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}
$\square$
