Let $X$ and $Y$ be two continuous random variables with support $\mathbb{R}^{+}$ and with PDF $f(x)$ and $g(y)$. If the Bhattacharyya distance of $f$ and $g$ is less than $\epsilon$, then is there any bound (lower or upper) on $\int_{0}^{\infty}\log(1+x)(f(x)-g(x))\,\mathrm{d}x$?

*Definition of Bhattacharyya distance: https://en.wikipedia.org/wiki/Bhattacharyya_distance