I would gladly use a hint to get from $\displaystyle a^4+b^4+c^4 \geq a^3+b^3+c^3$ to $\displaystyle \frac{a^3}{\sqrt{b^4 + b^2 c^2 + c^4}} + \frac{b^3}{\sqrt{a^4 + a^2 c^2 + c^4}} + \frac{c^3}{\sqrt{a^4 + a^2 b^2 + b^4}} \geq \sqrt{3}$. Thanks in advance. (in positive reals)