we have

$\begin{align*}(x^2+ y^2 + z^2)(x + y + z) + x^3+ y^3+ z^3 &\ge (x^2+ y^2 + z^2)(x + y + z)+ x^2+y^2+ z^2 \\& =(x^2+ y^2 + z^2)(x + y + z+1)\\&>4(x^2+y^2+z^2) \\&=4(xy+yz+zx)\,\,\,\,\,\,\text{(Q.E.D.)} \end{align*}$