Let $a,b,c > 0$. Prove that (using Hölder's inequality):
$\frac{ab}{\sqrt{ab+2c^2}}+\frac{bc}{\sqrt{bc+2a^2}}+\frac{ca}{\sqrt{ca+2b^2}} \geq \sqrt{ab+bc+ca}.$
Thanks :)
I tried to apply Hölder's inequality how I apply in this exercise but I didn't obtained anything.