The weighted AM-GM inequality

Logarithm. Since $\ln$ is concave, Jensen's inequality gives
$$\ln\left(\sum_i\lambda_i a_i\right)\ge\sum_i\lambda_i\ln(a_i)=\ln\left(\prod_i a_i^{\lambda_i}\right).$$

Exponentiation. Since $\exp$ is increasing:
$$\sum_i\lambda_i a_i\ge\prod_i a_i^{\lambda_i}.$$

Equality case. Equality holds iff all $a_i$ are equal (the concavity is strict).

Check. With $\lambda_i=1/n$, we recover the classical AM-GM.