1. ## An interesting inequality

There are such positive numbers a, b, c that:

$a^4 + b^4 + c^4 \ge a^3 + b^3 + c^3$

Prove that

$\frac {a^3}{\sqrt{b^4 + b^2 \cdot c^2 + c^4}} + \frac {b^3}{\sqrt{a^4 + a^2 \cdot c^2 + c^4}} + \frac {c^3}{\sqrt{a^4 + a^2 \cdot b^2 + b^4}} \ge \sqrt{3}$

All I know for now is that

$\frac {a^3}{\sqrt{b^4 + b^2 \cdot c^2 + c^4}} + \frac {b^3}{\sqrt{a^4 + a^2 \cdot c^2 + c^4}} + \frac {c^3}{\sqrt{a^4 + a^2 \cdot b^2 + b^4}} \ge \frac {a^3}{b^2 + c^2} + \frac {b^3}{a^2 + c^2} + \frac {c^3}{a^2 + b^2}$

While proving the second inequality we should take into consideration only those numbers that satisfy the first one.

I would be very grateful if anyone could at least give me any hint on how to do this inequality.

It would be even nicer if somebody, even partially, solved it for me

Thank you!