arqady wrote:

jokehim wrote:

Problem:Prove that the following inequality : $\left(\sqrt{a}+\sqrt{b}+\sqrt{c}\right)^2\ge\sqrt{3}\left(\sqrt[4]{\frac{5ab}{c}+4a}+\sqrt[4]{\frac{5bc}{a}+4b}+\sqrt[4]{\frac{5ca}{b}+4c}\right)$ holds for all positive real numbers such that: $abc=1$ .

By Holder $\sqrt3\sum_{cyc}\sqrt[4]{\frac{5ab}{c}+4a}=\sqrt3\sum_{cyc}\sqrt[4]{ab(5ab+4ac)}\leq$ $\leq\sqrt3\sqrt[4]{\left(\sum\limits_{cyc}\sqrt[4]{ab}\right)^3\sum_{cyc}\sqrt[4]{ab}(5ab+4ac)}\leq(\sqrt a+\sqrt b+\sqrt c)^2.$

Please help me tell more about the last one.