An Inequality with Two Triples of Variables

Problem

An Inequality with Two Triples of Variables

Solution

Let $u+b+c,$ $v=c+a,$ $w=a+b,$ with $a,b,c\gt 0.$ The required inequality becomes

$\displaystyle\left(\sum_{cycl}a\right)\left(\sum_{cycl}x\right)-\sum_{cycl}ax\ge 2\sqrt{\left(\sum_{cycl}ab\right)\left(\sum_{cycl}xy\right)}.$

With the Cauchy-Schwarz inequality,

$-(ax+by+cz)\ge -\sqrt{(a^2+b^2+c^2)(x^2+y^2+z^2)}.$

Thus suffice it to show that

$\displaystyle\left(\sum_{cycl}a\right)\left(\sum_{cycl}x\right)-\sqrt{\left(\sum_{cycl}a^2\right)\left(\sum_{cycl}x^2\right)}\ge 2\sqrt{\left(\sum_{cycl}ab\right)\left(\sum_{cycl}xy\right)},$

which has been dealt with elsewhere.

Acknowledgment

I am grateful to Leo Giugiuc for mailing me this problem, along with a solution of his. Leo suggested that the problem is well known.

 

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