Prove an inequality

[daogiauvang]

Let $a, b, c >0 $ and $a^2+b^2+c^2=3$. Prove that, $\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a} +3(a+b+c) \geq 12$.

[worm2]

daogiauvang писал(а):

$a^2+b^2=c^2=3$

$a^2+b^2+c^2=3$

[arqady]

$$\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a}+4(a+b+c)\geq15$$

is also true.

[daogiauvang]

arqady в сообщении #1300331 писал(а):

$$\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a}+4(a+b+c)\geq15$$

is also true.

Do you have any proof?arqady

[arqady]

Yes of course! The last inequality has a nice proof.