An Inequality with Constraint in Four Variables VI

Solution

The required inequality is equivalent to

$\displaystyle 3\left(\sum_{cycl}a\right)\left(\sum_{cycl}a^2\right)+18\sum_{cycl}abc\ge 5\left(\sum_{cycl}a\right)\left(\sum_{sym}ab\right)$

which, in turn, is equivalent to

(1)

$\displaystyle 3\sum_{cycl}a^3+3\sum_{cycl}abc\ge 2\sum_{sym}ab^2.$

Using Schur's inequality,

(2)

$\displaystyle a^3+b^3+c^3+3abc\ge a^2b+ab^2+b^2c+bc^2+c^2a+ca^2,$

(3)

$\displaystyle b^3+c^3+d^3+3bcd\ge b^2c+bc^2+c^2d+cd^2+d^2b+db^2,$

(4)

$\displaystyle c^3+d^3+a^3+3cda\ge c^2d+cd^2+d^2a+da^2+a^2c+ac^2,$

(5)

$\displaystyle d^3+a^3+b^3+3dab\ge d^2a+da^2+a^2b+ab^2+b^2d+bd^2.$

Summing up (2)-(5) gives (1) and, thus, proves the required inequality.

Acknowledgment

This problem by Leo Giugiuc was communicated to me by Marian Cucoaneş, along with a solution of his.