# Advanced Inequality

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• Jun 26th 2012, 10:55 AM
richard1234
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

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