对数求和不等式

任给 ${\displaystyle a_1,\cdots ,a_n,b_1,\cdots ,b_n>0}$，有 $$\begin{array}{r}\sum _{i=1}^n{a}_i\ln \frac{a_i}{b_i}\ge \left(\sum _{i=1}^n{a}_i\right)\ln \frac{\sum _{i=1}^n{a}_i}{\sum _{i=1}^n{b}_i}.\end{array}$$ 等号成立当且仅当 ${\displaystyle \frac{a_1}{b_1}=\cdots =\frac{a_n}{b_n}}$.

记 ${\displaystyle A=\sum _{i=1}^n{a}_i}$, ${\displaystyle B=\sum _{i=1}^n{b}_i}$.
设 ${\displaystyle f(x)=x\ln x}$. 则 ${\displaystyle f^{\prime }(x)=1+\ln x}$, ${\displaystyle f^{″}(x)=\frac{1}{x}>0}$, 所以这是一个下凸函数. 于是$$\begin{array}{rl}\text{左边}& =\sum _{i=1}^n{b}_{i}f\left(\frac{a_i}{b_i}\right)=B\sum _{i=1}^n\frac{b_i}{B}f\left(\frac{a_i}{b_i}\right)\\ & \stackrel{\ast }{\ge }Bf(\sum _{i=1}^n\frac{b_i}{B}\cdot \frac{a_i}{b_i})=Bf\left(\frac{1}{B}\sum _{i=1}^n{a}_i\right)\\ & =Bf\left(\frac{A}{B}\right)=A\ln \frac{A}{B}=\text{右边}.\end{array}$$ 这里 * 用了琴生不等式. 取等条件是每个 ${\displaystyle \frac{a_i}{b_i}}$ 都相等.