$$
\lambda_1=\cdots=\lambda_n=\tfrac1n $$
$$
\ln\!\Bigl(\tfrac1n\sum_{i=1}^n x_i\Bigr) \;\ge\;\sum_{i=1}^n\tfrac1n\,\ln x_i \;=\;\tfrac1n\sum_{i=1}^n\ln x_i.
$$
$$
\tfrac1n\sum_{i=1}^n x_i \;\ge\;\exp\!\Bigl(\tfrac1n\sum_{i=1}^n\ln x_i\Bigr) \;=\;\bigl(x_1x_2\cdots x_n\bigr)^{1/n}. $$
