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].