Let $x_1, \dots, x_N \in [0, \infty)$ each be an D4767: Unsigned real number.
Let $\lambda_1, \dots, \lambda_N \in [0, \infty)$ each be an D4767: Unsigned real number such that

(i)	\begin{equation} \sum_{n = 1}^N \lambda_n = 1 \end{equation}

Then

(1)	\begin{equation} \prod_{n = 1}^N x_n^{\lambda_n} \leq \sum_{n = 1}^N \lambda_n x_n \end{equation}
(2)	\begin{equation} \prod_{n = 1}^N x_n^{\lambda_n} = \sum_{n = 1}^N \lambda_n x_n \quad \iff \quad x_1 = x_2 = \cdots = x_N \end{equation}