Relative entropy bound for the unnormalized Hellinger distance ($H^2(P,Q)\le K(P,Q)$)

The scalar inequality $-\log u\ge 1-u$, applied to $u=\sqrt{q/p}$ and integrated against $P$, gives $K(P,Q)\ge 2-2\int \sqrt{pq}d\mu$. The right side is the unnormalized squared Hellinger distance. Infinite divergence and zero density regions are handled by the usual extended-integral conventions.