On inequalities for products of power sums (Q1061256)

One of the main results of the paper is the following theorem: Let \(a_ 1,...,a_ k\), \(\alpha_ 1,...,\alpha_ k\) be real parameters with \(\alpha_ 1+...+\alpha_ k=0\). Then the inequality \[ 1\leq (x_ 1^{a_ 1}+...+x_ n^{a_ 1})^{\alpha_ 1}...(x_ 1^{a_ k}+...+x_ n^{a_ k})^{\alpha_ k} \] is satisfied for all \(n\in {\mathbb{N}}\) and \(x_ 1,...,x_ n>0\) if and only if \(0\leq \alpha_ 1| a_ 1-a_ i| +...+\alpha_ k| a_ k-a_ i|\) for \(i=1,...,k\). Many classical inequalities, for instance the Lyapunov's inequality, easily follow from this result.