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}

Applying result R1557: Weighted real AM-GM inequality to the real numbers $y_n : = 1 / x_n$, we have the inequality \begin{equation} \begin{split} \sum_{n = 1}^N \frac{\lambda_n}{x_n} = \sum_{n = 1}^N \lambda_n y_n \geq \prod_{n = 1}^N y_n^{\lambda_n} = \prod_{n = 1}^N \frac{1}{x^{\lambda_n}_n} = \frac{1}{\prod_{n = 1}^N x^{\lambda_n}_n} \end{split} \end{equation} Inverting both sides, we conclude \begin{equation} \frac{1}{\sum_{n = 1}^N \frac{\lambda_n}{x_n}} \leq \prod_{n = 1}^N x_n^{\lambda_n} \end{equation} $\square$