Since we can write $x^{\lambda_n}_n = e^{\lambda_n \log x_n}$, we can apply results
to conclude
\begin{equation}
\begin{split}
\prod_{n = 1}^N x_n^{\lambda_n} = \prod_{n = 1}^N e^{\lambda_n \log x_n} = \exp \left( \sum_{n = 1}^N \lambda_n \log x_n \right) \leq \sum_{n = 1}^N \lambda_n e^{\log x_n} = \sum_{n = 1}^N \lambda_n x_n
\end{split}
\end{equation}
The second claim is a consequence of the above results together with the result
R5181: Subconvex real function preserves convex combination iff convex combination elements are all equal. $\square$
