Question 19
Let $a_1, a_2, \dots, a_n$ be positive integers ($a_1 \lt a_2 \lt \dots \lt a_n$), and let $k$ be a positive real number with $k \ge 1$.
Prove that:
$$
\sum_{i=1}^{n} a_i^{2k+1} \ge \left( \sum_{i=1}^{n} a_i^k \right)^2 .
$$
Prove that:
$$
\sum_{i=1}^{n} a_i^{2k+1} \ge \left( \sum_{i=1}^{n} a_i^k \right)^2 .
$$