Bernoulli relative entropy lower bound ($d(a,b)\ge 2(a-b{)}^2$)

For $d(a,b)=a\log (a/b)+(1-a)\log ((1-a)/(1-b))$, the second derivative with respect to $a$ is $1/[a(1-a)]\ge 4$. Its value and first derivative at $a=b$ vanish, so $d(a,b)\ge 2(a-b{)}^2$. Boundary values follow by continuity or the infinite-divergence convention. Together with the binary partition bound for relative entropy, this gives Pinsker's inequality with the natural-logarithm constant.