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Algebra Difficulty 6.2 National olympiad Prove it

Example 2.2.9. Let a,b,ca, b, c be positive real numbers. Prove that
(a2+ab+b2)(b2+bc+c2)(c2+ca+a2)≥(ab+bc+ca)3. \left(a^{2}+a b+b^{2}\right)\left(b^{2}+b c+c^{2}\right)\left(c^{2}+c a+a^{2}\right) \geq (a b+b c+c a)^{3}.

Solution

(a2+ab+b2)(b2+bc+c2)(c2+ca+a2)=(ab+a2+b2)(a2+ac+c2)(b2+c2+bc)≥(ab+ac+bc)3\begin{array}{c} \left(a^{2}+a b+b^{2}\right)\left(b^{2}+b c+c^{2}\right)\left(c^{2}+c a+a^{2}\right) \\ =\left(a b+a^{2}+b^{2}\right)\left(a^{2}+a c+c^{2}\right)\left(b^{2}+c^{2}+b c\right) \geq(a b+a c+b c)^{3} \end{array}

Applying Hölder inequality, we obtain
(a2+ab+b2)(b2+bc+c2)(c2+ca+a2)=(ab+a2+b2)(a2+ac+c2)(b2+c2+bc)≥(ab+ac+bc)3\begin{array}{c} \left(a^{2}+a b+b^{2}\right)\left(b^{2}+b c+c^{2}\right)\left(c^{2}+c a+a^{2}\right) \\ =\left(a b+a^{2}+b^{2}\right)\left(a^{2}+a c+c^{2}\right)\left(b^{2}+c^{2}+b c\right) \geq(a b+a c+b c)^{3} \end{array}

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.