A three-variable inequality
For all real numbers \(x,y,z\), prove that
\[\begin{aligned} &(x^2+yz)(y^2+zx)(z^2+xy)\\ &\qquad\leq (x^2+y^2)(y^2+z^2)(z^2+x^2). \end{aligned}\]Hint
Try writing the difference between the two sides as a sum of squared differences.
Solution
Write \(L\) and \(R\) for the left and right sides. Expanding and regrouping gives
\[\begin{aligned} 2(R-L) &=(x-y)^2(x^2y^2+z^4)\\ &\quad+(y-z)^2(y^2z^2+x^4)\\ &\quad+(z-x)^2(z^2x^2+y^4). \end{aligned}\]Every term is nonnegative for real \(x,y,z\), so \(L\leq R\).
Extension
Equality holds exactly when \(x=y=z\) or at least two variables are zero.
If two variables are nonzero, their squared difference must vanish; the remaining terms then force the third variable to equal them. If at most one variable is nonzero, all three terms vanish.