Power Mean Inequality

For \(N\) positive real numbers \(a_n\), the power mean with exponent \(t \in \mathbb{R}\) is defined as:

\[M(t) = \begin{cases} \prod_n a_n^{w_n} & t = 0\\ \Big(\sum_n w_n a_n^t \Big)^{1/t} \end{cases}\]

The Power Mean Inequality states that for all real numbers \(k_1 > k_2\), \(M(k_1) > M(k_2)\).

Proof: Jen