Advanced Inequality

Given $x,y,z$ are real numbers greater than or equal to zero, denote

$[\alpha, \beta, \gamma]= \sum_{sym} x^{\alpha} y^{\beta}z^{\gamma}$ (the symmetric sum contains all 3! = 6 permutations of $\alpha, \beta, \gamma$ in the exponents)

If $a^2 + b^2 = c^2$ where $a,b \ge 0$ , prove that

$[a+b, 0, -a-b] + [c, 0, -c] \ge [a+b, -a, -b] + [a, b, -a-b]$ .