<s>[INST] Proving inequality $\sqrt{\frac{2a}{b+c}}+\sqrt{\frac{2b}{c+a}}+\sqrt{\frac{2c}{a+b}} \leq \sqrt{3 \left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\right)}$ In the pdf which you can download here I found the following inequality which I can't solve it.

Exercise 2.1.11 Let $a,b,c  \gt 0$. Prove that 
  $$\sqrt{\frac{2a}{b+c}}+\sqrt{\frac{2b}{c+a}}+\sqrt{\frac{2c}{a+b}} \leq \sqrt{3 \left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\right)}.$$

Thanks :)
 [/INST] By C-S 
$$\left(\sum_{cyc}\sqrt{\frac{a}{b+c}}\right)^2\leq\sum_{cyc}\frac{a}{a+c}\sum_{cyc}\frac{a+c}{b+c}.$$
Thus, it remains to prove that
$$\frac{3}{2}\left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\right)\geq\sum_{cyc}\frac{a}{a+c}\sum_{cyc}\frac{a+c}{b+c}$$ or
$$\sum_{cyc}(3a^6c^3+3a^5b^4+6a^5c^4+a^6b^2c+2a^6c^2b+4a^5b^3c+4a^5c^3b+7a^4b^4c+$$
$$+a^5b^2c^2-11a^4b^3c^2-12a^4c^3b^2-8a^3b^3c^3)\geq0,$$
which is obviously true.
Done!
</s>