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Let $\sum_\text{sym}a^xb^yc^z = [x,y,z]$. Prove that $\frac32[7/3,1/3,1/3]+2[4/3,4/3,1/3] \geq \frac32[1,1,1] + 2[5/3,2/3,2/3]$. I got this while trying to prove that for positive reals $a,b,c$ where $abc=1$ prove that $a+b+c \geq \sqrt{\frac{(a+2)(b+2)(c+2)}3}$. By squaring, expanding and homogenizing I got this and yet neither muirhead nor schur seem to crack it. Any help would be much appreciated

John Marty
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1 Answers1

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With $p=a+b+c, q = ab+bc+ca$, the inequality is (on squaring):

$$3p^2\ge 4p+2q+9$$ which follows from adding the obvious inequalities $p^2 \ge 3q$, $q\ge 3$ and $2p^2 \ge 4p+6 \iff 2(p-3)(p+1) \ge 0$ which is true as $p \ge 3$.

Macavity
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