Let $x_1, x_2, ... ,x_n$ be positive real numbers and define $S(k)$ to be the power sum $S(k) = x_1^k + x_2^k +... + x_n^k$ . It is given that $S(3) = 3$ and that $S(5) = 5 $. Find the best lower bound for $S(1) $.
(1) An application of Cauchy Schwarz shows that $S(1) > 9/5$ is an admissible lower bound.
(2) The problem appears (with a smaller bound) in the book "Inequalities" by Cvetkovski (p. 360).