lean_workbook_plus_10279

the idea was an identity $a^3+b^3+c^3-3abc=(a+b+c)(a^2+b^2+c^2-ab-ac-bc)$ \n\n your inequality is \n\n $\frac {2(a^3+b^3+c^3)} {abc} -6+\frac {9(a+b+c)^2}{a^2+b^2+c^2} -27 \ge 0 \Leftrightarrow$ \n\n $2\frac {a^3+b^3+c^3-3abc} {abc}-18\frac {a^2+b^2+c^2-ab-bc-ca}{a^2+b^2+c^2}\ge 0 \Leftrightarrow$ \n\n $2\frac {(a+b+c)(a^2+b^2+c^2-ab-bc-ca)} {abc}-18\frac {a^2+b^2+c^2-ab-bc-ca}{a^2+b^2+c^2}\ge 0 \Leftrightarrow$ \n\n $(a^2+b^2+c^2-ab-bc-ca) (2\frac {a+b+c} {abc}-\frac {18}{a^2+b^2+c^2})\ge 0 \Leftrightarrow$ \n\n $(a^2+b^2+c^2-ab-bc-ca)\frac{2(a+b+c)(a^2+b^2+c^2)-18abc} {abc(a^2+b^2+c^2)}\ge 0$