Problem. Let $a,b,c> 0.$ Prove that$$\dfrac{a}{\sqrt{a+b}}+\dfrac{b}{\sqrt{b+c}}+\dfrac{c}{\sqrt{c+a}} \le \dfrac{3\sqrt{3}}{4}\cdot\sqrt{\dfrac{(a+b)(b+c)(c+a)}{ab+bc+ca}}.$$
I used Cauchy-Schwarz $$\sum_{cyc}\frac{a}{\sqrt{a+b}}=\sum_{cyc}\sqrt{a(a+c)}\sqrt{\frac{a}{(a+b)(a+c)}}\le \sqrt{(a^2+b^2+c^2+ab+bc+ca).\frac{2(ab+bc+ca)}{(a+b)(b+c)(c+a)}}.$$ It implies to prove $$[(a+b)(b+c)(c+a)]^2\ge \frac{32}{27}(ab+bc+ca)^2(a^2+b^2+c^2+ab+bc+ca).$$The last inequality is wrong.
I hope we can find a better idea. Thank you.
