Prove that $\sum_{\text{cyc}} \sqrt{(a^2 + 5b^2)(b^2 + 5c^2)} \geq 2(a + b + c)^2.$

Question

inequality

For all reals, prove that: \begin{align*} & ab(a+b)+ac(a+c)+bc(b+c) \\ & \hspace{5mm} + 2\sqrt{(a^2+b^2+c^2-ab-ac-bc)(a^4+b^4+c^4-a^2b^2-a^2c^2-b^2c^2)} \\ & \hspace{15mm} \geq 2(a^3+b^3+c^3). \end{align*}

My attempt

I have considered a few methods. One method is to square both sides and simplify. After doing so the remaining inequality to prove is $$\sum_{\text{cyc}} \sqrt{(a^2 + 5b^2)(b^2 + 5c^2)} \geq 2(a + b + c)^2.$$

I am stuck at this point. Any help, or hints, would be appreciated.

Answer 1

It's an old problem of mine (here). The proof uses an identity $$6(a^2+b^2+c^2-ab-bc-ca) = \sum (2a-b-c)^2.$$