Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 224 / 1 / a / Solution

For nonnegative $a_i,b_i$, with $a={\sum }_i{a}_i$ and $b={\sum }_i{b}_i$, the log-sum inequality iswith the usual extended-value conventions. Equality holds when $a_i/b_i$ is constant wherever $a_i>0$.