Let \(a_1, a_2, \cdots, a_n\) be positive real numbers, and let \(S_k\) be the sum of products of \(a_1, a_2, \cdots, a_n\) taken \(k\) at a time. Show that $$S_kS_{n-k}\geq {n\choose k}^2a_1a_2\cdots a_n,$$ for \(k=1, 2,\cdots, n-1\).

This topic was modified 3 months, 3 weeks ago by swastik pramanik.

Now here we shall use second inequality to prove the given inequality which is equivalent to $$\sigma_{k}\sigma_{n-k}\geq \sigma_n$$. Now observe that-