<< inequality

Log Sum Inequality

\[ \sum _{{i=1}}^{n}a_{i}\log {\frac {a_{i}}{b_{i}}}\geq a\log {\frac {a}{b}}, \]

Proof. Applying Jensen’s inequality \[ \begin{aligned} \sum_{i=1}^{n} a_{i} \log \frac{a_{i}}{b_{i}} &=\sum_{i=1}^{n} b_{i} f\left(\frac{a_{i}}{b_{i}}\right)=b \sum_{i=1}^{n} \frac{b_{i}}{b} f\left(\frac{a_{i}}{b_{i}}\right) \\ & \geq b f\left(\sum_{i=1}^{n} \frac{b_{i}}{b} \frac{a_{i}}{b_{i}}\right)=b f\left(\frac{1}{b} \sum_{i=1}^{n} a_{i}\right)=b f\left(\frac{a}{b}\right) \\ &=a \log \frac{a}{b} \end{aligned} \]