Suppose we have binary distributions $P,Q$ with probabilities $p\geq q$. Is it true that $D(p||q) \geq \frac12(2(p-q))^2$?
What I have so far is that $p\log\frac pq+(1-p)\log\frac{1-p}{1-q} \geq \log \min(\frac pq,\frac{1-p}{1-q}) = \log(\frac{1-p}{1-q})$ as $p\geq q$ implies $1-q \geq 1-p$, and that log is equal to $\log (1-p) - \log (1-q)$, but I don't think I can go on from there... any hints from here?