Let $x_1, \dots, x_N \in (0, \infty)$ each be a D5407: Positive 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} \frac{1}{\sum_{n = 1}^N \frac{\lambda_n}{x_n}} \leq \prod_{n = 1}^N x_n^{\lambda_n} \end{equation}
(2)	\begin{equation} \frac{1}{\sum_{n = 1}^N \frac{\lambda_n}{x_n}} = \prod_{n = 1}^N x_n^{\lambda_n} \quad \iff \quad \frac{1}{x_1} = \frac{1}{x_n} = \cdots = \frac{1}{x_N} \end{equation}