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