Theorem (Log-sum inequality)

Let $a_1,\cdots,a_n$ and $b_1,\cdots,b_n$ be positive real numbers. Then $$\sum_{i=1}^na_i\log\frac{a_i}{b_i}\ge\left(\sum_{i=1}^na_i\right)\log\frac{\sum_{i=1}^na_i}{\sum_{i=1}^nb_i}$$ with equality if and only if $\frac{a_i}{b_i}=\text{ constant }\forall i=1,\cdots,n$