Consider a polynomial 'f(x)' with 'n' positive real roots given by $$\alpha_1,alpha_2,\cdots,\alpha_n$$. Define-
$$\sigma_k=\frac{S_k}{n\choose k}$$.Now we following two results-
Newton's inequality- $$\sigma_{k-1}\sigma_{k+1}\leq \sigma_k^2$$
Maclaurian inequality- $$(\sigma_{1})^\frac{1}{1}\geq(\sigma_{2})^\frac{1}{2}\geq(\sigma_{3})^\frac{1}{3}\geq\cdots\geq(\sigma_{n})^\frac{1}{n}$$
For proof you may consult following link
https://artofproblemsolving.com/wiki/index.php/Newton%27s_Inequality
Now its clear that the given case is just a special case of maclaurian inequality (n=4)
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