There are many well-known relationships between the KL divergence, chi-squared ($\chi^2$) divergence, and the Hellinger metric. In the paper "Assouad, Fano, and Le Cam" by Bin Yu, the author appears to use the following relationship (see Example 2, p. 433):
$$
\text{KL}(f,g)
\le \int \frac{(\sqrt{f}-\sqrt{g})^2}{f}\,dx.
$$
This looks *almost* like the standard relationship $\text{KL}(f,g)\le \chi^2(f,g)$, but differs in the square roots taken in the numerator. I have not seen this general relationship previously.

My question is: *Does this inequality hold for general densities $f,g$?*

(Note: In the paper, this inequality is applied to special choices of $f$ and $g$, and so maybe the author only means it for these functions. It is not clear to me based on the writing.)