<s>[INST] Inequality 6 deg For $a,b,c\ge 0$ Prove that $$4(a^2+b^2+c^2)^3\ge 3(a^3+b^3+c^3+3abc)^2$$
My attempt: $$LHS-RHS=12(a-b)^2(b-c)^2(c-a)^2+2(ab+bc+ca)\sum_{sym} a^2(a-b)(a-c)$$
$$+\left(\sum_{sym} a(a-b)(a-c)\right)^2+14\left(\sum_{sym} a^3b^3+3a^2b^2c^2-abc\sum_{sym} ab(a+b)\right)+\sum_{sym} c^2(a-b)^2[2(a+b)^2+(a-b)^2]$$
Since $\sum_{sym} x^3+3xyz-\sum_{sym} xy(x+y)\ge 0$, set $(x,y,z)\rightarrow (ab,bc,ca)$ we obtain $$\sum_{sym} a^3b^3+3a^2b^2c^2-abc\sum_{sym} ab(a+b)\ge 0$$
Thus $LHS-RHS\ge 0$
I think this's a complicated solution and hard to find it by hand :">
Please give me a simplier solution. Thank you very much.
 [/INST] By the Cauchy-Schwarz inequality we have
$$(a^3+b^3+c^3+3abc)^2 = \left[\sum a(a^2+bc)\right] \leqslant \sum a^2 \sum (a^2+bc)^2.$$
Therefore we will show that
$$4(a^2+b^2+c^2)^2 \geqslant 3[(a^2+bc)^2+(b^2+ca)^2+(c^2+ab)^2],$$
equivalent to
$$a^4+b^4+c^4 + 5(a^2b^2+b^2c^2+c^2a^2) \geqslant 6abc(a+b+c). \quad (1)$$
Which is true because
$$a^4+b^4+c^4 \geqslant a^2b^2+b^2c^2+c^2a^2 \geqslant abc(a+b+c).$$
Note. The sum of squares of $(1)$
$$(a^2+b^2+c^2+ab+bc+ca)(a^2+b^2+c^2-ab-bc-ca)+2 \sum a^2(b-c)^2 \geqslant 0.$$
Zhaobin have posted it before
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