Prove that $a^7 + b^7 + c^7 \geq a^4b^3 + b^4c^3 + c^4a^3$ 
Prove that $a^7 + b^7 + c^7 \geq a^4b^3 + b^4c^3 + c^4a^3$

Values $a,b,c$ are all positive reals. I tried Muirhead and a few AM $\geq$ GM. 
This problem is equivalent to proving $a^4b^3 + b^4c^3 + c^4a^3 \geq c^2a^5 + b^2c^5+a^2b^5$.

The inequality does not change if $(a, b, c)$ are permuted
cyclically, therefore we can assume
that $a \le b \le c$ or $a \ge b \ge c$.
In the first case $a \le b \le c$ we can
apply the Rearrangement inequality to
$$
x_1 = a^4, x_2 = b^4, x_3 = c^4 \\ y_1 = a^3, y_2 = b^3, y_3 = c^3 \, .
$$
The rearrangement inequality states that
$$
x_1y_2 + x_2y_3 + x_3y_1 \le 
x_1y_1 + x_2y_2 + x_3y_3 
$$
which is exactly the desired inequality.
In the second case $a \ge b \ge c$ we can do the same with
$$
x_1 = c^3, x_2 = b^3, x_3 = a^3 \\ y_1 = c^4, y_2 = b^4, y_3 = a^4 \, .
$$
