<s>[INST] How can we not use Muirhead's Inequality for proving the following inequality? There was a question in the problem set in my math team training homework:

Show that $∀a, b, c ∈ \mathbb{R}_{≥0}$ s.t. $a + b + c = 1, 7(ab + bc + ca) ≤ 2 + 9abc.$

I used Muirhead's inequality to do the question (you can try out yourself):

 By Muirhead's inequality, $$\begin{align}7(ab+bc+ca)&=7(a+b+c)(ab+bc+ca)\\&=21abc+6\big(a^2b+b^2c+c^2a+ab^2+bc^2+ca^2\big)+\sum_{sym}a^2b\\&\le21abc+6\big(a^2b+b^2c+c^2a+ab^2+bc^2+ca^2\big)+\sum_{sym}a^3\\&=2(a+b+c)^3+9abc\\&=2+9abc\end{align}$$As $(3,0,0)$ majorizes $(2,1,0)$. 

Is above proof correct? Also, can we find a proof not using Muirhead's inequality?
Any help is appreciated!
 [/INST] Schur's inequality:
$$a^3+b^3+c^3+3abc \ge a^2(b+c)+b^2(c+a)+c^2(a+b) \iff \\
(a+b+c)^3+9abc\ge 4(a+b+c)(ab+bc+ca) \Rightarrow \\
1+9abc\ge 4(ab+bc+ca) \quad (1)$$
Rearrangement:
$$a+b+c=1 \Rightarrow a^2+b^2+c^2=1-2(ab+bc+ca)\ge ab+bc+ca \Rightarrow \\
1\ge 3(ab+bc+ca) \quad (2)$$
Now add $(1)$ and $(2)$.
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