Proving the inequality $\frac{9}4\ + \frac{3abc}4\ \ge ab+bc+ca$ If a,b,c are non-negative real numbers for which holds that $a+b+c=3$ then prove the following inequality:  $$\frac{9}4\ + \frac{3abc}4\ \ge ab+bc+ca$$
 A: If we consider the homogeneous form of our inequality:
$$\frac{9}{4}\left(\frac{a+b+c}{3}\right)^3+\frac{3}{4}abc\geq (ab+ac+bc)\left(\frac{a+b+c}{3}\right)$$
we just have to prove that for any triple $(a,b,c)$ of non-negative real numbers
$$ a^3+b^3+c^3+3abc \geq ab(a+b)+bc(b+c)+ac(a+c)\tag{1} $$
holds, but $(1)$ is exactly Schur's inequality.
A: So, we have Schur's inequality which says that for non-negative real numbers a,b,c the following inequality holds: $a^3+b^3+c^3+3abc\ge ab(a+b)+bc(b+c)+ca(c+a)$.
Now using the an identity which says: $a^3+b^3+c^3=(a+b+c)(a^2+b^2+c^2-ab-bc-ca)+3abc$ and the fact that $ab+c+=3$ we have:
$3(a^2+b^2+c^2-ab-bc-ca)+3abc+3abc\ge ab(3-c)+bc(3-a)+ca(3-b)$
$3(a^2+b^2+c^2)-3(ab+bc+ca)+6abc\ge 3(ab+bc+ca)-3abc$
$3(a^2+b^2+c^2)+9abc\ge 6(ab+bc+ca)$. Dividing both sides with 3 gives us:
$a^2+b^2+c^2+3abc\ge2(ab+bc+ca)$. Now adding to both sides $2(ab+bc+ca)$:
$(a^2+b^2+c^2+2(ab+bc+ca))+3abc\ge4(ab+bc+ca)$
$(a+b+c)^2+3abc\ge4(ab+bc+ca)$. Using now that $a+b+c=3$ and dividng both sides with 4 we get:
$\frac{9}{4}\ + \frac{3abc}4\ge ab+bc+ca$
