Muirhead's inequality -- equality occur

Question

An equality occurs in muirhead's inequality when all variables are the same. But is it possible for the equality to occur when not all variables are the same? (also all variables are non-zero) If so, is there a way to analyse it?

Answer 1

Yes, it's possible.

For example, for non-negatives $a$, $b$ and $c$ by Muirhead we have $$a^3b+a^3c+b^3a+b^3c+c^3a+c^3b\geq2(a^2b^2+a^2c^2+b^2c^2),$$ where the equality occurs also for $(a,b,c)=(1,0,0).$