Show that $b^{2} c^{2} + c^{2} a^{2} + a^{2} b^{2} > a b c (a + b + c)$ .

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Solution

Apply $A M \geq G M$ for $a^{2} b^{2}, b^{2} c^{2}$
$\Rightarrow \frac{a^{2} b^{2} + b^{2} c^{2}}{2} \geq \sqrt{a^{2} b^{2} \times b^{2} c^{2}}$
$\Rightarrow \frac{a^{2} b^{2} + b^{2} c^{2}}{2} \geq b^{2} a c$ $\to$ eqn $1$
Apply $A M \geq G M$ for $c^{2} b^{2}, a^{2} c^{2}$
$\Rightarrow \frac{c^{2} b^{2} + a^{2} c^{2}}{2} \geq \sqrt{c^{2} b^{2} \times a^{2} c^{2}}$
$\Rightarrow \frac{c^{2} b^{2} + a^{2} c^{2}}{2} \geq c^{2} b c$ $\to$ eqn $2$
Apply $A M \geq G M$ for a^{2}b^{2}, a^{2}c^{2}$
$\Rightarrow \frac{a^{2} b^{2} + a^{2} c^{2}}{2} \geq \sqrt{a^{2} b^{2} \times a^{2} c^{2}}$
$\Rightarrow \frac{a^{2} b^{2} + a^{2} c^{2}}{2} \geq a^{2} b c$ $\to$ eqn 3

Adding corresponding sides of eqns $1, 2, 3$ we get
$a^{2} b^{2} + b^{2} c^{2} + a^{2} c^{2} \geq a b c (a + b + c)$

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