2
$\begingroup$

This question already has an answer here:

If $a,b,c>0$, Then prove that $$\frac{a^3}{bc}+\frac{b^3}{ca}+\frac{c^3}{ab}\geq a+b+c$$

$\bf{My\; Try::}$ Using Cauchy- Schwarz Inequality

$$\frac{a^4}{abc}+\frac{b^4}{abc}+\frac{c^4}{abc}\geq \frac{a^2+b^2+c^2}{3abc}$$

Now How can i solve after that , Help required, Thanks

$\endgroup$

marked as duplicate by Arnaud D., mrp, Davide Giraudo, Stefan4024, John B Oct 9 '16 at 22:53

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

4
$\begingroup$

$$a^4+b^4+c^4\geq a^2bc+b^2ac+c^2 ab $$ holds by the rearrangement inequality or Muirhead's inequality.

$\endgroup$
4
$\begingroup$

You can actually use Holder's inequality for three variables in the following way:

$$(b+c+a)(c+a+b) \Big(\frac{a^3}{bc}+\frac{b^3}{ca}+\frac{c^3}{ab}\Big) \geqslant (a+b+c)^3 $$

From where you get: $$ \frac{a^3}{bc}+\frac{b^3}{ca}+\frac{c^3}{ab} \geqslant a+b+c$$

$\endgroup$

Not the answer you're looking for? Browse other questions tagged or ask your own question.