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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 $a_i>0$.

Let $0\le \lambda \le 1$ and put $P=\lambda P_1+(1-\lambda )P_2$ and $Q=\lambda Q_1+(1-\lambda )Q_2$. For each alphabet symbol $x$, apply the log-sum inequality to $a_1=\lambda P_1(x)$, $a_2=(1-\lambda )P_2(x)$ and the corresponding $b$ values. Summing over $x$ giveswhich is joint convexity in $(P,Q)$.

For nonnegative numbers $a_i,b_i$, put $a={\sum }_i{a}_i$ and $b={\sum }_i{b}_i$. The log-sum inequality iswith $0\log (0/b)=0$ and the usual extended-value convention when a denominator vanishes. Equality holds precisely when $a_i/b_i$ is constant over the indices with $a_i>0$.

Let $K(y\mid x)$ be a Markov kernel, and let $P_Y=PK$, $Q_Y=QK$ be the output probability distributions. Applying the log-sum inequality for each $y$ to $a_x=P(x)K(y\mid x)$ and $b_x=Q(x)K(y\mid x)$ givesSumming over $y$ and using ${\sum }_{y}K(y\mid x)=1$ yields the data processing inequality for relative entropy