Problem
ALG-B2-M07-P017 Fourth powers versus product
#17
★★★★★ Level 5 of 5
Prove for \(a,b,c\ge0\): \[a^4+b^4+c^4\ge abc(a+b+c).\]
Hint. The inequality is homogeneous of degree \(4\). Use the UVW module result, or Muirhead/AM-GM.
This is a symmetric homogeneous inequality of degree \(4\). By the UVW form already proved, it is enough to check \(b=c=1\), \(a=t\): \(t^4+2\ge t(t+2)\), equivalent to \((t-1)^2(t^2+2t+2)\ge0\). On the boundary, the right side is \(0\). Hence the inequality holds.
Connects this module with the previous UVW module.