Prove the following inequality for all real numbers $a, b, c$ and for $k \in [0, 1]$: $$a^4 + b^4 + c^4 + k(a^3b + b^3c + c^3a) \geq (k + 1)(ab^3 + bc^3 + ca^3)$$
I have tried to prove by Rearrangement inequality (supose $a \geq b \geq c$): $$aa^3 + bb^3 + cc^3 \geq ab^3 + bc^3 + ca^3$$
So, by simplifying $k$ we have got the following inequality: $$a^3b + b^3c + c^3a \geq ab^3 + bc^3 + ca^3$$
Which seems true by some rearrangement-type inequality, but I am not able to figure out and point it clearly. And also, what is $k \in [0, 1]$ for?