<s>[INST] Inequality $(a+b+c)(ab+bc+ca)(a^3+b^3+c^3)\le (a^2+b^2+c^2)^3$ For positive real numbers $a,b,c$ prove that $$(a+b+c)(ab+bc+ca)(a^3+b^3+c^3)\le (a^2+b^2+c^2)^3$$
My try :
Rewrite this as $$\frac{a^2+b^2+c^2}{ab+bc+ca}-1\ge \frac{(a+b+c)(a^3+b^3+c^3)}{a^2+b^2+c^2)^2}-1$$
Then I am stumped, I was trying to use SOS but I think an easier proof is possible. My teacher give this to me. He combined to well known inequalities $a^2+b^2+c^2\ge ab+bc+ca$ and $(a+b+c)(a^3+b^3+c^3)\ge (a^2+b^2+c^2)^2$ and changed the sign. This is easy by uvw method
but my teacher wouldn't accept it. Can some one help me? Thanks in advance.
 [/INST] Here's a proof without using the $uvw$ method. (I'm late to the party, and now having to clean the mess. Also, credit goes to Michael Rozenberg for this solution.)
Continue to where you left off, we have that $$\frac{a^2 + b^2 + c^2}{ab + bc + ca} - 1\ge \frac{(a^3 + b^3 + c^3)(a + b + c)}{(a^2 + b^2 + c^2)^2} - 1$$
$$\iff \frac{\displaystyle \sum_{cyc}(a - b)^2}{2(ab + bc + ca)} \ge \frac{\displaystyle \sum_{cyc}ab(a - b)^2}{(a^2 + b^2 + c^2)^2}$$
Furthermore, because of $ab + bc + ca \le a^2 + b^2 + c^2$
$$\iff \frac{\displaystyle \sum_{cyc}ab(a - b)^2}{(a^2 + b^2 + c^2)^2} \le \frac{\displaystyle \sum_{cyc}ab(a - b)^2}{(ab + bc + ca)(a^2 + b^2 + c^2)}$$
Now, what needs to be proven is $$\sum_{cyc}(a - b)^2 \ge \frac{\displaystyle \sum_{cyc}ab(a - b)^2}{2(a^2 + b^2 + c^2)} \iff \sum_{cyc}\left(\frac{1}{2} - \frac{ab}{a^2 + b^2 + c^2}\right)(a - b)^2 \ge 0$$
$$\iff \sum_{cyc}\left[\frac{(a - b)^2 + c^2}{2(a^2 + b^2 + c^2)}\right](a - b)^2 \ge 0$$
In conclusion, the Sum of Square method does work. 
</s>