Question 27
Let $a_1, a_2, \dots, a_n, b_1, b_2, \dots, b_n, p$ be real numbers with $p \gt -1$. Prove that
$$
\sum_{i=1}^{n} (a_i - b_i)(a_i(a_1^2 + a_2^2 + \dots + a_n^2)^{p/2} - b_i(b_1^2 + b_2^2 + \dots + b_n^2)^{p/2}) \ge 0.
$$
$$
\sum_{i=1}^{n} (a_i - b_i)(a_i(a_1^2 + a_2^2 + \dots + a_n^2)^{p/2} - b_i(b_1^2 + b_2^2 + \dots + b_n^2)^{p/2}) \ge 0.
$$