Inequality 6 deg

Question

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.

Answer 1

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

Answer 2

For example $$4(a^2+b^2+c^2)^3-(a^3+b^3+c^3+3abc)^2=$$ $$=\sum_{cyc}(a^6+12a^4b^2+12a^4c^2-6a^3b^3-18a^4bc-a^2b^2c^2)\geq0,$$ which is true by Muirhead.

Answer 3

We have$:$ \begin{align*} \text{LHS}-\text{RHS}\\&=\frac{1}{3} \sum\limits_{cyc} \left( 2\,{a}^{4}+10\,{a}^{2}{b}^{2}+2\,{a}^{2}bc+5\,{a}^{2}{c}^ {2}+2\,a{b}^{3}+4\,ab{c}^{2}+{b}^{4}+4\,{b}^{3}c+24\,{c}^{4} \right) \left( a-b \right) ^{2}\end{align*} Another solution.

Let $p=a+b+c,\,q=ab+bc+ca,\, r=abc.$ We have$:$

\begin{align*} \text{LHS}-\text{RHS}&=\frac{4}{3}\, \left( 5\,{p}^{2}-9\,q \right) \left( {q}^{2} -3\,pr\right) \\&\,\,\,+\frac{1}{3}\, \left( {p}^{2}-3\,q \right) \left( 3\,{p}^{4}-9\,{p}^{2}q+4\,{q} ^{2} \right)\\&\,\,\, -16\,{p}^{3}r+4\,{p}^{2}{q}^{2}+72\,pqr-16\,{q}^{3}-108\, {r}^{2}\\&\geqq 0\end{align*}

Note that$:$ $$-16\,{p}^{3}r+4\,{p}^{2}{q}^{2}+72\,pqr-16\,{q}^{3}-108\, {r}^{2}=4(a-b)^2(b-c)^2(c-a)^2 \geqq 0$$