Total variation–Hellinger–relative entropy inequality ($\mathrm{TV}(P,Q)\le h(P,Q)\le \sqrt{\mathrm{KL}(P,Q)}$)

For the unnormalized Hellinger distance, $\mathrm{TV}(P,Q)\le h(P,Q)\le \sqrt{\mathrm{KL}(P,Q)}$. Factor $|p-q|=|\sqrt{p}-\sqrt{q}|(\sqrt{p}+\sqrt{q})$ and use the Cauchy-Schwarz inequality to bound the total variation distance. For the other bound, $\log t\le t-1$ gives $p\log (p/q)\ge 2(p-\sqrt{pq})$, whose integral is the squared unnormalized Hellinger distance. Infinite Kullback-Leibler divergence is allowed.