Problem
ALG-B2-M06-P010 Schur degree 3
#10
★★★☆☆ Level 3 of 5
Prove for \(a,b,c\ge0\): \[\sum a^3+3abc\ge\sum_{\mathrm{sym}}a^2b.\]
Hint. Use the form \(p^3-4pq+9r\ge0\), i.e. Schur degree \(3\).
By Problem 5, the difference between the left and right sides is \(p^3-4pq+9r\). By Schur degree \(3\), this expression is nonnegative for \(a,b,c\ge0\). Hence the inequality follows.
This is the central basic inequality of the module.