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.


Using Hölder's inequality, we get $$ \left(\sum_{cyc} \frac {ab}{\sqrt{ab + 2c^2}}\right) \left(\sum_{cyc} \frac {ab}{\sqrt{ab + 2c^2}}\right) \left(\sum_{cyc} ab (ab + 2c^2)\right) \geq (ab + bc + ca)^3 $$ The above inequality can be rewritten as $$ \left(\sum_{cyc} \frac {ab}{\sqrt{ab + 2c^2}}\right)^2 (ab + bc +ca)^2 \geq (ab + bc + ca)^3 $$ And so $$ \sum_{cyc} \frac {ab}{\sqrt{ab + 2c^2}} \geq \sqrt{ab + bc + ca} $$


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